The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday January 21, 2020 at 15:00, Penn (DRL 4C8)

  Fast randomized iterative numerical linear algebra for quantum chemistry (and other applications)

  Jonathan Weare, Courant Institute, NYU
  
  I will discuss a family of recently developed stochastic techniques for linear algebra problems involving very large matrices.  These methods can be used to, for example, solve linear systems, estimate eigenvalues/vectors, and apply a matrix exponential to a vector, even in cases where the desired solution vector is too large to store.  The first incarnations of this idea appear for dominant eigenproblems arising in statistical physics and in quantum chemistry and were inspired by the real space diffusion Monte Carlo algorithm which has been used to compute chemical ground states since the 1970's.  I will discuss our own general framework for fast randomized iterative linear algebra as well share a very partial explanation for their effectiveness.  I will also report on the progress of an ongoing collaboration aimed at developing fast randomized iterative schemes specifically for applications in quantum chemistry.  This talk is based on joint work with Lek-Heng Lim, Timothy Berkelbach, Sam Greene, and Rob Webber.

• Tuesday January 28, 2020 at 15:00, Temple (Wachman 617)

  The KPZ fixed point

  Konstantin Matetski, Columbia University
  
  The KPZ universality class is a broad collection of models, which includes directed random polymers, interacting particle systems and random interface growth, characterized by unusual scale of fluctuations which also appear in the random matrix theory. The KPZ fixed point is a scaling invariant Markov process which is the conjectural universal limit of all models in the class. A complete description of the KPZ fixed point was obtained in a joint work with Jeremy Quastel and Daniel Remenik. In this talk I will describe how the KPZ fixed point was derived by solving a special model in the class called TASEP.

• Tuesday February 4, 2020 at 15:00, Penn (DRL 4C8)

  Scaling limit of a directed polymer among a Poisson field of independent walks

  Jian Song, Shandong University
  
  We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb{Z}$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation with a Gaussian noise whose space-time covariance is given by the heat kernel.

• Tuesday February 11, 2020 at 15:00, Temple (Wachman 617)

  Entropy of ribbon tilings

  Vladislav Kargin, Binghamton University
  
  I will talk about ribbon tilings, which have been originally introduced and studied by Pak and Sheffield. These are a generalization of the domino tilings which, unfortunately, lacks relations to determinants and spanning trees but still retains some of the nice properties of domino tilings. I will explain how ribbon tilings are connected to multidimensional heights and acyclic orientations, and present some results about enumeration of these tilings on simple regions. Joint work with Yinsong Chen.

• Tuesday February 18, 2020 at 15:00, Penn (DRL 4C8)

  Extreme eigenvalue distributions of sparse random graphs

  Jiaoyang Huang, Institute for Advanced Study
  
  I will discuss the extreme eigenvalue distributions of adjacency matrices of sparse random graphs, in particular the Erdős-Rényi graphs $G(N,p)$ and the random $d$-regular graphs. For Erdős-Rényi graphs, there is a crossover in the behavior of the extreme eigenvalues. When the average degree $Np$ is much larger than $N^{1/3}$, the extreme eigenvalues have asymptotically Tracy-Widom fluctuations, the same as Gaussian orthogonal ensemble. However, when $N^{2/9}\ll Np\ll N^{1/3}$ the extreme eigenvalues have asymptotically Gaussian fluctuations. The extreme eigenvalues of random $d$-regular graphs are more rigid, we prove on the regime $N^{2/9}\ll d\ll N^{1/3}$ the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by the Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have the second-largest eigenvalue strictly less than $2\sqrt{d-1}$. These are based on joint works with Roland Bauerschmids, Antti Knowles, Benjamin Landon and Horng-Tzer Yau.

• Tuesday February 25, 2020 at 15:00, Temple (Wachman 617)

  Universality of extremal eigenvalue statistics of random matrices

  Benjamin Landon, MIT
  
  The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory.  However, the behavior of certain "extremal" or "critical" observables is not fully understood.  Towards the former, we discuss progress on the universality of the largest gap between consecutive eigenvalues.  With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.

• Tuesday March 3, 2020 at 15:00, Penn (DRL 4C8)

  Estimation of Wasserstein distances in the spiked transport model

  Jonathan Niles-Weed, Courant Institute, NYU
  
  We propose a new statistical model, generalizing the spiked covariance model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study various probabilistic and statistical features of this model, including the estimation of the Wasserstein distance, which we show can be accomplished by an estimator which avoids the "curse of dimensionality" typically present in high-dimensional problems involving the Wasserstein distance. However, this estimator does not seem possible to compute in polynomial time, and we give evidence that any computationally efficient estimator is bound to suffer from the curse of dimensionality. Our results therefore suggest the existence of a computational-statistical gap.
   

  Joint work with Philippe Rigollet.

   

• Tuesday March 17, 2020 at 15:00, Penn (DRL 4C8)

  (POSTPONED)

  Robert Hough, Stony Brook University
  
   

• Tuesday March 24, 2020 at 15:00, Wachman 617

  (POSTPONED)

  Kavita Ramanan, Brown University
  
   

• Tuesday March 31, 2020 at 15:00, Penn (DRL 4C8)

  (POSTPONED)

  Yilin Wang, MIT
  
   

• Tuesday April 7, 2020 at 15:00, Temple (Wachman 617)

  (POSTPONED)

  Hoi Nguyen, Ohio State University
  
   

• Tuesday April 14, 2020 at 15:00, Penn (DRL 4C8)

  (POSTPONED)

  Louis Fan, Indiana University
  
   

• Tuesday April 21, 2020 at 15:00, Temple (Wachman 617)

  (POSTPONED)

  Kyle Luh, Harvard University
  
   

• Tuesday April 28, 2020 at 15:00, Penn (DRL 4C8)

  (POSTPONED)

  Gourab Ray, University of Victoria
  
   

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday January 22, 2019 at 15:00, Penn (DRL 4C8)

  One-point function estimates and natural parametrization for loop-erased random walk in three dimensions

  Xinyi Li, University of Chicago
  
  In this talk, I will talk about loop-erased random walk (LERW) in three dimensions. I will first give an asymptotic estimate on the probability that 3D LERW passes a given point (commonly referred to as the one-point function). I will then talk about how to apply this estimate to show that 3D LERW as a curve converges to its scaling limit in natural parametrization. If time permits, I will also talk about the asymptotics of non-intersection probabilities of 3D LERW with simple random walk. This is a joint work with Daisuke Shiraishi (Kyoto).

• Tuesday January 29, 2019 at 15:00, Temple (Wachman 617)

  Fractional Gaussian fields in geometric quantization and the semi-classical analysis of coherent states

  Alexander Moll, Northeastern University
  
  The Born Rule (1926) formalized in von Neumann's spectral theorem (1932) gives a precise definition of the random outcomes of quantum measurements as random variables from the spectral theory of non-random matrices. In [M. 2017], the Born rule provided a way to derive limit shapes and global fractional Gaussian field fluctuations for a large class of point processes from the first principles of geometric quantization and semi-classical analysis of coherent states. Rather than take a point process as a starting point, these point process are realized as auxiliary objects in an analysis that starts instead from a classical Hamiltonian system with possibly infinitely-many degrees of freedom that is not necessarily Liouville integrable. In this talk, we present these results with a focus on the case of one degree of freedom, where the core ideas in the arguments are faithfully represented.

• Tuesday February 5, 2019 at 15:00, Penn (DRL 4C8)

  Conformal embedding and percolation on the uniform triangulation

  Xin Sun, Columbia University
  
  Following Smirnov’s proof of Cardy’s formula and Schramm’s discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice has been achieved. Smirnov’s proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk, I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations. I will also explain how this result fits in the larger picture of random planar maps and Liouville quantum gravity.

• Tuesday February 19, 2019 at 15:00, Penn (DRL 4C8)

  Asymptotic zero distribution of random polynomials

  Duncan Dauvergne, University of Toronto
  
  It is well known that the roots of a random polynomial with i.i.d. coefficients tend to concentrate near the unit circle. In particular, the zero measures of such random polynomials converge almost surely to normalized Lebesgue measure on the unit circle if and only if the underlying coefficient distribution satisfies a particular moment condition. In this talk, I will discuss how to generalize this result to random sums of orthogonal (or asymptotically minimal) polynomials.

• Tuesday February 26, 2019 at 15:00, Penn (DRL 4C8)

  Distances between random orthogonal matrices and independent normals

  Tiefeng Jiang, University of Minnesota
  
  We study the distance between Haar-orthogonal matrices and independent normal random variables. The distance is measured by the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. Optimal rates are obtained. This is a joint work with Yutao Ma.

• Tuesday March 19, 2019 at 15:00, Penn (DRL 4C8)

  Delocalization of random band matrices

  Fan Yang, UCLA
  
  We consider Hermitian random band matrices $H$ in dimension $d$, where the entries $h_{xy}$, indexed by $x,y \in [1,N]^d$, vanish if $|x-y|$ exceeds the band width $W$. It is conjectured that a sharp transition of the eigenvalue and eigenvector statistics occurs at a critical band width $W_c$, with $W_c=\sqrt{N}$ in $d=1$, $W_c=\sqrt{\log N}$ in $d=2$, and $W_c=O(1)$ in $d\ge 3$. Recently, Bourgade, Yau and Yin proved the eigenvector delocalization for 1D random band matrices with generally distributed entries and band width $W\gg N^{3/4}$. In this talk, we will show that for $d\ge 2$, the delocalization of eigenvectors in a certain averaged sense holds under the condition $W\gg N^{2/(2+d)}$. Based on joint work with Bourgade, Yau and Yin.

• Tuesday March 26, 2019 at 15:00, Penn (DRL 4C8)

  Large deviations for functionals of Gaussian processes

  Xiaoming Song, Drexel University
  
  We prove large deviation principles for $\int_0^t \gamma(X_s)ds$, where $X$ is a $d$-dimensional Gaussian process and $\gamma(x)$ takes the form of the Dirac delta function $\delta(x)$, $|x|^{-\beta}$ with $\beta\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-\beta_i}$ with $\beta_i\in(0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.

• Tuesday April 2, 2019 at 15:00, Temple (Wachman 617)

  Geometry of the corner growth model

  Timo Seppalainen, UW-Madison
  
  The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah). 

• Tuesday April 9, 2019 at 15:00, Temple (Wachman 617)

  Eigenvectors of non-Hermitian random matrices

  Guillaume Dubach, Courant Institute, NYU
  
  Eigenvectors of non-Hermitian matrices are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables also quantify the stability of the spectrum, and characterize the joint eigenvalue increments under Dyson-type dynamics. Overlaps first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and some have been established for quaternionic Gaussian matrices.

• Tuesday April 16, 2019 at 15:00, Temple (Wachman 617)

  Stochastic homogenization for reaction-diffusion equations

  Jessica Lin, McGill University
  
  I will present several results concerning the stochastic homogenization for reaction-diffusion equations. We consider reaction-diffusion equations with nonlinear, heterogeneous, stationary-ergodic reaction terms. Under certain hypotheses on the environment, we show that the typical large-time, large-scale behavior of solutions is governed by a deterministic front propagation. Our arguments rely on analyzing a suitable analogue of “first passage times” for solutions of reaction-diffusion equations. In particular, under these hypotheses, solutions of heterogeneous reaction-diffusion equations with front-like initial data become asymptotically front-like with a deterministic speed. This talk is based on joint work with Andrej Zlatos.

• Tuesday April 30, 2019 at 15:00, Temple (Wachman 617)

  The geometry of the last passage percolation problem

  Tom Alberts, University of Utah
  
  Last passage percolation is a well-studied model in probability theory that is simple to state but notoriously difficult to analyze. In recent years it has been shown to be related to many seemingly unrelated things: longest increasing subsequences in random permutations, eigenvalues of random matrices, and long-time asymptotics of solutions to stochastic partial differential equations. Much of the previous analysis of the last passage model has been made possible through connections with representation theory of the symmetric group that comes about for certain exact choices of the random input into the last passage model. This has the disadvantage that if the random inputs are modified even slightly then the analysis falls apart. In an attempt to generalize beyond exact analysis, recently my collaborator Eric Cator (Radboud University, Nijmegen) and I have started using tools of tropical geometry to analyze the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model and our work to use it to produce probabilistic information.

• Tuesday September 3, 2019 at 15:00, Penn (DRL 4C8)

  Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$

  Ewain Gwynne, University of Cambridge
  
  We show that for each $\gamma \in (0,2)$, there is a unique metric associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the Gaussian free field $h$ on a planar domain $U$, there is a unique random metric $D_h =$ "$e^{\gamma h} (dx^2 + dy^2)$" on $U$ which is uniquely characterized by a list of natural axioms.
  
  The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the Gaussian free field. Earlier work by Ding, Dubédat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. We show that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield.
  
  Based on four joint papers with Jason Miller, one joint paper with Julien Dubédat, Hugo Falconet, Josh Pfeffer, and Xin Sun, and one joint paper with Josh Pfeffer.
   

• Tuesday September 10, 2019 at 15:00, Temple (Wachman 617)

  Semigroups for one-dimensional Schrödinger operators with multiplicative white noise

  Pierre Yves Gaudreau Lamarre, Princeton University
  
  In this talk, we are interested in the semigroup theory of continuous one-dimensional random Schrödinger operators with white noise. We will begin with a brief reminder of the rigorous definition of these operators as well as some of the problems in which they naturally arise. Then, we will discuss the proof of a Feynman-Kac formula describing their semigroups. In closing, we will showcase an application of this new semigroup theory to the study of rigidity (in the sense of Ghosh-Peres) of random Schrödinger eigenvalue point processes.
  
  Some of the results discussed in this talk are joint work with Promit Ghosal (Columbia) and Yuchen Liao (Michigan).
   

• Tuesday September 17, 2019 at 15:00, Penn (DRL 4C8)

  Hard-core models in discrete 2D

  Izabella Stuhl, Pennsylvania State University
  
  Do hard disks in the plane admit a unique Gibbs measure at high density? This is one of the outstanding open problems of statistical mechanics, and it seems natural to approach it by requiring the centers to lie in a fine lattice; equivalently, we may fix the lattice, but let the Euclidean diameter $D$ of the hard disks tend to infinity. Unlike most models in statistical physics, we find non-universality and connections to number theory, with different new phenomena arising in the triangular lattice $\mathbb{A}_2$, the square lattice $\mathbb{Z}^2$ and the hexagonal tiling $\mathbb{H}_2$.
  
  In particular, number-theoretic properties of the exclusion diameter $D$ turn out to be important. We analyze high-density hard-core Gibbs measures via Pirogov-Sinai theory. The first step is to identify periodic ground states, i.e., maximal density disk configurations which cannot be locally 'improved'. A key finding is that only certain 'dominant' ground states, which we determine, generate nearby Gibbs measures. Another important ingredient is the Peierls bound separating ground states from other admissible configurations.
  
  Answers are provided in terms of Eisenstein primes for $\mathbb{A}_2$ and norm equations in the ring $\mathbb{Z}[\sqrt{3}]$ for $\mathbb{Z}^2$. The number of high-density hard-core Gibbs measures grows indefinitely with $D$ but non-monotonically. In $\mathbb{Z}^2$ we analyze the phenomenon of 'sliding' and show it is rare.
  
  This is a joint work with A. Mazel and Y. Suhov.
   

• Tuesday September 24, 2019 at 15:00, Temple (Wachman 617)

  Stationary dynamics in finite time for the totally asymmetric simple exclusion process

  Axel Saenz, University of Virginia
  
  The totally asymmetric simple exclusion process (TASEP) is a Markov process that is the prototypical model for transport phenomena in non-equilibrium statistical mechanics. It was first introduced by Spitzer in 1970, and in the last 20 years, it has gained a strong resurgence in the emerging field of "Integrable Probability" due to exact formulas from Johansson in 2000 and Tracy and Widom in 2007 (among other related formulas and results). In particular, these formulas led to great insights regarding fluctuations related to the Tracy-Widom distribution and scalings to the Kardar-Parisi-Zhang (KPZ) stochastic differential equation. 
  
  In this joint work with Leonid Petrov (University of Virginia), we introduce a new and simple Markov process that maps the distribution of the TASEP at time $t >0$ , given step initial time data, to the distribution of the TASEP at some earlier time $t-s>0$. This process "back in time" is closely related to the Hammersley process introduced by Hammersley in 1972, which later found a resurgence in the longest increasing subsequence problem in the work of Aldous and Diaconis in 1995. Hence, we call our process the backwards Hammersley-type process (BHP). As a fun application of our results, we have a new proof of the limit shape for the TASEP. The central objects in our constructions and proofs are the Schur point processes and the Yang-Baxter equation for the $sl_2$ quantum affine Lie algebra. In this talk, we will discuss the background in more detail and will explain the main ideas behind the constructions and proof. 
   

• Tuesday October 1, 2019 at 15:00, Penn (DRL 4C8)

  Dynamics for spherical spin glasses: Disorder dependent initial conditions

  Amir Dembo, Stanford University
  
  In this talk, based on a joint work with Eliran Subag, I will explain how to rigorously derive the integro-differential equations that arise in the thermodynamic limit of the empirical correlation and response functions for Langevin dynamics in mixed spherical p-spin disordered mean-field models.
  
  I will then compare the large-time asymptotic of these equations in case of a uniform (infinite-temperature)  starting point, to what one obtains when starting within one of the spherical bands on which the Gibbs measure concentrates at low temperature, commenting on the existence of an aging phenomenon, and on the relations with the recently discovered geometric structure of the Gibbs measures at low temperature.
   

• Tuesday October 8, 2019 at 15:00, Temple (Wachman 617)

  Lower-tail large deviations of the KPZ equation

  Li-Cheng Tsai, Rutgers University
  
  Consider the solution of the KPZ equation with the narrow wedge initial condition. We prove the one-point, lower-tail Large Deviation Principle (LDP) of the solution, with time $t\to\infty$ being the scaling parameter, and with an explicit rate function. This result confirms existing physics predictions. We utilize a formula from Borodin and Gorin (2016) to convert the LDP of the KPZ equation to calculating an exponential moment of the Airy point process, and analyze the latter via the stochastic Airy operator and Riccati transform.
   

• Tuesday October 15, 2019 at 15:00, Penn (DRL 4C8)

  Local regime of random band matrices

  Tatyana Shcherbina, Princeton University
  
  Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.
   

• Tuesday October 29, 2019 at 15:00, Penn (DRL 4C8)

  Geometric TAP approach for spherical spin glasses

  Eliran Subag, Courant Institute, NYU
  
  The celebrated Thouless-Anderson-Palmer approach suggests a way to relate the free energy of a mean-field spin glass model to the solutions of certain self-consistency equations for the local magnetizations. In this talk I will first describe a new geometric approach to define free energy landscapes for general spherical mixed p-spin models and derive from them a generalized TAP representation for the free energy. I will then explain how these landscapes are related to various concepts and problems: the pure states decomposition, ultrametricity, temperature chaos, and optimization of full-RSB models.
   

• Tuesday November 5, 2019 at 15:00, Temple (Wachman 617)

  Absence of backward infinite paths in first-passage percolation in arbitrary dimension

  Michael Damron, Georgia Tech
  
  In first-passage percolation (FPP), one places weights $(t_e)$ on the edges of $\mathbb{Z}^d$ and considers the induced metric. Optimizing paths for this metric are called geodesics, and infinite geodesics are infinite paths all whose finite subpaths are geodesics. It is a major open problem to show that in two dimensions, with i.i.d. continuous weights, there are no bigeodesics (doubly-infinite geodesics). In this talk, I will describe work on bigeodesics in arbitrary dimension using "geodesic graph'' measures introduced in '13 in joint work with J. Hanson. Our main result is that these measures are supported on graphs with no doubly-infinite paths, and this implies that bigeodesics cannot be constructed in a translation-invariant manner in any dimension as limits of point-to-hyperplane geodesics. Because all previous works on bigeodesics were for two dimensions and heavily used planarity and coalescence, we must develop new tools based on the mass transport principle. Joint with G. Brito (Georgia Tech) and J. Hanson (CUNY).
   

• Tuesday November 12, 2019 at 15:00, Penn (DRL 4C8)

  Sharp threshold for the Ising perceptron model

  Changji Xu, University of Chicago
  
  Consider the discrete cube $\{-1,1\}^N$ and a random collection of half spaces which includes each half space $H(x) := \{ y \in \{-1,1\}^N: x \cdot y \geq \kappa \sqrt{N} \}$ for $x \in \{-1,1\}^N$ independently with probability $p$. Is the intersection of these half spaces empty? This is called the Ising perceptron model under Bernoulli disorder. We prove that this event has a sharp threshold, that is, the probability that the intersection is empty increases quickly from $\epsilon$ to $1- \epsilon$ when $p$ increases only by a factor of $1 + o(1)$ as $N \to \infty$.

• Tuesday November 19, 2019 at 15:00, Temple (Wachman 617)

  The Edwards-Wilkinson limit of the KPZ equation in $d>1$

  Yu Gu, Carnegie Mellon University
  
  In this talk, I will explain some recent work where we prove that in a certain weak disorder regime, the KPZ equation scales to the Edwards-Wilkinson equation in $d>1$.

• Tuesday December 3, 2019 at 15:00, Penn (DRL 4C8)

  Maximum of 3D Ising interfaces

  Eyal Lubetzky, Courant Institute, NYU
  
  Consider the random surface separating the plus and minus phases, above and below the $xy$-plane, in the low temperature Ising model in dimension $d\geq 3$. Dobrushin (1972) showed that if the inverse-temperature $\beta$ is large enough then this interface is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height on a box of side length $n$ is $O_P ( \log n )$.
  
  We study the large deviations of the interface in Dobrushin’s setting, and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to obtain a law of large numbers for the maximum height $M_n$ of the interface: $M_n/ \log n$ converges to $c_\beta$ in probability, where $c_\beta$ is given by a large deviation rate in infinite volume. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tail bounds.
  
  Joint work with Reza Gheissari.

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday September 4, 2018 at 15:00, Penn (DRL 4C8)

  In between random walk and rotor walk in the square lattice

  Swee Hong Chen, Cornell
  
  How much randomness is needed to prove a scaling limit result? In this talk we consider this question for a family of random walks on the square lattice. When the randomness is turned to the maximum, we have the symmetric random walk, which is known to scale to a two-dimensional Brownian motion. When the randomness is turned to zero, we have the rotor walk, for which its scaling limit is an open problem. This talk is about random walks that lie in between these two extreme cases and for which we can prove their scaling limit. This is a joint work with Lila Greco, Lionel Levine, and Boyao Li.

• Tuesday September 11, 2018 at 15:00, Temple (Wachman 617)

  The Sine-beta process: DLR equations and applications

  Thomas Leblé, NYU Courant
  
  One-dimensional log-gases, or Beta-ensembles, are statistical physics toy models finding their incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE’s. 
  We give a new description of Sine-beta as an « infinite volume Gibbs measure », using the Dobrushin-Lanford-Ruelle (DLR) formalism, and use it to prove the “rigidity” of the process, in the sense of Ghosh-Peres. If time permits, I will mention another application to the study of fluctuations of linear statistics. Joint work with David Dereudre, Adrien Hardy, and Mylène Maïda.

• Tuesday September 18, 2018 at 15:00, Temple (Wachman 617)

  Stationary coalescing walks on the lattice

  Arjun Krishnan, University of Rochester
  
  Consider a measurable dense family of semi-infinite nearest-neighbor paths on the integer lattice in d dimensions. If the measure on the paths is translation invariant, we completely classify their collective behavior in d=2 under mild assumptions. We use our theory to classify the behavior of semi-infinite geodesics in random translation invariant metrics on the lattice; it applies, in particular, to first- and last-passage percolation. We also construct several examples displaying unexpected behaviors. (Joint work with Jon Chaika.)

• Tuesday September 25, 2018 at 15:00, Penn (DRL 4C8)

  Zeros of polynomials, the distribution of coefficients, and a problem of J.E. Littlewood

  Julian Sahasrabudhe, Cambridge University

  While it is an old and fundamental fact that every (nice enough) even function $f : [-\pi,\pi] \rightarrow \mathbb{C}$ may be uniquely expressed as a cosine series \[ f(\theta) = \sum_{r \geq 0 } C_r\cos(r\theta), \] the relationship between the sequence of coefficients $(C_r)_{r \geq 0 }$ and the behavior of the function $f$ remains mysterious in many aspects. We mention two variations on this theme. First a more probabilistic setting: what can be said about a random variable if we constrain the roots of the probability generating function? We then settle on our main topic; a solution to a problem of J.E. Littlewood about the behavior of the zeros of cosine polynomials with coefficients $C_r \in \{0,1\}$.

• Tuesday October 2, 2018 at 15:00, Temple (Wachman 617)

  Shifted weights and restricted path length in first-passage percolation

  Firas Rassoul-Agha, University of Utah
  
  We study standard first-passage percolation via related optimization problems that restrict path length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of geodesic length due to Hammersley, Smythe and Wierman, and Kesten. We study the regularity of the time constant as a function of the shift of weights. For unbounded weights, this function is strictly concave and in case of two or more atoms it has a dense set of singularities. For any weight distribution with an atom at the origin there is a singularity at zero, generalizing a result of Steele and Zhang for Bernoulli FPP. The regularity results are proved by the van den Berg-Kesten modification argument. This is joint work with Arjun Krishnan and Timo Seppäläinen.
   

• Tuesday October 9, 2018 at 15:00, Temple (Wachman 617)

  The coin turning walk and its scaling limit

  Janos Englander, CU Boulder
  
  Given a sequence of numbers $p_n ∈ [0, 1]$, consider the following experiment. First, we flip a fair coin and then, at step $n$, we turn the coin over to the other side with probability $p_n, n > 1$, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as $n \to \infty$? We show that a number of phase transitions take place as the turning gets slower (i.e. $p_n$ is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $p_n = \textrm{const}/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws. The critical regime is particularly interesting: when the corresponding random walk is considered, an interesting process emerges as the scaling limit; also, a connection with Polya urns will be mentioned. This is joint work with S. Volkov (Lund) and Z. Wang (Boulder).

• Tuesday October 23, 2018 at 15:00, Penn (DRL 4C8)

  Isoperimetric shapes in supercritical bond percolation

  Julian Gold, Northwestern University
  
  We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ of supercritical bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove a shape theorem for these random graphs, showing that upon rescaling they tend almost surely to a deterministic shape. This limit shape is itself an isoperimetric set for a norm we construct. In addition, we obtain sharp asymptotics for a modification of the Cheeger constant of $\textbf{C}_\infty \cap [-n,n]^d$, settling a conjecture of Benjamini for this modified Cheeger constant. Analogous results are shown for the giant component in dimension two, where we use the original definition of the Cheeger constant, and a more complicated continuum isoperimetric problem emerges as a result.

• Tuesday October 30, 2018 at 15:00, Temple (Wachman 617)

  Sharp transition of invertibility of sparse random matrices

  Anirban Basak, Tata Institute
  
  Consider an $n \times n$ matrix with i.i.d. Bernoulli($p$) entries. It is well known that for $p= \Omega(1)$, i.e., $p$ bounded below by some positive constant, the matrix is invertible with high probability. If $p \ll \frac{\log n}{n}$ then the matrix contains zero rows and columns with high probability and hence it is singular with high probability. In this talk, we will discuss the sharp transition of the invertibility of this matrix at $p =\frac{\log n}{n}$. This phenomenon extends to the adjacency matrices of directed and undirected Erdös-Rényi graphs, and random bipartite graphs. Joint work with Mark Rudelson.

• Tuesday November 6, 2018 at 15:00, Penn (DRL 4C8)

  Applications of CLTs and homogenization for Dyson Brownian Motion to Random Matrix Theory

  Philippe Sosoe, Cornell University
  
   I will explain how two recent technical developments in Random Matrix Theory allow for a precise description of the fluctuations of single eigenvalues in the spectrum of large symmetric matrices. No prior knowledge of random matrix theory will be assumed. (Based on joint work with B. Landon and H.-T. Yau.)

• Tuesday November 13, 2018 at 15:00, Penn (DRL 4C8)

  Inference and compression problems on dynamic networks

  Abram Magner, Purdue University
  
  Networks in the real world are dynamic -- nodes and edges are added and removed over time, and time-varying processes (such as epidemics) run on them. In this talk, I will describe mathematical aspects of some of my recent work with collaborators on statistical inference and compression problems that involve this time-varying aspect of networks. I will focus on two related lines of work: (i) network archaeology -- broadly concerning problems of dynamic graph model validation and inference about previous states of a network given a snapshot of its current state, and (ii) structural compression -- for a given graph model, exhibit an efficient algorithm for invertibly mapping network structures (i.e., graph isomorphism types) to bit strings of minimum expected length. For both classes of problems, I give both information-theoretic limits and efficient algorithms for achieving those limits. Finally, I briefly describe some ongoing projects that continue these lines of work.

• Tuesday November 27, 2018 at 15:00, Temple (Wachman 617)

  Stick breaking processes, clumping, and Markov chain occupation laws

  Sunder Sethuraman, University of Arizona 
  
  A GEM (Griffiths-Engen-McCloskey) sequence specifies the (random) proportions in splitting a `resource' infinitely many ways. Such sequences form the backbone of `stick breaking' representations of Dirichlet processes used in nonparametric Bayesian statistics. In this talk, we consider the connections between a class of generalized `stick breaking' processes, an intermediate structure via `clumped' GEM sequences, and the occupation laws of certain time-inhomogeneous Markov chains.

• Tuesday December 4, 2018 at 15:00, Temple (Wachman 617)

  The algorithmic hardness threshold for continuous random energy models

  Pascal Maillard, Orsay/CRM
  
  I will report on recent work with Louigi Addario-Berry on algorithmic hardness for finding low-energy states in the continuous random energy model of Bovier and Kurkova. This model can be regarded as a toy model for strongly correlated random energy landscapes such as the Sherrington-Kirkpatrick model. We exhibit a precise and explicit hardness threshold: finding states of energy above the threshold can be done in linear time, while below the threshold this takes exponential time for any algorithm with high probability. I further discuss what insights this yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.

The seminar is jointly organized between Temple and Penn, by Brian Rider (Temple) and Robin Pemantle (Penn).

For a chronological listing, click the year above.

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn).

You can also check out the seminar website at Penn.

• Tuesday January 24, 2017 at 15:00, UPenn (David Riitenhouse Lab 3C8)

  Abelian squares and their progenies

  Charles Burnette, Drexel University

  A polynomial P ∈ C[z1, . . . , zd] is strongly Dd-stable if P has no zeroes in the closed unit polydisc D d . For such a polynomial define its spectral density function as SP (z) = P(z)P(1/z) −1 . An abelian square is a finite string of the form ww0 where w0 is a rearrangement of w. We examine a polynomial-valued operator whose spectral density function’s Fourier coefficients are all generating functions for combinatorial classes of con- strained finite strings over an alphabet of d characters. These classes generalize the notion of an abelian square, and their associated generating functions are the Fourier coefficients of one, and essentially only one, L2 (T d)-valued operator. Integral representations and asymptotic behavior of the coefficients of these generating functions and a combinatorial meaning to Parseval’s equation are given as consequences.

   

• Tuesday January 31, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

  How round are the complementary components of planar Brownian motion?

  Nina Holden, MIT

  Consider a Brownian motion $W$ in the complex plane started from $0$ and run for time $1$. Let $A(1), A(2),...$ denote the bounded connected components of $C-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i \in N$. Our main result is that $E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty$ for any $\theta <1$. We also prove that $\sum_i r(i)^2|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape. Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.

   

• Tuesday February 7, 2017 at 15:00, Temple (Wachmann Hall 617)

  Stochastic areas and Hopf fibrations

  Fabrice Baudoin, University of Connecticut

  We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces ℂℙn and ℂℍn. The characteristic functions of those processes are computed and limit theorems are obtained. For ℂℙn the geometry of the Hopf fibration plays a central role, whereas for ℂℍn it is the anti-de Sitter fibration. This is joint work with Jing Wang (UIUC).

   

• Tuesday February 14, 2017 at 15:00, Temple (Wachmann Hall 617)

  Intermediate disorder limits for multi-layer random polymers

  Mihai Nica, NYU

  The intermediate disorder regime is a scaling limit for disordered systems where the inverse temperature is critically scaled to zero as the size of the system grows to infinity. For a random polymer given by a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this talk, I consider polymers made up of multiple non-intersecting walkers and consider the same type of limit. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. This result proves a conjecture about the KPZ line ensemble. Part of this talk is based on joint work with I. Corwin.

   

• Tuesday February 21, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

  Large deviation and counting problems in sparse settings

  Shirshendu Ganguly, Berkeley

  The upper tail problem in the Erdös-Rényi random graph $G \sim G(n,p)$, where every edge is included independently with probability $p$, is to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1 + d$. The arithmetic analog considers the count of arithmetic progressions in a random subset of $Z/nZ$, where every element is included independently with probability $p$. In this talk, I will describe some recent results regarding the solution of the upper tail problem in the sparse setting i.e. where $p$ decays to zero, as $n$ grows to infinity. The solution relies on non-linear large deviation principles developed by Chatterjee and Dembo and more recently by Eldan and solutions to various extremal problems in additive combinatorics.

   

• Tuesday February 28, 2017 at 15:00, Temple (Wachmann Hall 617)

  Bounds on the maximum of the density for certain linear images of independent random variables

  James Melbourne, University of Delaware

  We will present a generalization of a theorem of Rogozin that identifies uniform distributions as extremizers of a class of inequalities, and show how the result can reduce specific random variables questions to geometric ones. In particular, by extending "cube slicing" results of K. Ball, we achieve a unification and sharpening of recent bounds on densities achieved as projections of product measures due to Rudelson and Vershynin, and the bounds on sums of independent random variable due to Bobkov and Chistyakov. Time permitting we will also discuss connections with generalizations of the entropy power inequality.

   

• Tuesday March 21, 2017 at 15:00, Temple (Wachmann Hall 617)

  Local extrema of random matrices and the Riemann zeta function 

  Paul Bourgade, NYU

  Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to conjecture the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on a joint works with Arguin, Belius, Radziwill and Soundararajan.

   

• Tuesday March 28, 2017 at 15:00, Temple (Wachman Hall 617)

  Majority dynamics on the infinite 3-regular tree

  Arnab Sen, University of Minnesota

  The majority dynamics on the infinite 3-regular tree can be described as follows. Each vertex of the tree has an i.i.d. Poisson clock attached to it, and when the clock of a vertex rings, the vertex looks at the spins of its three neighbors and flips its spin, if necessary, to come into agreement with majority of its neighbors. The initial spins of the vertices are taken to be i.i.d. Bernoulli random variables with parameter p. In this talk, we will discuss a couple of new results regarding this model. In particular, we will show that the limiting proportion of ‘plus’ spins in the tree is continuous with respect to the initial bias p. A key tool in our argument is the mass transport principle. The talk is based on an ongoing work with M. Damron.

   

• Tuesday April 4, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

  Galton-Watson fixed points, tree automata, and interpretations

  Tobias Johnson, NYU

  onsider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

   

• Tuesday April 11, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

  Biased random permutations are predictable (proof of an entropy conjecture of Leighton and Moitra)

  Patrick Devlin, Rutgers

  Suppose F is a random (not necessarily uniform) permutation of {1, 2, ... , n} such that |Prob(F(i) < F(j)) -1/2| > epsilon for all i,j. We show that under this assumption, the entropy of F is at most (1-delta)log(n!), for some fixed delta depending only on epsilon [proving a conjecture of Leighton and Moitra]. In other words, if (for every distinct i,j) our random permutation either noticeably prefers F(i) < F(j) or prefers F(i) > F(j), then the distribution inherently carries significantly less uncertainty (or information) than the uniform distribution.

  Our proof relies on a version of the regularity lemma, a combinatorial bookkeeping gadget, and a few basic probabilistic ideas. The talk should be accessible for any background, and we will gently recall any relevant notions (e.g., entropy) as needed. Those unhappy with the talk are welcome to form an unruly mob to depose the speaker, and pitchforks and torches will be available for purchase.

  This is from a recent paper joint with Huseyin Acan and Jeff Kahn.

   

• Tuesday April 25, 2017 at 15:00, Temple (Wachman Hall 617)

  An introduction to p-adic electrostatics

  Christopher Sinclair, University of Oregon

  We consider the distribution of N p-adic particles with interaction energy proportional to the log of the p-adic distance between two particles. When the particles are constrained to the ring of integers of a local field, the distribution of particles is proportional to a power of the p-adic absolute value of the Vandermonde determinant. This leads to a first question: What is the normalization constant necessary to make this a probability measure? This sounds like a triviality, but this normalization constant as a function of extrinsic variables (like number of particles, or temperature) holds much information about the statistics of the particles. Viewed another way, this normalization constant is a p-adic analog of the now famous Selberg integral. While a closed form for this seems out of reach, I will derive a remarkable identity that may hold the key to unlocking more nuanced information about p-adic electrostatics. Along the way we’ll find an identity for the generating function of probabilities that a degree N polynomial with p-adic integer coefficients split completely. Joint work with Jeff Vaaler.

   

• Tuesday May 2, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

  Percolation in Weighted Random Connection Model

  Milan Bradonjic, Bell Labs

  When modeling the spread of infectious diseases, it is important to incorporate risk behavior of individuals in a considered population. Not only risk behavior, but also the network structure created by the relationships among these individuals as well as the dynamical rules that convey the spread of the disease are the key elements in predicting and better understanding the spread. We propose the weighted random connection model, where each individual of the population is characterized by two parameters: its position and risk behavior. A goal is to model the effect that the probability of transmissions among individuals increases in the individual riskfactors, and decays in their Euclidean distance. Moreover, the model incorporates a combined risk behavior function for every pair of theindividuals, through which the spread can be directly modeled or controlled. The main results are conditions for the almost sure existence of an infinite cluster in the weighted random connection model. We use results on the random connection model and sitepercolation in Z^2.

   

• Tuesday September 5, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

  Large deviations for first passage percolation

  Allan Sly, Princeton

  We establish a large deviation rate function for the upper tail of first passage percolation answering a question of Kesten who established the lower tail in 1986. Moreover, conditional on the large deviation event, we show that the minimal cost path is delocalized, that is it moves linearly far from the straight line path. Joint work with Riddhipratim Basu (Stanford/ICTS) and Shirshendu Ganguly (UC Berkeley).

• Tuesday September 12, 2017 at 15:00, Temple (Wachman Hall 617)

  Stability of phases and interacting particle systems

  Nick Crawford, Technion

  I will discuss recent work with W. de Roeck on the following natural question: Given an interacting particle system are the stationary measures of the dynamics stable to small (extensive) perturbations? In general, there is no reason to believe this is so and one must restrict the class of models under consideration in one way or another. As such, I will focus in this talk on the simplest setting for which one might hope to have a rigorous result: attractive Markov dynamics (without conservation laws) relaxing at an exponential rate to its unique stationary measure in infinite volume. In this case we answer the question affirmatively.

  As a consequence we show that ferromagnetic Ising Glauber dynamics is stable to small, non-equilibrium perturbations in the entire uniqueness phase of the inverse temperature/external field plane. It is worth highlighting that this application requires new results on the (exponential) rate of relaxation for Glauber dynamics defined with non-zero external field.

• Tuesday September 19, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

  Invasion percolation on Galton-Watson trees 

  Marcus Michelin, UPenn

  Given an infinite rooted tree, how might one sample, nearly uniformly, from the set of paths from the root to infinity? A number of methods have been studied including homesick random walks, or determining the growth rate of the number of self-avoiding paths. Another approach is to use percolation. The model of invasion percolation almost surely induces a measure on such paths in Galton-Watson trees, and we prove that this measure is absolutely continuous with respect to the limit uniform measure as well as other properties of invasion percolation. This work in progress is joint with Robin Pemantle and Josh Rosenberg.

• Tuesday September 26, 2017 at 16:00, UPenn (David Rittenhouse Lab 4C8)

  Cutoff for random to random 

  Evita Nestoridi, Princeton

  Random to random is a card shuffling model that was created to study strong stationary times. Although the mixing time of random to random has been known to be of order n log n since 2002, cutoff had been an open question for many years, and a strong stationary time giving the correct order for the mixing time is still not known. In joint work with Megan Bernstein, we use the eigenvalues of the random to random card shuffling to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4} n log n$, answering a conjecture of Diaconis.

• Tuesday October 3, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

  Lattice path enumeration, multivariate singularity analysis, and probability theory

  Stephen Melczar, UPenn

  The problem of enumerating lattice paths with a fixed set of allowable steps and restricted endpoint has a long history dating back at least to the 19th century. For several reasons, much research on this topic over the last decade has focused on two dimensional lattice walks restricted to the first quadrant, whose allowable steps are "small" (that is, each step has coordinates +/- 1, or 0). In this talk we relax some of these conditions and discuss recent work on walks in higher dimensions, with non-small steps, or with weighted steps. Particular attention will be given to the asymptotic enumeration of such walks using representations of the generating functions as diagonals of rational functions, through the theory of analytic combinatorics in several variables. Several techniques from computational and experimental mathematics will be highlighted, and open conjectures of a probabilistic nature will be discussed.

• Tuesday October 10, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

  Rigidity of the 3D hierarchical Coulomb gas

  Sourav Chatterjee, Stanford

  The mathematical analysis of Coulomb gases, especially in dimensions higher than one, has been the focus of much recent activity. For the 3D Coulomb, there is a famous prediction of Jancovici, Lebowitz and Manificat that if N is the number of particles falling in a given region, then N has fluctuations of order cube-root of E(N). I will talk about the recent proof of this conjecture for a closely related model, known as the 3D hierarchical Coulomb gas. I will also try to explain, through some toy examples, why such unusually small fluctuations may be expected to appear in interacting gases.

• Tuesday October 17, 2017 at 15:00, Temple (Wachman Hall 617)

  Homogenization of a class of 1-D nonconvex viscous Hamilton-Jacobi equations with random potential

  Atilla Yilmaz, NYU & Koc University

  There are general homogenization results in all dimensions for (inviscid and viscous) Hamilton-Jacobi equations with random Hamiltonians that are convex in the gradient variable. Removing the convexity assumption has proved to be challenging. There was no progress in this direction until two years ago when the 1-D inviscid case was settled positively and several classes of (mostly inviscid) examples for which homogenization holds were constructed as well as a 2-D inviscid counterexample. Methods that were used in the inviscid case are not applicable to the viscous case due to the presence of the diffusion term.

  In this talk, I will present a new class of 1-D viscous Hamilton-Jacobi equations with nonconvex Hamiltonians for which homogenization holds. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial data have representations involving exponential expectations of controlled Brownian motion in random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in random potential. The proof relies on (i) analyzing the large deviation behavior of the controlled Brownian particle which assumes the role of one of the players in an emergent two-player game, (ii) identifying asymptotically optimal control policies and (iii) constructing correctors which lead to exponential martingales.

  Based on recent joint work with Elena Kosygina and Ofer Zeitouni.

• Tuesday October 24, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

  Extreme level sets of branching Brownian motion

  Lisa Hartung, NYU

  We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida. (Joint work with A. Cortines, O Louidor.)

• Tuesday November 7, 2017 at 15:00, 617 Wachman Hall

  Spectrum of random band matrices 

  Indrajit Jana, Temple University

  We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.

• Tuesday November 14, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

  How fragile are information cascades? 

  Miklos Racz, Berkeley

  It is well known that sequential decision making may lead to information cascades. If the individuals are choosing between a right and a wrong state, and the initial actions are wrong, then the whole cascade will be wrong. We show that if agents occasionally disregard the actions of others and base their action only on their private information, then wrong cascades can be avoided. Moreover, we obtain the optimal asymptotic rate at which the error probability at time t can go to zero. This is joint work with Yuval Peres, Allan Sly, and Izabella Stuhl.

• Tuesday November 28, 2017 at 15:00, Temple (Wachman Hall 617)

  (Postponed)

  Thomas Leblé, NYU

• Tuesday December 5, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

  TBA

  Konstantinos Karatapanis, UPenn

   

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday February 12, 2024 at 16:00, Wachman 617

  Rowing, waving, and worms: using computational methods to study how natural variation affects swimming performance in Tomopteridae

  Nick Battista, The College of New Jersey
  
  The ocean is home to an incredible diversity of animals of many shapes and sizes. Living life in a water-based environment presents unique challenges that vary based on the size and shape of each organism. Animals have evolved a variety of morphological structures, locomotor mechanisms, and swimming strategies that help reduce their energy expenditure by favoring more energetically efficient modes. Comprehensive studies that consider multiple morphological and kinematics traits and their influence on swimming performance are needed to investigate these differing strategies. Computational modeling gives us a tool to glean insight into how morphological or kinematics variation affects performance across different scales. For example, validated models can be used to thoroughly explore how varying multiple traits affects performance, where conducting an empirical study may be unrealistic due to finding enough organisms to test across the landscape of multiple traits. In addition, models can assess how natural variation affects performance and identify where trade-offs occur. In today's talk, I will describe my undergraduate lab's approach to studying the swimming behaviors for a variety of animals through a blend of math modeling, computational fluid dynamics, and machine learning. I will walk through our modeling process using Tomopteris, a polychaete, as an example, while also touching upon our own set of challenges, limitations, and future directions.
   

• Monday February 26, 2024 at 16:00, Wachman 617

  Weak KAM theory and homogenization for Hamiltonian ODEs and Hamilton-Jacobi PDEs

  Fraydoun Rezakhanlou, University of California, Berkeley
  
  Traditionally homogenization asks whether average behavior can be discerned from Hamilton-Jacobi equations that are subject to high-frequency fluctuations in spatial variables. A similar question can be asked for the associated Hamiltonian ODEs. When the Hamiltonian function is convex in momentum variable, these two questions turn out to be equivalent. This equivalence breaks down for general Hamiltonian functions. In this talk I will give a dynamical system formulation for homogenization and address some results concerning weak and strong homogenization phenomena.

• Monday March 18, 2024 at 16:00, Wachman 617

  Geometric Structures associated to Higher Teichmuller Theory

  Sara Maloni, University of Virgina
  
  
  Abstract: The Teichmuller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, this space can also be seen as a connected component of representations from the fundamental group of S into Isom(H^2). Generalizing this point of view, Higher Teichmuller Theory studies connected components of representations from the fundamental group of S into Lie groups of rank greater than 1.
  
  We will discuss parts of the classical theory of deformations of geometric structures, Higher Teichmuller Theory and the notion of Anosov representation. We will then describe how Anosov representations correspond to deformation of certain geometric structures, and a joint work with Alessandrini, Tholozan and Wienhard about their topology.

• Monday April 15, 2024 at 16:00, Wachman 617

  On the Higgs mechanism for mass generation

  Sourav Chatterjee, Stanford/IAS
  
  I will talk about the Higgs mechanism for mass generation and some recent progress on this topic. No background is necessary. I will start by introducing lattice gauge theories coupled to Higgs fields. After a survey of existing results, I will discuss what is needed to prove rigorously that the Higgs mechanism can indeed generate mass in the continuum limit of these theories. Finally, I will present a result which shows that in a certain scaling limit in any dimension three or higher, SU(2) Yang-Mills-Higgs theory converges to a continuum limit object which has an explicit description as a scale-invariant random distribution. This allows an exact computation of the mass generated by the Higgs mechanism in the continuum limit of this theory.

• Monday September 23, 2024 at 16:00,

  Global regularity for the Rayleigh-Taylor unstable Muskat bubble problem with critical regularity

  Robert Strain, University of Pennsylvania
  
  This talk concerns the Muskat problem with surface tension, modeling the filtration of two incompressible immiscible fluids in porous media. This non-local and non-linear partial differential equation is a basic mathematical model in petroleum engineering; it was formulated by the petroleum engineer M. Muskat in 1934 to describe the mixture of water into an oil sand. Given its origins and its equivalence with Hele-Shaw flows, the Muskat problem has received a lot of attention from the physics community.
  
  We consider the case in which the fluids have different constant densities together with different constant viscosities. The Rayleigh-Taylor condition cannot hold for a closed curve, which makes this situation unstable. In this case the equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available to bound the amplitude and the slopes of the interface. We prove global in time existence and uniqueness results for medium size initial stable data in critical functional spaces. In particular we prove for the first time the global in time stability of star shaped bubbles influenced by Gravity. This is joint work with Gancedo, Garcia-Juarez, and Patel.
  
   

• Monday October 7, 2024 at 16:00, Wachman 617

  Exotic four-manifolds

  Tye Lidman, North Carolina State University
  
  Geometric topology studies properties of manifolds, objects which locally look like Euclidean space. While we know a lot about manifold topology in most dimensions, four-dimensional manifolds remain particularly elusive and have some very unique properties compared to all other dimensions. One of the coolest ways this manifests is through exotic manifolds: manifolds which are homeomorphic (continuously equivalent) but not diffeomorphic (differentiably equivalent). In this talk I will discuss some of the strange phenomena of four-manifolds, as well as some new approaches to building and distinguishing exotic four-manifolds. This is joint work with Adam Levine and Lisa Piccirillo.

• Monday October 21, 2024 at 16:00, Wachman 617

  Data-driven modeling and simulation of the human tear film

  Toby Driscoll, University of Delaware
  
  The tear film is a complex fluid that plays critical roles in the optical function and health of the eye. Its detailed dynamics can vary greatly between individuals and over time, and quantitative assessment of many important tear film properties is difficult. Mathematical and computational simulation of the physics and chemistry of tear film dynamics has become increasingly sophisticated, but matching the modeling with clinical observation remains a major challenge. Techniques from deep learning and inverse problems are now enabling quantitative insights into tear film function of individuals at previously unreachable scales, and scientific machine learning methods show promise for future developments.
  
   

• Monday November 4, 2024 at 16:00, Wachman 617

  Tropical Geometry

  Diane Maclagan, University of Warwick/IAS
  
  Tropical geometry is a combinatorial shadow of algebraic geometry. It is geometry over the tropical semiring, where multiplication is replaced by addition, and addition is replaced by minimum. This turns (algebraic)-geometric questions into questions from polyhedral combinatorics. I will give a gentle introduction to this twenty-first century field, giving some idea of where it can be applied, both inside and outside algebraic geometry. No knowledge of algebraic geometry will be assumed.

• Friday November 22, 2024 at 15:00, Wachman 617

  Rational points and algebraic cycles on Jacobians of curves

  Ari Shnidman, Hebrew University/IAS
  
  I'll survey the arithmetic of algebraic curves and their Jacobians. Euclid showed how to find all rational points on conics such as $x^2 + y^2 = n$, but even the case of cubics such as $x^3 + y^3 = n$ is very much open and is the topic of the Birch and Swinnerton-Dyer conjecture. In general, to find rational points on a curve $C$ of genus $g$, one first finds the rational points on its Jacobian variety $J$, a $g$-dimensional complex torus. Efforts to understand higher "rank" cases of the Birch and Swinnerton-Dyer conjecture lead us to study the more general question of determining the Chow group of algebraic cycles (deformation classes of formal sums of subvarieties of a given codimension). Little is known once the codimension is at least 2, but I'll discuss some recent work on the Ceresa cycle, which is a canonical 1-cycle living on $J$, namely $[C] - [-C]$. I will not assume prior knowledge of number theory or algebraic geometry.

• Monday December 2, 2024 at 16:00, Wachman 617

  Prime distribution and arithmetic of curves

  Wanlin Li, Washington University of St. Louis
  
  The distribution of primes among congruence classes is one of the most classical and influential problems in number theory. The question of whether there are more primes of the form 4k+1 or 4k+3 leads to the construction of Dirichlet characters, L-functions, and the study of analytic number theory. In this talk, I will discuss the study of Chebyshev's bias and the set of zeros of Dirichlet L-functions over global function fields. These studies can be viewed from a geometric perspective as studying the arithmetic of algebraic curves defined over finite fields. I will introduce the notion of ``supersingular'' and discuss the distribution of supersingular curves in algebraic families and in reductions of curves defined over number fields.
  
   

• Wednesday December 4, 2024 at 16:30, Wachman 617

  Shimura varieties and their canonical models: new directions

  Alexander Youcis, National University of Singapore
  
  Shimura varieties are a class of spaces that sit at the intersection of number theory, algebraic geometry, differential geometry, and harmonic analysis, and that generalize the classically-studied modular curves. Shimura varieties have played a central part in a large number of advances in number theory and algebraic geometry since their introduction in the 1960s. In this talk, I will give a high-level introduction to these spaces, focusing on their role within the broader goals of arithmetic geometry. I will end by indicating how recent advances in our understanding of the cohomology of varieties promise to put the guiding principles motivating the study of Shimura varieties on firmer footing. This is based on joint work with Naoki Imai and Hiroki Kato.
   

• Tuesday December 17, 2024 at 15:00, Wachman 617

  Cubic fourfolds and their Fano varieties of lines

  Sarah Frei, Dartmouth College
  
  Identifying which cubic fourfolds are rational is a famous open problem in classical algebraic geometry. It has been approached from numerous perspectives, one of which is using the variety parametrizing lines on the cubic fourfold, called the Fano variety of lines. There is a natural relationship between the geometry of a cubic fourfold and of its Fano variety of lines, but a better understanding of this relationship is expected to shed light on the birational geometry of cubic fourfolds. In this talk, I will discuss joint work with C. Brooke and L. Marquand, in which we investigate this circle of ideas.
  
   

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday February 6, 2023 at 16:00, Wachman 617

  An aspect of the geometry of p-Schatten spaces and its relation to the hyperplane conjecture

  Kavita Ramanan, Brown University
  
  The hyperplane conjecture in convex geometry is a statement about the volume of a convex body and that of its hyperplane sections. Taking a measure-theoretic perspective to this problem, Bourgain highlighted the importance of the notion of a $\psi_2$-convex body, which captures integrability properties of linear images of the volume measure on the body. Despite this notion being introduced more than a quarter century ago, there are not many examples of such bodies. We describe several results on the $\psi_2$ (or more generally, $\psi_\alpha$) behavior of Schatten balls and their marginals, and their relation to the hyperplane conjecture. Along the way, we also establish some properties of the Haar measure on the orthogonal group that may be of independent interest. This is joint work with Grigoris Paouris.

• Monday February 13, 2023 at 16:00, Wachman 617

  From graphs to algebra and geometry

  Ralph Kaufmann, Purdue University
  
  Graphs are a universal tool in mathematics. Their study is interesting in its own right. They also naturally appear in the geometry of surfaces. When coupled with additional structures such as local orders or other local information in the form of so-called ribbon graphs. We will present a unified approach to these structures and explore their relationship with algebra, topology and geometry.

• Monday March 13, 2023 at 16:00, Wachman 617

  Stark's Conjectures and Hilbert's 12th Problem

  Samit Dasgupta, Duke University
  
  In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field; this question lies at the core of Hilbert's 12th Problem. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe two recent joint results with Mahesh Kakde on these topics. The first is a proof of the Brumer-Stark conjecture. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years. We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to the question of explicit class field theory for these fields. 
   

• Monday March 27, 2023 at 16:00, Wachman 617

  Lifting Galois representations to the Grothendieck-Teichmueller group and beyond

  Frauke Bleher, University of Iowa
  
  This is joint work with A. Lubotzky and T. Chinburg. It is a longstanding problem in Galois theory to give an explicit description of the absolute Galois group G_Q of the rationals. A famous theorem by Belyi from 1979 says that G_Q can be embedded into the automorphism group A of a free profinite group on two generators. This theorem led Grothendieck, Drinfel'd and others to try to identify G_Q inside A as a subgroup satisfying various extra relations. Each set of such relations defines a so-called Grothendieck-Teichmueller group GT inside A that is a candidate for being G_Q. It has been an open problem since the 1990's to identify natural non-abelian representations of G_Q that can be shown to lift to such GT groups. We will show that after passing to subgroups of finite index, this can be done for various natural families of representations, e.g. for the actions of subgroups of finite index in G_Q on all of the finite subgroups of the points of an elliptic curve defined over a number field. The main tool is a construction of linear representations of the automorphism group of a free profinite group on finitely many generators that generalizes work of Grunewald and Lubotzky on linear representations of the automorphism group of a finitely generated free discrete group. 
   

• Monday April 3, 2023 at 16:00, Wachman 617

  Algebraic methods in extremal graph theory

  Michael Tait, Villanova University
  
  How many edges may a graph with no triangle have? Given a graph F, the Turan problem asks to maximize the number of edges in a graph on n vertices subject to the constraint that it does not contain F as a subgraph. In this talk, we will discuss constructions for this problem coming from finite geometry (eg using projective planes), additive combinatorics, and "random polynomials". 
   

• Monday April 10, 2023 at 16:00, Wachman 617

  Rational points on elliptic curves (and their p-adic construction)

  Kazim Buyukboduk, University College Dublin
  
  Negative answer to Hilbert's 10th problem tells us that determining whether or not an algebraic variety should carry any rational points is impossibly hard (literally!). The same problem even for curves is very difficult: For elliptic curves, this is the subject of the celebrated Birch and Swinnerton-Dyer conjecture. I will survey recent results on this problem, and explain briefly an explicit p-adic analytic construction of rational points of infinite order on elliptic curves of rank one (settling a conjecture of Perrin-Riou). These final bits of the talk will be a report on joint works with Rob Pollack & Shu Sasaki, and with Denis Benois.

• Monday April 24, 2023 at 16:00, Wachman 617

  Kac's random walk and its variants: Mixing times and applications

  Natesh Pillai, Harvard University
  
  Determining the mixing time of Kac's random walk on the n-sphere was a long-standing open problem. In this talk, I will discuss my joint work with Aaron Smith on obtaining the optimal mixing time bounds for this walk and its variants. In addition to discussing the key coupling construction underlying our proof, I will discuss its connections and applications to random matrix theory, dimension reduction methods, and other statistical applications. In particular, we will exhibit a Johnson-Lindenstrauss (JL) transform using Kac's walk that is memory-optimal and outperforms existing algorithms in certain regimes, confirming a conjecture of Ailon and Chazelle.
  
  This is joint work with Aaron Smith, Vishesh Jain, Ashwin Sah and Mehtaab Sawhney.

• Monday September 11, 2023 at 16:00, Wachman 617

  Liquid Crystal Variational Problems

  Ricardo Nochetto, University of Maryland
  
  We discuss modeling, numerical analysis and computation of liquid crystal networks (LCNs). These materials couple a nematic liquid crystal with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCNs are natural candidates for soft robotics applications. We start from the classical 3D trace energy formula and derive a reduced 2D membrane energy as the formal asymptotic limit of vanishing thickness and characterize the zero energy deformations. We design a sound numerical method and prove its Gamma convergence despite the strong nonlinearity and lack of convexity properties of the membrane energy. This work is joint with L. Bouck and S. Yang.

• Monday September 18, 2023 at 16:00, Wachman 617

  Special event: Using Inclusive Language to Create a Sense of Belonging

   

  Valerie C. Dudley

  Director of Multicultural Education and Training

  Institutional Diversity, Equity, Advocacy, and Leadership, Temple University

  Inclusive language puts our humanity at the center; it allows everyone to feel recognized, valued, invited, and motivated to contribute at their highest level. To become an anti-racist institution, we must take concrete action to change our culture and the experiences of members of our community. In this workshop, we will consider the impact of using respectful, identity-affirming language in creating an antiracist welcoming environment that embraces diversity as a whole.

   

• Monday October 9, 2023 at 16:00, Wachman 617

  Space vectors forming rational angles

  Kiran Kedlaya, UCSD/IAS
  
  We classify all possible configurations of vectors in three-dimensional space with the property that any two of the vectors form an angle whose measure is a rational multiple of pi. As a corollary, we find all tetrahedra whose six dihedral angles are all rational multiples of pi. While these questions (and their answers) are of an elementary nature, their resolution will take us on a tour through cyclotomic number fields, computational algebraic geometry, and an amazing fact about the geometry of tetrahedra discovered by two physicists in the 1960s. Joint work with Sasha Kolpakov, Bjorn Poonen, and Michael Rubinstein. 
   

• Monday October 23, 2023 at 16:00, Wachman 617

  Towards homogenization of mechanism-based mechanical metamaterials

  Ian Tobasco, University of Illinois Chicago
  
  Mechanical metamaterials are many-body elastic systems that deform in unusual ways, due to the interactions of essentially rigid building blocks. Examples include origami patterns with many folds, or kirigami patterns made by cutting material from a thin elastic sheet. In either case, the local deformations of the pattern involve internal degrees of freedom which must be matched with the usual global Euclidean invariances--- e.g., groups of four origami panels move by coordinated rotations and translations, but it is still possible to bend the whole origami pattern into an overall curved shape. This talk will introduce the homogenization problem for kirigami and origami metamaterials to a broad audience and describe our recent results. Our goal is to explain the link between the design of the individual cuts/folds and the bulk deformations and geometries they can produce. This is joint work with Paul Plucinsky (U. Southern California, Aerospace and Mechanical Engineering) and Paolo Celli (Stony Brook U., Civil Engineering).
   

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday January 24, 2022 at 16:00, Wachman 617

  Morrey's problem and quasiconformal mappings

  Andre Guerra, Institute for Advanced Study
  
  Quasiconvexity is a fundamental notion in the vectorial Calculus of Variations and is essentially equivalent to the applicability of the Direct Method. A fundamental problem, considered by Morrey in the 50s and 60s, is whether quasiconvexity is equivalent to ellipticity (in the sense of Legendre-Hadamard). In 1992 Vladimir Sverak showed that in 3 or higher dimensions they are not equivalent, but the two-dimensional case remains open. In this case one can expect a "complex analysis miracle", and we will discuss deep connections of Morrey's problem to old questions in Quasiconformal Analysis.

• Monday September 19, 2022 at 16:00, Wachman 617

  Integral points on elliptic curves

  Wei Ho, IAS, Princeton, Michigan
   
  Elliptic curves are fundamental and well-studied objects in arithmetic geometry. However, much is still not known about many basic properties, such as the number of rational points on a "random" elliptic curve. We will discuss some conjectures and theorems about this "arithmetic statistics" problem, and then show how they can be applied to answer a related question about the number of integral points on elliptic curves over Q. In particular, we show that the second moment (and the average) for the number of integral points on elliptic curves over any number field is bounded (joint work with Levent Alpöge).

• Monday September 26, 2022 at 16:00, Wachman 617

  Maps between configuration spaces and moduli spaces

  Lei Chen, University of Maryland
  
  In this talk, I will survey what is known or conjectured about maps between configuration spaces and moduli spaces of surfaces. Among them, we consider two categories: continuous maps and also holomorphic maps. We will also talk about some results and conjectures about maps between finite covers of those spaces. Unlike the case of the whole spaces, much less is known about their finite covers. 

• Monday October 17, 2022 at 16:00, Wachman 617

  Some congruences and consequences in number theory and beyond

  Ellen Eischen, University of Oregon
  
  In the middle of the nineteenth century, Kummer observed striking congruences between certain values of the Riemann zeta function, which have important consequences in number theory.  In spite of its potential, this topic lay mostly dormant for nearly a century until breakthroughs by Iwasawa in the middle of the twentieth century. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have revealed similarly consequential congruences in the context of other arithmetic data.  This remains an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude with some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art. 

• Monday October 24, 2022 at 16:00, Wachman 617

  Stochastic Nash evolution

  Govind Menon, Brown University
   
  We introduce a probabilistic formulation for the Nash embedding theorems. Our approach inverts the usual relation between mathematics and physics. We use rigorous mathematical results, including Nash’s work, results of De Lellis and Szekelyhidi, and work of the speaker, Rezakhanlou and co-workers, to guide the design of algorithms and evolution equations.
  
  We use relaxation as in Nash’s work, but replace his iteration (in low codimension) or continuous flow (in high codimension) with a stochastic flow. The main issue in the derivation of our flow is a principled resolution of a semidefinite program. The same fundamental structure applies to several hard constraint systems and nonlinear PDE.

• Monday November 14, 2022 at 16:00, Wachman 617

  Formation and development of singularities for the compressible Euler equations

  Vlad Vicol, Courant Institute
   
  We consider the compressible Euler equations of fluid dynamics, in multiple space dimensions. In this talk, we discuss the formation and development of stable singularities, from smooth initial conditions. The questions we address are: given smooth initial conditions, precisely how does the first singularity arise? is the mechanism stable? how can one geometrically characterize the preshock (the boundary of the space-time set on which the solution remains smooth)? precisely how does the entropy-producing shock wave instantaneously develop from the preshock? does uniqueness hold once the shock has formed? do other singularities instantaneously arise after the preshock? In this level of detail, the problem was previously open even in one space dimension. We discuss a sequence of joint works with Steve Shkoller, Tristan Buckmaster, and Theodore Drivas, in which we have developed a multidimensional theory to answer the above questions. 
   

• Tuesday November 29, 2022 at 16:00, Wachman 617

  Classification of divergent trajectories

  Nattalie Tamam, University of Michigan
  
  The study of group actions gained significant interest in the past several decades, as group actions are a powerful tool when approaching problems from number theory and geometry. We will focus on the dynamical equivalent to vectors with 'infinitely good' diophantine approximation. From this dynamical point of view, Weiss conjectured a complete classification of the relevant trajectories. We will discuss the steps and different tools used in proving this conjecture. This is in part a joint work with Omri Solan, and in part a joint work with Lingmin Liao, Ronggang Shi and Omri Solan.

• Friday December 2, 2022 at 16:00, Wachman 617

  Wave propagation on rotating cosmic string spacetimes

  Katrina Morgan, Northwestern University
   
  Energy methods have historically been a useful tool for studying waves on different background geometries. Under the right conditions, solutions to the wave equation satisfy energy estimates which state that the energy of the solution $u$ at time $t$ is controlled by the energy of the initial data. However, such techniques are not always available, such as in the case of rotating cosmic string spacetimes. These geometries are solutions to the Einstein field equations which exhibit a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. They have a notable unusual feature: they admit closed timelike curves near the so-called ``string" and are thus not globally hyperbolic. In joint work with Jared Wunsch, we show that \textit{forward in time} solutions to the wave equation (in an appropriate microlocal sense) do exist on rotating cosmic string spacetimes, despite the causality issues present in the geometry. Our techniques involve proving a statement on propagation of singularities which provides a microlocal version of an energy estimate that allows us to establish existence of solutions.

• Tuesday December 6, 2022 at 16:00, Wachman 617

  A PDE perspective on the hydrodynamics of flexible filaments

  Laurel Ohm, Princeton University
  
  Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics.
   
  First, we propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing.
  
  Second, we consider a classical elastohydrodynamic model for the motion of an immersed inextensible filament. We highlight how the analysis can help to better understand undulatory swimming at low Reynolds number. This includes the development of a novel numerical method to simulate inextensible swimmers.

• Thursday December 8, 2022 at 16:00, Wachman 617

  Designing low rank methods via rational functions 

  Heather Wilber, UT Austin
  
  Low rank approximation methods are a central pillar of modern scientific computing. They are the powerhouse behind many fast and superfast methods relied upon on for computing solutions to various partial differential equations, linear systems, and matrix equations.  In this talk, we focus on the role that rational approximation methods can play in the design of such algorithms. We illustrate how rational approximation tools can help us design highly effective low rank methods in the context of two very different (but surprisingly related!) kinds of problems: (1) the development of a direct solver for linear systems involving non-uniform discrete Fourier transform matrices, and (2) the development of solvers for the spectral fractional Poisson equation on geometrically complicated domains.  

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday October 11, 2021 at 16:00, Wachman 617

  Computational methods for high-dimensional stochastic optimal control

  Mete Soner, Princeton University
  
  Stochastic optimal control has been an effective tool for many problems in a wide range of fields including social sciences.  Although it provides the much needed quantitative modeling for such problems, until recently it has been intractable in high-dimensional settings.  However, several recent studies report impressive numerical results.  All these papers use a Monte-Carlo type algorithm combined with deep neural networks proposed by Han, E and Jentzen.  In this talk I will outline this approach.  Numerical results, while validating the power of the method in high dimensions, show the dependence on the dimension and the size of the training data.  Also, studies for the optimal stopping problem illustrate the potential difficulties. This is joint work with Max Reppen of Boston University and Valentin Tissot-Daguette from Princeton.

• Monday October 18, 2021 at 16:00, Wachman 617

  Flexibility in complex and symplectic geometry

  Emmy Murphy, Princeton University
  
  Abstract: We survey a number of recent results in symplectic geometry, related to h-principle type techniques. For one example, a Lagrangian submanifold of \(C^n\) is by definition a smooth real n-manifold embedded in \(C^n\) so that \(iTL\) is the orthogonal complement of \(TL\) at every point. Which diffeomorphism types admit Lagrangian embeddings into \(C^n\)? The answer is surprisingly nuanced, and gives a sort of "rigid-flexible dichotomy". We'll discuss related phenomena for Stein manifolds and some other geometric situations. No background in geometry will be assumed for the talk.
  
   

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday February 3, 2020 at 16:00, Wachman 617

  A tale of three curves

  Jennifer Balakrishnan, Boston University
  
  Let C be a smooth projective curve of genus at least 2 defined over the rational numbers. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points.
  
  In the case when the Jacobian of C has rank less than its genus, the Chabauty--Coleman method can often be used to find the rational points of C, using the construction of p-adic line integrals.  In certain cases of higher rank, p-adic heights can often be used to find rational or integral points on C. I will describe these "quadratic Chabauty" techniques (part of Kim's nonabelian Chabauty program) and will highlight some recent examples where the techniques have been used: this includes a 1700-year old problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13.  This talk is based on joint work with Amnon Besser, Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.  

• Monday February 24, 2020 at 16:00, Wachman 617

  3-Manifold groups, coherence, and fibering 

  Genevieve Walsh, Tufts University
  
  When is a group the fundamental group of a 3-manifold?  What properties of 3-manifold groups can we extract to better understand other groups?  A group is coherent if every finitely generated subgroup is finitely presented, and incoherent otherwise.  A group algebraically fibers if it admits a map to the integers with finitely generated kernel.  The fundamental groups of closed hyperbolic 3-manifolds are coherent, and they virtually algebraically fiber.  We will discuss the geometry and importance of these notions, and develop techniques to find witnesses to incoherence and algebraic fibers.  We apply these techniques to large classes of groups, including many free by free, surface by surface and surface by free groups.  Any new work mentioned is joint work with Rob Kropholler.

• Monday March 9, 2020 at 16:00, Wachman 617

  (POSTPONED) 

  Jean-Christophe Mourrat, Courant Institute, NYU
  
   

• Monday March 23, 2020 at 16:00, Wachman 617

  (POSTPONED)

  Kavita Ramanan, Brown University
  
   

• Monday March 30, 2020 at 16:00, Wachman 617

  (POSTPONED)

  Lai-Sang Young, Courant Institute, NYU
  
   

• Monday April 20, 2020 at 16:00, Wachman 617

  (POSTPONED)

  Marc Culler, University of Illinois at Chicago
  
   

• Monday April 27, 2020 at 16:00, Wachman 617

  (POSTPONED)

  Vlad Vicol, Courant Institute, NYU
  
   

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday January 29, 2018 at 16:00, Wachman 617

  Gravitational lensing and dark matter. What do we know?

  Jose Maria Diego Rodriguez, Instituto de Fisica de Cantabria

  Dark matter is arguably one of the main mysteries in modern physics. We know how much is there, we know where it is but we don't know what it is. Despite the numerous (and expensive) efforts on Earth to directly detect the alleged and elusive dark matter particle, experimental evidence remains as elusive as the dark matter particle itself. As of today, the strongest (and only) experimental evidence for dark matter still comes from astrophysical probes. One of such probes is gravitational lensing that can be used to map the distribution of dark matter on cosmological scales. I will briefly review the most popular candidates for dark matter and focus on our research that uses gravitational lensing to rule out some of these candidates.

   

• Monday February 5, 2018 at 16:00, Wachman 617

  Computing Belyi maps

  John Voight, Dartmouth College

  A Belyi map is a finite, branched cover of the complex projective line that is unramified away from 0, 1, and infinity. Belyi maps arise in many areas of mathematics, and their applications are just as numerous. They gained prominence in Grothendieck's program of dessins d'enfants, a topological/combinatorial way to study the absolute Galois group of the rational numbers.

  In this talk, we survey computational methods for Belyi maps, and we exhibit a uniform, numerical method that works explicitly with power series expansions of modular forms on finite index subgroups of Fuchsian triangle groups. This is joint work with Jeroen Sijsling and with Michael Klug, Michael Musty, and Sam Schiavone.

• Monday February 12, 2018 at 16:00, Wachman 617

  Affine Curvature in Harmonic Analysis

  Phil Gressman, University of Pennsylvania

  In the 1970s, E. Stein and other mathematicians studying fundamental questions related to pointwise convergence of Fourier series discovered surprising new links between this very old problem and the geometry of submanifolds of Euclidean space. These discoveries paved the way for many of the questions at the forefront of modern harmonic analysis. A common element in many of these areas is the role of a strange sort of curvature condition which arises naturally from Fourier-theoretic roots but is poorly understood outside the extreme cases of curves and hypersurfaces. In this talk, I will discuss recent work which combines elements of Geometric Invariant Theory, Convex Geometry, Signal Processing, and other areas to shed light on this problem in intermediate dimensions.

• Monday March 12, 2018 at 16:00, Wachman 617

  On non-physical free boundaries

  Eduardo Teixeira, University of Central Florida

  The development of modern free boundary theory has promoted major knowledge leverage across pure and applied disciplines and in this talk I will provide a panoramic overview of such endeavor. The goal of lecture, however, will be to explicate how geometric insights and powerful analytic tools pertaining to free boundary theory can be imported to investigate regularity issues in nonlinear diffusive partial differential equations. This new systematic approach has been termed non-physical free boundaries, and in the past few years has led us to a plethora of unanticipated results.

• Monday March 19, 2018 at 16:00, Wachman 617

  The relation (X+Y)^n = X^n + Y^n, and miraculous cancellations in quantum SL_2

  Francis Bonahon, USC

  The convenient formula (X+Y)^n = X^n + Y^n is (unfortunately) frequently used by our calculus students. Our more advanced students know that this relation does hold in some special cases, for instance in prime characteristic n or when YX=qXY with q a primitive n-root of unity. I will discuss similar ``miraculous cancellations’’ for 2-by-2 matrices, in the context of the quantum group U_q(sl_2).

• Monday April 2, 2018 at 16:00, Wachman 617

  Boundary obstacle problems in the applied sciences

  Donatella Danielli, Purdue University

  Obstacle problems play an ubiquitous role in the applied sciences, with applications ranging from linear elasticity to fluid dynamics, from temperature control to financial mathematics. In this talk we will show how seemingly different phenomena can be expressed in terms of the same mathematical model of obstacle type. We will also discuss some recent results concerning the regularity of the solution and of its free boundary. In particular, we will highlight the pervasive role played by some families of monotonicity formulas.

• Wednesday April 25, 2018 at 16:00, Wachman 617

  Exploiting Multiprecision Arithmetic

  
  Nick Higham, University of Manchester, UK
  
  
  There is a growing availability of multiprecision arithmetic: floating
  point arithmetic in multiple, possibly arbitrary, precisions.
  Demand in applications includes for both low precision (deep learning and
  climate modelling) and high precision (long-term simulations and solving
  very ill conditioned problems).  We discuss
  
  - Half-precision arithmetic: its characteristics, availability, attractions,
    pitfalls, and rounding error analysis implications.
  
  - Quadruple precision arithmetic: the need for it in applications, its
    cost, and how to exploit it.
  
  As an example of the use of multiple precisions we discuss iterative
  refinement for solving linear systems.  We explain the benefits of
  combining three different precisions of arithmetic (say, half, single, and
  double) and show how a new form of preconditioned iterative refinement can
  be used to solve very ill conditioned sparse linear systems to high
  accuracy.

• Monday September 10, 2018 at 16:00, Wachman 617

  Shape optimization for an eigenvalue problem on manifolds with boundary

  Ailana Fraser

  University of British Columbia and IAS

  When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and contrast this with some recent results in higher dimensions. This is joint work with R. Schoen.

• Monday September 24, 2018 at 16:00, Wachman 617

  Maple day

  Maple Day at Temple University

   

  Join us on Monday, September 24th at 4:00pm at Temple University for FREE Maple training from a Maple Product Manager! Learn some of the fundamental concepts for using Maple and also about the latest release of Maple 2018. We’ve made Maple more intuitive, while implementing an extensive collection of improvements to core functionality -- you’ll benefit from this release no matter how you use Maple. We want this to be interactive, so we are happy to tackle any issues, questions or even suggestions you have!

• Monday October 22, 2018 at 16:00, Wachman 617

  Inscribing rectangles in Jordan curves

   

  Richard Schwartz, Brown University

  The notorious Square Peg Problem asks if every Jordan curve has an inscribed square -- namely 4 points on the curve which are the vertices of a square. I'll demonstrate a computer program I wrote which investigates the Square Peg problem for polygonal Jordan curves. One thing I discovered using the program is the result that all but at most 4 points of any Jordan curve (polygonal or not) are vertices of inscribed rectangles. I'll illustrate this result (and some others) using the program and sketch proofs.

• Monday October 29, 2018 at 16:00, Wachman 617

  Counting holomorphic disks by means of bounding chains

  Sara Tukachinsky

  Institute for Advanced Study

  In 2003, Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which are related to counts of pseudo-holomorphic disks with boundary and interior point constraints (Solomon, 2006). The problem of extending the definition to higher dimensions remained open until recently (Georgieva, 2013, and Solomon-Tukachinsky, 2016-17). In the talk, I will give some background on the problem, and describe a generalization of Welschinger's invariants to higher dimensions called open Gromov-Witten invariants. Our generalization is constructed in the language of A-infinity algebras and bounding chains, where bounding chains play the role of boundary point constraints. The invariants satisfy a version of the open WDVV equations. In the example of $\mathbb{C}P^n$ with odd $n$, these equations give rise to recursive formulae that allow the computation of all invariants. This is joint work with Jake Solomon. No previous knowledge of any of the objects mentioned above will be assumed.

   

• Monday November 5, 2018 at 16:00, Wachman 617

  Multiple zeta values: crossroads between geometry and number theory

  Leila Schneps

  Institut de Mathématiques de Jussieu

  Grothendieck-Teichmüller theory was originated by Alexander Grothendieck as a way to study the absolute Galois group of the rationals by considering its action on fundamental groups of varieties, in particular of moduli spaces of curves with marked points: the special properties of the Galois action with respect to inertia generators and the fact of respecting the relations in the fundamental group gave rise to the definition of the group GT which contains G_Q.

  The group GT is profinite, but its defining relations can also be used to give a pro-unipotent avatar, and an associated graded Lie algebra grt. The study of the Lie algebra grt reveals many unexpected relations with number theory that are completely invisible in the profinite situation. We will show how Bernoulli numbers, cusp forms on SL_2(Z) and multiple zeta values arise in the Lie algebra context.