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.