• Wednesday January 31, 2024 at 16:00, 617 Wachman Hall

  Low rank time integrators for solving time-dependent PDEs

  Jingmei Qiu, University of Delaware
  
  I will provide overview of low rank time integrators for time dependent PDEs.These include an explicit scheme that involves a time stepping followed by a SVD truncation procedure with application to the Vlasov equations; two implicit schemes: Reduced Augmentation Implicit Low rank (RAIL) scheme and a Krylov subspace low rank scheme with applications to the heat equation and the Fokker-Planck equation; as well as implicit-explicit low rank integrators for advection-diffusion equations.

• Wednesday February 7, 2024 at 16:00, 617 Wachman Hall

  Minimal Quantization Model for an Active System

  Rodolfo Ruben Rosales, Massachusetts Institute of Technology
  
  A small liquid drop placed above the vibrating surface of a liquid, will not (under appropriate conditions) fall and merge. In fact it will bounce from the surface, and can be made to do so for very many bounces (hundreds of thousands). If the liquid below is just under the Faraday threshold, the drop excites waves with each bounce, and via these waves it can extract momentum from the fluid underneath it, and starts moving "walking" at some preferred speed. The drop-wave system then becomes a peculiar active system, where the active elements interact with each other (if there are many drops) via waves, as well as with their own past via the waves generated earlier in their history. This system have many special properties, some reminiscent of quantum mechanics. In this talk I will focus on one such property:
  If the drop is constrained to move in a bounded region by some external force (e.g.: Coriolis), then its path exhibits radial quantization: the statistics for the radius of curvature along the drop path is concentrated on a discrete set of values. The question is why? There are various models that predict this, but the question is not about the model(s), but about what is the mechanism behind the behavior. An obvious answer is that it is because the drop motion is caused via waves. This is, basically, correct; but too vague, even misleading. First of all, the drop does not move on some "external" wave field, but on a self-generated one. Second, the waves decay, hence the wave field is dominated by the waves produced in the recent path. Yet, if one discards all but the recent past, the quantization disappears --- the recent past selects the preferred speed, but does not quantize. It turns out that the effect is caused by (exponentially suppressed) waves emitted in the past at "special" regions where constructive interference magnifies their effect. As I hope to show, this gives a simple and intuitive explanation of how the radii selection occurs.

• Wednesday February 14, 2024 at 16:00, 617 Wachman Hall

  Combining metabolic models, large data sets, and deep learning to improve systems biology simulators

  Sean McQuade, Rutgers University Camden
  
  Chemical networks, such as metabolism, can be simulated to assist in an array of research including new drug discovery, personalized medicine, and testing high-risk treatment before applying it to humans. Improved biochemical simulations can also reduce our dependence on animal testing before clinical trials. This talk demonstrates a mathematical framework for biochemical systems that was designed with two goals in mind: 1. improved early phase drug discovery and 2. personalized medicine. The talk also addresses a particular contribution that can be made by deep learning models.

• Wednesday February 28, 2024 at 16:00, 617 Wachman Hall

  Dynamic Boundary Conditions and Motion of Grain Boundaries 

  Chun Liu, Illinois Institute of Technology
  
  I will present the dynamic boundary conditions in the general
  energetic variational approaches. The focus is on the coupling between
  the bulk effects with the active boundary conditions.
  In particular, we will study applications in the evolution of grain
  boundary networks, in particular, the drag of trip junctions. This is a
  joint work with Yekaterina Epshteyn (University of Utah) and Masashi
  Mizuno (Nihon University).

• Monday March 11, 2024 at 16:00,

  Randomized Numerical Linear Algebra

  Erik Boman, Sandia National Laboratory
  
  Randomization has become a popular technique in numerical linear algebra in recent years, with applications in several areas from scientific computing to machine learning. We review some problems where it works well. Sketching is a powerful way to reduce a high-dimensional problem to a lower-dimensional problem. Sketch-and-solve and sketch-and-precondition are the two main approaches for linear systems and least squares problems. Finally, we describe two recent applications in more detail: Fast and stable orthogonalization (QR on tall, skinny matrices), and spectral graph partitioning.

• Wednesday March 27, 2024 at 16:00, 617 Wachman Hall

  Numerical Solution of Double Saddle-Point Systems

  Chen Greif, University of British Columbia
  
  Double saddle-point systems are drawing increasing attention in the past few years, due to the importance of multiphysics and other relevant applications and the challenge in developing efficient iterative numerical solvers. In this talk we describe some of the numerical properties of the matrices arising from these problems. We derive eigenvalue bounds and analyze the spectrum of preconditioned matrices, and it is shown that if Schur complements are effectively approximated, the eigenvalue structure gives rise to rapid convergence of Krylov subspace solvers. A few numerical experiments illustrate our findings.

• Wednesday April 3, 2024 at 16:00, 617 Wachman Hall

  Decentralized Stochastic Bilevel Optimization

  Hongchang Gao, Temple University
  
  Stochastic Bilevel Optimization (SBO) has widespread applications in machine learning, such as meta learning, hyperparameter optimization, and network architecture search. To train those machine learning models on large-scale distributed data, it is necessary to develop distributed SBO algorithms. Therefore, Decentralized Stochastic Bilevel Optimization (DSBO) has been actively studied in recent years due to the efficiency and robustness of decentralized communication. However, it is challenging to estimate the stochastic hypergradient on each worker due to the loss function's bilevelstructure and decentralized communication.
  In this talk, I will present our recent work on decentralized stochastic bilevel gradient descent algorithms. On the algorithmic design side, I will talk about how to estimate the hypergradient without incurring large communication overhead under both homogeneous and heterogeneous settings. On the theoretical analysis side, I will describe the convergence rate of our algorithms, showing how the communication topology, the number of workers, and heterogeneity affect the theoretical convergence rate. Finally, I will show the empirical performance of our algorithms.

• Wednesday April 10, 2024 at 16:00, 617 Wachman Hall

  New Perspectives on Multiscale Modeling, Simulation, and Analysis of Grain Growth in Polycrystalline Materials

  Yekaterina Epshteyn, University of Utah
  
  Many technologically useful materials are polycrystals composed of small monocrystalline grains that are separated by grain boundaries of crystallites with different lattice orientations. One of the central problems in materials science is to design technologies capable of producing an arrangement of grains that delivers a desired set of material properties.
  A method by which the grain structure can be engineered in polycrystalline materials is through grain growth (coarsening) of a starting structure. Grain growth in polycrystals is a very complex multiscale multiphysics process. It can be regarded as the anisotropic evolution of a large cellular network and can be described by a set of deterministic local evolution laws for the growth of individual grains combined with stochastic models for the interaction between them. In this talk, we will present new perspectives on mathematical modeling, numerical simulation, and analysis of the evolution of the grain boundary network in polycrystalline materials. Relevant recent experiments will be discussed as well.

• Wednesday April 17, 2024 at 16:00, 617 Wachman Hall

  A Mixed Sparse-Dense BLR Solver for Electromagnetics

  Francois-Henry Rouet, Ansys
  
  Element-by-element preconditioners were an active area of research in the 80s and 90s, and they found some success for problems arising from Finite Element discretizations, in particular in structural mechanics and fluid dynamics (e.g., the "EBE" preconditioner of Hughes, Levit, and Winget). Here we consider problems arising from Boundary Element Methods, in particular the discretization of Maxwell's equations in electromagnetism. The matrix comes from a collection of elemental matrices defined over all pairs of elements in the problem and is therefore dense. Inspired by the EBE idea, we select subsets of elemental matrices to define different sparse preconditioners that we can factor with a direct method. Furthermore, the input matrix is rank-structured ("data sparse") and is compressed to accelerate the matrix-vector products. We use the Block Low-Rank approach (BLR). In the BLR approach, a given dense matrix (or submatrix, in the sparse case) is partitioned into blocks following a simple, flat tiling; off-diagonal blocks are compressed into low-rank form using a rank-revealing factorization, which reduces storage and the cost of operating with the matrix. We demonstrate results for industrial problems coming from the LS-DYNA multiphysics software.
  Joint work with Cleve Ashcraft and Pierre L'Eplattenier

• Wednesday September 4, 2024 at 16:00, 617 Wachman Hall

  Mathematical and computational epidemiology of antimalarial drug resistance evolution

  Maciej Boni, Temple University

• Wednesday September 11, 2024 at 16:00, 617 Wachman Hall

  Trustworthy Machine Learning for Biomedicine

  Xinghua (Mindy) Shi, Temple University
  
  Recent biomedical data deluge has fundamentally transformed biomedical research into a data science frontier. The unprecedented accumulation of biomedical data presents a unique yet challenging opportunity to develop novel methods leveraging artificial intelligence and machine learning to further our understanding of biology and advance medicine. In this talk, I will first introduce the cutting-edge research in characterizing human genetic variation and their associations with disease. I will then present statistical and machine learning methods for robust modeling of medical data. Finally, I will overview recent development in trustworthy machine learning to combat model overfitting, privacy and biases.

• Wednesday September 18, 2024 at 16:00, 617 Wachman Hall

  On the lack of external response of a nonlinear medium in the second-harmonic generation process.

  Narek Hovsepyan, Rutgers University 
  
  Second Harmonic Generation (SHG) is a process inwhich the input wave (e.g. laser beam) interacts with a nonlinearmedium and generates a new wave, called the second harmonic, atdouble the frequency of the original input wave. Weinvestigate whether there are situations in which the generatedsecond harmonic wave does not scatter and is localizedinside the medium, i.e., the nonlinearinteraction of the medium with the probing wave isinvisible to an outside observer. This leadsto the analysis of a semilinear elliptic system formulatedinside the medium with non-standard boundary conditions. Moregenerally, we set up a mathematical framework needed to investigate amultitude of questions related to the nonlinear scatteringproblem associated with SHG (or other similar multi-frequency opticalphenomena). This is based on a joint work with Fioralba Cakoni, MattiLassas and Michael Vogelius. 

• Wednesday September 25, 2024 at 16:00, 617 Wachman Hall

  Mode switching in organisms for solving explore-versus-exploit problems

  Kathleen Hoffman, University of Maryland Baltimore County
  
  Fish use active sensing to constantly re-evaluate their position in space. The weakly electric glass knifefish, Eigenmannia virescens, incorporates an electric field as one of its active sensing mechanisms. The motion of the knifefish in a stationary refuge is captured using high-resolution motion tracking and illustrates many small amplitude oscillations inside the refuge coupled with high amplitude “jumps”. We show that this active sensing mechanism is not reflected by a Gaussian distribution of the velocities. Instead, we show that the velocities are more accurately reflected by a mixture of Gaussians because of the number of high amplitude jumps in the tails of the velocity distribution. The experimental position measurements were taken in both the light and the dark showing more frequent bursts of faster movement in the dark, where presumably the fish are relying more on their electric sensor than their vision. Computational models of active state estimation with noise injected into the system based on threshold triggers exhibit velocity distributions that resemble those of the experimental data, more so than with pure noise or zero noise inputs. Similar distributions have been observed in a variety of different senses and species. 
  This is joint work with Debojyoti Biswas (JHU), Noah Cowan (JHU), John Guckenheimer (Cornell), Andrew Lamperski (UMN), Yu Yang (JHU)

• Wednesday October 2, 2024 at 16:00, 617 Wachman Hall

  The Average Rate of Convergence of the Exact Line Search Gradient Descent Method, with applications to polynomial optimization problems in data sciences 

  Thomas P.Y. Yu, Drexel University.
   
   

  It is very well known that when the exact line search gradient descent method is applied to a convex quadratic objective, the worst-case rate of convergence (ROC), among all seed vectors, deteriorates as the condition number of the Hessian of the objective grows. By an elegant analysis due to H. Akaike, it is generally believed -- but not proved -- that in the ill-conditioned regime the ROC for almost all initial vectors, and hence also the average ROC, is close to the worst-case ROC. We complete Akaike's analysis using the theorem of center and stable manifolds. Our analysis also makes apparent the effect of an intermediate eigenvalue in the Hessian by establishing the following somewhat amusing result: In the absence of an intermediate eigenvalue, the average ROC gets arbitrarily fast -- not slow -- as the Hessian gets increasingly ill-conditioned.

  This work is motivated by contemporary applications in data sciences. We shall discuss some of the surprising properties of the polynomial optimization problems involved in these applications.

   

• Wednesday October 16, 2024 at 16:00, 617 Wachman Hall

  Seminar postponed

   

• Wednesday October 23, 2024 at 16:00, 617 Wachman Hall

  Modified Patankar-Runge-Kutta Methods: Introduction, Analysis and Numerical Applications

  Andreas Meister, University of Kassel
  
  Mathematical modeling leads to so-called convection-diffusion-reaction equations in the form of systems of partial differential equations in numerous practical applications. Examples are turbulent air flows or algae growth in oceans orlakes. After discretization of the spatial derivatives, an extremely large systemof ordinary differential equations occurs. A reasonable numerical time integration scheme must reflect present properties like the positivity of single balancequantities or also the conservativity of the initial model.In the talk we will present so-called modified Patankar-Runge-Kutta (MPRK)schemes. They adapt explicit Runge-Kutta schemes in a way to ensure positivity and conservativity irrespective of the time step size. Thereby, we introducea general definition of MPRK schemes and present a thorough investigationof necessary as well as sufficient conditions to derive first, second and thirdorder accurate MPRK schemes. The theoretical results will be confirmed bynumerical experiments in which MPRK schemes are applied to solve non-stiffand stiff systems of ordinary differential equations. Furthermore, we investigatethe efficiency of MPRK schemes in the context of convection-diffusion-reactionequations with source terms of production-destruction type.

• Wednesday November 20, 2024 at 16:00, 617 Wachman Hall

  Smart Data, Smarter Models: Enhancing the Predictive Power of Mathematical Models of Cancer

  Jana Gevertz, The College of New Jersey
  
  Mathematical models are powerful tools that can vastly improve our understanding of cancer dynamics and treatment response. However, to be useful, experimental or clinical data are necessary to both train and validate such predictive models, and not all data are created equal. Here I present two methodologies that improve upon model-informed experimental design and model-based predictions. First, I will introduce a multi-objective optimization algorithm to identify combination protocols that maximize synergy from the perspective of both efficacy and potency (toxicity), while simultaneously reconciling sometimes contradictory assessments made by different synergy metrics. Second, using the notion of parameter identifiability, I will address the question of what is the minimal amount of experimental data that needs to be collected, and when it should be collected, to have confidence in a model's predictions. Real-world applications of both methodologies will be presented.

Current contacts: Vasily Dolgushev, Ed Letzter, Martin Lorenz or Chelsea Walton

The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.

• Wednesday September 5, 2001 at 11:30, Wachman 617

  Seymour Lipschutz, Temple University, "Groups and semisymmetric graphs"

• Friday September 14, 2001 at 11:30, Wachman 617

  Dennis Spellman, Temple University "On discriminating groups" (Part 1)

• Friday September 21, 2001 at 11:30, Wachman 617

  Dennis Spellman, Temple University, "On discriminating groups" (Part 2)

• Friday September 28, 2001 at 11:30, Wachman 617

  Martin Lorenz, Temple University, "On Euler classes of crystallographic groups" (Part 1)

• Friday October 5, 2001 at 11:30, Wachman 617

  talk cancelled

• Friday October 12, 2001 at 11:30, Wachman 617

  Martin Lorenz, Temple University, "On Euler classes of crystallographic groups" (Part 2)

• Friday October 19, 2001 at 11:30, Wachman 617

  Leonard Greendlinger, Project SEED and Temple U, Martin Greendlinger, AIPH and Temple U, "Tilings and group diagrams" (Part 1)

• Friday October 26, 2001 at 11:30, Wachman 617

  Leonard Greendlinger, Project SEED and Temple U, Martin Greendlinger, AIPH and Temple U, "Tilings and group diagram" (Part 2)

• Friday November 9, 2001 at 11:30, Wachman 617

  Edward Letzter, Temple University, "Effective representation theory" (Part 1)

• Friday November 16, 2001 at 11:30, Wachman 617

  Edward Letzter, Temple University, "Effective representation theory" (Part 2)

• Tuesday November 27, 2001 at 13:30, Wachman 617

  Special Seminar: Daniel Sage, Louisiana State U, "Affine flag manifolds and representation theory"

• Friday November 30, 2001 at 11:30, Wachman 617

  Seymour Lipschutz, Temple University, "Graphs and groups"

Current contacts: Jaclyn Lang, Catherine Hsu, Ian Whitehead, and Djordje Milicevic

The Philadelphia Area Number Theory Seminar rotates between Bryn Mawr, Swarthmore, and Temple. In Fall 2022, we meet on Tuesday afternoons, usually with tea at 3pm and then the talk 3:30-5pm. Please pay careful attention to the times and locations of the talks as they change from week to week! If you would like to be added to our mailing list or if you are interested in being a speaker, please contact one of the organizers. In future semesters, we anticipate that the seminar will be on Wednesday afternoons.

Click on title for abstract.

• Wednesday January 25, 2023 at 14:30, Temple University, Wachman 414

  Serre curves relative to obstructions modulo 2

  Rakvi, University of Pennsylvania 
  
  Let 𝐸 be an elliptic curve defined over Q. Fix an algebraic closure Q of Q. We get a Galois representation 𝜌𝐸 : Gal(Q/Q) → GL2(Z) associated to 𝐸 by choosing a compatible bases for the 𝑁 -torsion subgroups of 𝐸 (Q). In this talk, I will discuss my recent work joint with Jacob Mayle where we consider elliptic curves 𝐸 defined over Q for which the image of the adelic Galois representation 𝜌𝐸 is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their l-adic images, compute all examples of conductor at most 500,000, precisely describe the image of 𝜌𝐸 , and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.

• Wednesday February 1, 2023 at 15:30, Bryn Mawr College, Park Science Center 328

  Relative Trace Formula and L-functions for GL(n+1) x GL(n)

  Liyang Yang, Princeton University
  
  We will introduce a relative trace formula on GL(n + 1) weighted by cusp forms on GL(n) over number fields. The spectral side is an average of Rankin– Selberg L-functions for GL(n + 1) × GL(n) over the full spectrum, and the geometric side consists of Rankin–Selberg L-functions for GL(n) × GL(n), and certain explicit meromorphic functions. The formula yields new results towards central L-values for GL(n + 1) × GL(n): the second moment evaluation, and simultaneous nonvnanish- ing in the level aspect. Further applications to the subconvexity problem will be discussed if time permits. 

• Wednesday February 15, 2023 at 15:30, Bryn Mawr College, Park Science Center 328

  Consecutive tuples of multiplicatively dependent integers

  Ingrid Vukusic, University of Salzburg
  
   

  An n-tuple of integers (a1 , . . . , an ) is called multiplicatively dependent, if it allows you to win the “Cancelling Game”, i.e. if there exist integers k1, . . . , kn ∈ Z, not all zero, such that

  a^{k1} ···a^{kn} =1. 

  After an unconventional introduction, we will ask many questions related to con- secutive tuples of multiplicatively dependent integers, and answer some of them. For example, do there exist integers 1 < a < b such that (a,b) and (a+1,b+1) are both multiplicatively dependent? It turns out that this question is easily answered, and after briefly discussing some more general properties of pairs, we will move on to triples. The proof of the main result relies on lower bounds for linear forms in logarithms. This talk is based on joint work with Volker Ziegler, as well as some work in progress. 

   

• Wednesday March 1, 2023 at 15:30, Bryn Mawr College, Park Science Center 328

  Arithmetic Quantum Unique Ergodicity for 3-dimensional hyperbolic manifolds

  Zvi Shgem-Tov, IAS
  
  The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on a compact manifold of negative curvature become equidistributed as the eigenvalue tends to infinity. In the talk I will discuss a recent work on this problem for arithmetic quotients of the three dimensional hyperbolic space. I will present a rather detailed proof of our key result that these eigenfunctions cannot concentrate on certain proper submanifolds. Joint work with Lior Silberman. 

• Wednesday March 15, 2023 at 14:30, Temple University, Tuttleman 404

  Ribet’s Lemma, the Brumer-Stark Conjecture, and the Main Conjecture

  Samit Dasgupta, Duke University
  
  In 1976, Ken Ribet used modular techniques to prove an important relationship between class groups of cyclotomic fields and special values of the zeta function. Ribet’s method was generalized to prove the Iwasawa Main Conjecture for odd primes p by Mazur-Wiles over Q and by Wiles over arbitrary totally real fields.
  
  Central to Ribet’s technique is the construction of a nontrivial extension of one Galois character by another, given a Galois representation satisfying certain properties. Throughout the literature, when working integrally at p, one finds the assumption that the two characters are not congruent mod p. For instance, in Wiles’ proof of the Main Conjecture, it is assumed that p is odd precisely because the relevant characters might be congruent modulo 2, though they are necessarily distinct modulo any odd prime.
  
  In this talk I will present a proof of Ribet’s Lemma in the case that the characters are residually indistinguishable. As arithmetic applications, one obtains a proof of the Iwasawa Main Conjecture for totally real fields at p=2. Moreover, we complete the proof of the Brumer-Stark conjecture by handling the localization at p=2, building on joint work with Mahesh Kakde for odd p. Our results yield the full Equivariant Tamagawa Number conjecture for the minus part of the Tate motive associated to a CM abelian extension of a totally real field, which has many important corollaries.
  
  This is joint work with Mahesh Kakde, Jesse Silliman, and Jiuya Wang.

• Wednesday March 22, 2023 at 15:30, Bryn Mawr College, Park Science Center 328

  Fine scale properties of sequences modulo 1

  Christopher Lutsko, Rutgers University
  
  Given a sequence of numbers, a key question one can ask is how is this sequence distributed? In particular, does the sequence exhibit any pseudo-random properties? (i.e., properties shared by random sequences). For example one can ask if the sequence is uniformly distributed modulo 1 (macroscopic scale), or if the pair correlation or gap distribution is Poissonian (fine scale). In this talk I will introduce these concepts, and discuss a set of examples where this behavior is fully understood. The techniques used are common tools in analytic number theory, and the question relates to problems in quantum chaos, and relates to the study of the zeros of the Riemann zeta function (although I will refrain from presenting my proof of RH...). This is joint work with Athanasios Sourmelidis and Nichlas Technau. 

• Wednesday March 29, 2023 at 14:30, Temple University, Tuttleman 1A

  Massey products and elliptic curves

  Frauke Bleher, University of Iowa
  
  This is joint work with T. Chinburg and J. Gillibert. The application of Massey products to understand the Galois groups of extensions of number fields is a longstanding research topic. In 2014, Minac and Tan showed that triple Massey products vanish for the absolute Galois group of any field F. In 2019, Harpaz and Wittenberg showed that this remains true for all higher Massey products in the case when F is a number field. The first natural case to consider beyond fields is that of Massey products for curves over fields. I will discuss some known and new vanishing and non-vanishing results in this case. In particular, for elliptic curves I will provide a classification for the non-vanishing of triple Massey products under various natural assumptions. The main tool is the representation theory of etale fundamental groups into upper triangular unipotent matrix groups. I will begin with background about Massey products, which first arose in topology, and about the relevant representation theory, before discussing our results. 

• Wednesday April 5, 2023 at 15:30, Bryn Mawr College, Park Science Center 328

  Applications of the Endoscopic Classification to Statistics of Cohomological Automorphic Representations on Unitary Groups

  Rahul Dalal, Johns Hopkins University
  
  Starting from the example of classical modular modular forms, we mo- tivate and describe the problem of computing statistics of automorphic representa- tions. We then describe how techniques using or built off of the Arthur–Selberg trace formula help in studying it.
   

  Finally, we present recent work on one particular example: consider the family of automorphic representations on some unitary group with fixed (possibly non- tempered) cohomological representation π0 at infinity and level dividing some finite upper bound. We compute statistics of this family as the level restriction goes to in- finity. For unramified unitary groups and a large class of π0, we are able to compute the exact leading term for both counts of representations and averages of Satake parameters. We get bounds on our error term similar to previous work by Shin– Templier that studied the case of discrete series at infinity. We also discuss corollar- ies related to the Sarnak–Xue density conjecture, average Sato–Tate equidistribution in families, and growth of cohomology for towers of locally symmetric spaces. The specific new technique making this unitary example tractable is an extension of an inductive argument that was originally developed by Ta ̈ıbi to count unramified rep- resentations on Sp and SO and used the endoscopic classification of representations (which our case requires for non-quasisplit unitary groups).

  This is joint work with Mathilde Gerbelli-Gauthier. 

   

• Wednesday April 12, 2023 at 14:30, Temple University, Tuttleman 1A

  The nonvanishing of Selmer groups for certain symplectic Galois representations

  Sam Mundy, Princeton University
  
  
  Given an automorphic representation π of SO(n,n+1) with certain nice properties at infinity, one can nowadays attach to π a p-adic Galois representation R of dimension 2n. The Bloch--Kato conjectures then predict in particular that if the L-function of R vanishes at its central value, then the Selmer group attached to a particular twist of R is nontrivial.
  
  
  I will explain work in progress proving the nonvanishing of these Selmer groups for such representations R, assuming the L-function of R vanishes to odd order at its central value. The proof constructs a nontrivial Selmer class using p-adic deformations of Eisenstein series attached to π, and I will highlight the key new input coming from local representation theory which allows us to check the Selmer conditions for this class at primes for which π is ramified. 

• Wednesday April 26, 2023 at 14:30, Temple University, Tuttleman 404

  Intersection of components for Emerton-Gee stack for GL2

  Kalyani Kansal, Johns Hopkins University
  
  The Emerton-Gee stack for GL2 is a stack of (phi, Gamma)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and it can be viewed as a moduli stack of mod p representations of a p-adic Galois group. We compute criteria for codimension one intersections of the irreducible components of X. We interpret these criteria in terms motivated by conjectural categorical p-adic and mod p Langlands correspondence. We also give a cohomological criterion for the number of top-dimensional components in a codimension one intersection. 

• Wednesday September 13, 2023 at 14:30, Temple, Wachman Hall 413

  On the Universal Deformation Ring of Residual Galois Representations with Three Jordan Holder Factors

  Xiaoyu (Coco) Huang, CUNY Graduate Center
  
  In this work, we study Fontaine-Laffaille, essentially self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions on certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke side, we also prove an R=T theorem. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. In particular, our result identifies the special L-value conditions for the uniqueness of the abelian surface isogeny class, and assuming the Bloch-Kato conjecture, an R=T theorem for the 6-dimensional representations. 

• Wednesday September 20, 2023 at 14:30, Temple, Wachman Hall 413

  Modularity of trianguline Galois representations

  Rebecca Bellovin, IAS
  
  The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation $\rho:Gal_Q\rightarrow GL_2(\overline{Q_p})$ is modular if it is unramified outside finitely many places and de Rham at $p$. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations $\rho:Gal_Q\rightarrow GL_2(L)$ when $L$ is instead a non-archimedean local field of characteristic $p$. 

• Wednesday September 27, 2023 at 14:30, Temple, Wachman Hall 413

  On 3-adic Galois images associated to isogeny torsion graphs of non-CM elliptic curves defined over Q

  Rakvi, University of Pennsylvania
  
  Let E be a non CM elliptic curve defined over ℚ. There is an isogeny torsion graph associated to E and there is also a Galois representation ρE,l:Gal(ℚ¯/ℚ)→GL2(ℤl) associated to E for every prime l. In this talk, I will discuss a classification of 3 adic Galois images associated to vertices of isogeny torsion graph of E.

• Wednesday October 4, 2023 at 15:00, Swarthmore College, Science Center 128

  Functional Equations for Axiomatic Multiple Dirichlet Series

  Ian Whitehead, Swarthmore College
  
  Recent work of Sawin gives a very general construction of multiple Dirichlet series over function fields satisfying a set of geometric axioms. This construction should encompass all Weyl group multiple Dirichlet series, as well the multiple Dirichlet series associated to higher moments of L-functions and Kac-Moody root systems. The analytic properties of Sawin's multiple Dirichlet series are not yet fully understood. In this talk, I will describe the groups of functional equations satisfied by these series, some proven and some still conjectural. This is joint work in progress with Sawin.

• Wednesday October 11, 2023 at 15:00, Swarthmore College, Science Center 128

  Rigid analytic diamonds

  Sean Howe, University of Utah
  
  In classical p-adic geometry, the fundamental objects are rigid analytic spaces built out of convergent power series rings reminiscent of those appearing in complex geometry. These are sufficient for many purposes, but they do not allow for a useful theory of infinite covering spaces. One remedy is to embed rigid analytic spaces in Scholze's category of diamonds by considering the functor of points on perfectoid algebras. The category of diamonds satisfies nice stability properties, including the existence of infinite covering spaces, but these perfectoid test objects behave very differently than the classical convergent power series rings: for example, because perfectoid algebras contain approximate p-power roots, they admit no continuous derivations and thus no tangent space in the classical sense. In this talk, we will survey some interesting phenomena and examples that arise while studying the relation between rigid analytic varieties and more general diamonds, especially in the context of period maps. 

• Wednesday October 25, 2023 at 15:30, Bryn Mawr College, Park Science Center 336

  Statistics of roots of polynomial congruences

  Matthew Walsh, University of Maryland
  
  For an integer, monic, irreducible polynomial F, we call the x (mod m) satisfying F(x) ≡ 0 (mod m) the roots of the congruence, and we consider the se- quence of normalized roots x/m ordered by increasing m. For quadratic F , statistical information about this sequence and certain subsequences has proven to be valuable input to many problems in analytic number theory. In joint work with Jens Marklof, we found a dynamical realization of the roots as return times to a specific section for the horocycle flow on SL(2, Z)\SL(2, R), analogous to Athreya and Cheung’s in- terpretation of the BCZ map for Farey fractions. Our realization of the roots leads to limit theorems for the pair correlation and other fine-scale statistics. Similar in- terpretations can be found for cubic and higher degree F but give weaker statistical information than can be obtained in the quadratic setting.

• Wednesday November 8, 2023 at 15:30, Bryn Mawr College, Park Science Center 336

  Root Number Correlation Bias of Fourier Coefficients of Modular Forms

  Nina Zubrilina, Princeton University 
  
  In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the p-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number. In my talk, I will discuss this root number correlation bias when the average is taken over all weight k modular newforms. I will point to a source of this phenomenon in this case and compute the correlation function exactly. 

• Wednesday November 15, 2023 at 15:00, Bryn Mawr College, Park Science Center 336

  Tiling, Sudoku, Domino, and Decidability

  Rachel Greenfeld, IAS
  
  Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a "translational tile", without any positive measure overlaps. Can we determine whether a given set is a translational tile? Does any translational tile admit a periodic tiling? A well known argument shows that these two questions are closely related. In the talk, we will discuss this relation and present some new developments, joint with Terence Tao, establishing answers to both questions.

• Wednesday November 29, 2023 at 15:00, Temple University, Wachman Hall 413

  Derived K-invariants and the derived Satake transform

  Karol Koziol, CUNY Baruch
  
  The classical Satake transform gives an isomorphism between the complex spherical Hecke algebra of a p-adic reductive group G, and the Weyl-invariants of the complex spherical Hecke algebra of a maximal torus of G. This provides a way for understanding the K-invariant vectors in smooth irreducible complex representations of G (where K is a maximal compact subgroup of G), and allows one to construct instances of unramified Langlands correspondences. In this talk, I'll present work in progress with Cédric Pépin in which we attempt to understand the analogous situation with mod p coefficients, and working at the level of the derived category of smooth G-representations. 

Philadelphia Area Number Theory Seminar 2024

2022 | 2023 | 2024

Current contacts: Jaclyn Lang, Catherine Hsu, Ian Whitehead, and Djordje Milicevic

The Philadelphia Area Number Theory Seminar rotates between Bryn Mawr, Swarthmore, and Temple. In Fall 2022, we meet on Tuesday afternoons, usually with tea at 3pm and then the talk 3:30-5pm. Please pay careful attention to the times and locations of the talks as they change from week to week! If you would like to be added to our mailing list or if you are interested in being a speaker, please contact one of the organizers. In future semesters, we anticipate that the seminar will be on Wednesday afternoons.

Click on title for abstract.

• Wednesday February 7, 2024 at 14:30, Temple University, Wachman Hall 413

  Distribution of the successive minima of the Petersson norm on cusp forms

  Souparna Purohit, University of Pennsylvania
  
  Given an arithmetic variety 𝒳 and a hermitian line bundle ℒ⎯⎯⎯⎯⎯⎯, the arithmetic Hilbert-Samuel theorem describes the asymptotic behavior of the co-volumes of the lattices H0(𝒳,ℒ⊗k) in the normed spaces H0(𝒳,ℒ⊗k)⊗ℝ as k→∞. Using his work on quasi-filtered graded algebras, Chen proved a variant of the arithmetic Hilbert-Samuel theorem which studies the asymptotic behavior of the successive minima of the lattices above. Chen's theorem, however, requires that the metric on ℒ⎯⎯⎯⎯⎯⎯ is continuous, and hence does not apply to automorphic vector bundles for which the natural metrics are often singular. In this talk, we discuss a version of Chen's theorem for the line bundle of modular forms for a finite index subgroup Γ⊆PSL2(ℤ) endowed with the logarithmically singular Petersson metric. This generalizes work of Chinburg, Guignard, and Soul\'{e} addressing the case Γ=PSL2(ℤ).

• Wednesday February 21, 2024 at 14:30, Temple University, Wachman Hall 413

  The Schottky problem in characteristic p

  Steven Groen, Lehigh University
  
  The Schottky problem is a classical problem that asks which Abelian varieties are isomorphic to the Jacobian of a (smooth) curve. If the dimension exceeds 3, not every Abelian variety can be a Jacobian. In characteristic p, there are additional tools that shed light on this question. In particular, the Ekedahl-Oort stratification partitions Abelian varieties by their p-torsion group scheme. An example of this is the distinction between ordinary elliptic curves and supersingular elliptic curves. The Ekedahl-Oort stratification leads to the following question: which p-torsion group schemes arise from Jacobians of (smooth) curves? Although this question is still wide open, I will present some progress on it, in particular when the curves in question are Artin-Schreier covers. Part of this is joint work with Huy Dang.

• Wednesday February 28, 2024 at 15:30, Bryn Mawr College, Park Science Center 328

  On the Density Hypothesis for Families of Lattices

  Djordje Milicevic, Bryn Mawr College
  
  Selberg’s celebrated Eigenvalue Conjecture states that all nonzero Lapla- cian eigenvalues on congruence quotients of the upper half-plane are at least 41 . This particularly strong form of the “spectral gap” property can be thought of as the archimedean counterpart of the Ramanujan–Petersson conjecture for Hecke eigen- values of cusp forms, is expected to suitably hold for more general Lie groups and their arithmetic quotients, and remains far from resolution.
   

  For analytic applications in a family of automorphic forms, in the absence of Selberg’s conjecture, the non-tempered spectrum can often be satisfactorily handled if the exceptions in the family are known to be “sparse” and “not too bad”, in a sense made precise by the so-called density hypothesis evoking the classical density estimates of prime number theory.

  In this friendly talk, we will first talk about the density hypothesis in general and how one can go about proving such an estimate. We will then present our recent result (joint with Fra ̧czyk, Harcos, and Maga) establishing the density hypothesis for a broad natural “horizontal” family of not necessarily commensurable arithmetic orbifolds, with uniform power-saving estimates in the volume and spectral aspects.

   

• Wednesday March 20, 2024 at 14:30, Temple University, Wachman Hall 413

  Upper and lower bounds for the large deviations of Selberg's central limit theorem

  Emma Bailey, CUNY
  
  Suppose we form a complex random variable by evaluating the Riemann zeta function at a random uniform height on the critical line, 1/2 + i U. Selberg’s central limit theorem informs us that the real (or indeed the imaginary) part of the logarithm of this random variable behaves, as T grows, like a centred Gaussian with a particular variance.  It is of interest, in particular in relation to the moments of the Riemann zeta function, to understand the large deviations of this random variable.  In this talk I will discuss the case for the right tail, presenting upper and lower bounds in work joint with L-P Arguin.

• Monday April 1, 2024 at 16:00, Temple University, Tuttleman Learning Center, 101

  Archimedes' giant troll

  Akshay Venkatesh, Institute for Advanced Study
  
   

  A 22-line poem written around 200 B.C. and attributed to Archimedes challenges the diligent and wise recipient to compute the number of cattle in the herds of the Sun God, after enumerating various properties of these herds. In mathematical language, the lines of the poem translate into a system of constraints amounting to a quadratic equation in two variables; but the resulting question required a further 2000 years to solve. (It turns out that the Sun God has no shortage of cows.) I will tell the story of this puzzle and its underlying mathematics, which is amazingly rich, inspiring mathematicians from Ancient Greece and medieval India to the present.

  This lecture is intended for a general audience. 

   

• Wednesday April 3, 2024 at 11:00, Temple University, Wachman Hall 617

  The duality paradigm for compact groups

  Akshay Venkatesh, Institute for Advanced Study
  
   

  It is a remarkable fact that there is a duality on the set of compact connected Lie groups; this duality interchanges, for example, the rotation group in three dimensions, and the group of unitary two-by-two matrices with determinant 1. This duality emerged in mathematics in the 1960s and, independently, in physics in the 1970s. In mathematics, it has served as an organizing principle for a great variety of phenomena related to Lie group theory, much of which falls under the heading of the “Langlands program”. I will describe some of the history and then two more recent developments: the realization that the mathematical and physics contexts for the duality are actually related to one another, and my recent work with Ben-Zvi and Sakellaridis where we seek to incorporate into the duality spaces upon which the group acts.

   

• Wednesday April 3, 2024 at 14:00, Temple University, Wachman Hall 617

  The AI mathematician

  Akshay Venkatesh, Institute for Advanced Study
  
  It is likely that developments in automated reasoning will transform research mathematics. I will discuss some ways in which we mathematicians might think about and approach this. The talk will *not* be about current or potential abilities of computers to do mathematics — rather I will look at topics such as the history of automation and mathematics, and related philosophical questions.

• Wednesday April 10, 2024 at 15:00, Swarthmore College, Science Center Room 181

  A refined random matrix model for function field L-functions

  Will Sawin, Columbia University
  
  The moments of the absolute value of the Riemann zeta function, up to height T, are expected to be a certain polynomial in log T. Since Keating and Snaith, the random matrix model for the Riemann zeta function has been used not just to model the distribution of its zeroes but the distribution of its values as well, which should include the moments. A modified random matrix model due to Gonek, Hughes, and Keating predicts the leading term of the moment polynomial but not the lower-order terms. In, for now, the function field case, I propose a different modification of the random matrix model. In work in progress, I show this model predicts all terms of the moment polynomial when q is sufficiently large. 

• Wednesday April 17, 2024 at 15:30, Bryn Mawr College, Park Science Center 245

  Arithmetic and Topology of Modular Knots

  Christopher-Lloyd Simon, Pennsylvania State University
  
  We study several arithmetic and topological structures on the set of conjugacy classes of the modular group PSL(2;Z), such as equivalence relations or bilinear functions.
   

  A) The modular group PSL(2; Z) acts on the hyperbolic plane with quotient the modular orbifold M, whose oriented closed geodesics correspond to the hyperbolic conjugacy classes in PSL(2; Z). For a field K containing Q, two matrices of PSL(2; Z) are said to be K-equivalent if they are conjugated by an element of PSL(2;K). For K = C this amounts to grouping modular geodesics of the same length. For K = Q we obtain a refinement of this equivalence relation which we will relate to genus- equivalence of binary quadratic forms, and we will give a geometrical interpretation in terms of the modular geodesics (angles at the intersection points and lengths of the ortho-geodesics).

  T) The unit tangent bundle U of the modular orbifold M is a 3-dimensional manifold homeomorphic to the complement of trefoil in the sphere. The modular knots in U are the periodic orbits for the geodesic flow, lifts of the closed oriented geodesics in M , and also correspond to the hyperbolic conjugacy classes in PSL(2; Z). Their linking number with the trefoil is well understood as it has been identified by E. Ghys with the Rademacher cocycle. We are interested in the linking numbers between two modular knots. We will show that the linking number with a modular knot minus that with its inverse yields a quasicharacter on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate. We will also associate to a pair of modular knots a function defined on the character variety of PSL(2;Z), whose limit at the boundary recovers their linking number.

  
   

   

• Wednesday April 24, 2024 at 14:30, Temple University, Wachman Hall 413

  Rational torsion in modular Jacobians

  Preston Wake, Michigan State University
  
  For a prime number N, Ogg's conjecture states that the torsion in the Jacobian of the modular curve X0(N) is generated by the cusps. Mazur proved Ogg's conjecture as one of the main theorems in his "Eisenstein ideal" paper. I'll talk about a generalization of Ogg's conjecture for squarefree N and a proof using the Eisenstein ideal. This is joint work with Ken Ribet. 

• Wednesday September 11, 2024 at 14:45, Swarthmore College Science Center 149

  Preview of Lectures 1 & 2 in mini-course on local fields

  Catherine Hsu, Swarthmore College
  
  In these lectures, we explore the theory of discrete valuations, including both arithmetic and topological properties. We conclude by defining non-archimedean local fields.
   

• Wednesday September 18, 2024 at 14:30, Temple University, Wachman Hall 413

  The geometry and arithmetic of the Ceresa cycle

  Ari Shnidman, IAS
  
  An algebraic curve C of genus g > 1 can be embedded in its Jacobian variety J in two different ways: {x - e : x in C} and {e - x : x in C}.  The Ceresa cycle k(C) is the formal difference of these two curves-- it vanishes if and only if these two curves can be algebraically deformed to each other within J. To date, the only known cases of k(C) = 0 are hyperelliptic curves. However, many examples of k(C) being torsion have recently been found, among curves/Jacobians with symmetries. I'll survey these recent examples, with an emphasis on what number theory has to say about this (a priori) purely algebro-geometric question.  

• Wednesday September 25, 2024 at 14:30, Temple University, Wachman Hall 413

  Convergent Decomposition Groups and an S-adic Shafarevich Conjecture

  Andrew Kwon, UPenn
  
  In the first part of this talk, I will survey some of the landmark results that demonstrate how closely intertwined arithmetic, Galois theory, and model theory can be (especially situations when "Galois theory = arithmetic" and "model theory = arithmetic" for nice fields). Special attention will be given to decomposition groups and local-global principles, which are the main characters for the second half.
  
  In the second part of the talk, I will discuss known and new results about decomposition groups that generate (generalized) profinite free products and how this relates to a certain Shafarevich Conjecture. Time permitting, I will also discuss future directions pertaining to reconstruction theorems and decidability.

• Wednesday October 2, 2024 at 14:45, Swarthmore College Science Center 149

  Preview of Lectures 3 & 4 in PAWS mini-course on local fields

  Catherine Hsu, Swarthmore College
  
  In these lectures, we will state and prove Hensel's lemma and explore some applications of this important tool. We'll then give a classification of non-archimedean local fields and describe the structure of the multiplicative group of p-adic number fields.

• Wednesday October 9, 2024 at 14:45, Swarthmore College Science Center 149

  Experiments with L-functions

  Nathan Ryan, Bucknell College 
  
  L-functions are central to many questions in modern number theory in the sense that they encode a lot of information about complex number theoretic objects.  While L-functions can be attached to primes, to primes in arithmetic progressions, to modular forms, to elliptic curves, etc., at the end of the day they're just complex analytic functions and they can be evaluated numerically at different points on the complex plane.  In this talk, I'll describe two projects that I've worked on.  The first is using random matrix theory to model the zeros of L-functions and the second is a method to evaluate L-functions at points in the plane that uses surprisingly little number theoretic information.

• Wednesday October 23, 2024 at 14:30, Temple University, Wachman Hall 413

  Generalised Eigenvalues and Flach classes

  Alice Pozzi, University of Bristol
  
  The connection between eigenvalues of Hecke operators acting on spaces of automorphic forms and representations of Galois groups is the central theme of the Langlands program. When the action of the Hecke algebra is not semisimple, one can define the finer notion of generalised Hecke eigenvalues; these turn out to be closely related to deformations of Galois representations. In this talk, I will discuss instances of this phenomenon arising for weight 2 mod p modular forms exploiting certain cohomology classes for the symmetric square representation constructed by Flach. This is joint work with Henri Darmon.

• Wednesday October 30, 2024 at 15:00, Temple University, Wachman Hall 413

  Geometric aspects of general multiple Dirichlet series over function fields

  Matthew Case-Liu, Columbia University
  
  Multiple Dirichlet series were originally defined as multi-variate Dirichlet series satisfying certain functional equations with conjecturally nice analytic properties that would give precise asymptotics for moments of L-functions. Generalizing an observation of Chinta, Sawin recently gave an axiomatic characterization of a general class of multiple Dirichlet series over function fields that is independent of their functional equations. Moreover, he proved their existence as formal power series by exhibiting the coefficients as trace functions of explicit perverse sheaves.
  
  In this talk, I'll explain how to 1. prove analyticity of these series in a suitable non-empty region of convergence, and 2. establish some (but not all) of the functional equations that they satisfy. The methods for both are completely geometric: analyticity is a consequence of bounding the cohomology of local systems on a compactification of a configuration space, and the functional equations follow from a density trick for irreducible perverse sheaves.

• Wednesday November 6, 2024 at 14:45

  Preview of Lectures 5 & 6 in mini-course on local fields

  Catherine Hsu, Swarthmore College
  
  
   

• Wednesday November 20, 2024 at 14:30, Temple University, Wachman Hall 413

  Tate classes and automorphic forms: a GSp(4) example

  Naomi Sweeting, IAS/Princeton
  
  The Hodge and Tate Conjectures aim to describe the part of cohomology coming from algebraic cycles on an algebraic variety. In this talk, I'll describe some Tate classes on the product of a modular curve and a Siegel 3-fold, that can be constructed using automorphic forms. As it turns out, one can prove that some -- but not all -- of these classes come from an algebraic subvariety with a simple moduli interpretation. One can also prove that all the Tate classes in this family of examples are Hodge classes, a result which is a kind of cohomological shadow of Tate's conjecture.

  The talk will start with a review of the Hodge and Tate Conjectures, the expected relation between them, and what is already known about the conjectures for Shimura varieties.