5/16/24 Davide parise - Monotonicity formulae in analysis and geometry

When trying to solve partial differential equations, a common practice is to enlarge the space of possible solutions to the class of non-differentiable functions, where it is easier to find “weak" solutions (i.e. potentially very irregular). As we are usually interested in “strong” solutions (i.e very regular), one is then confronted with the following problem: how do we upgrade the regularity? A fundamental tool in these situations is a monotonicity formula, an object that allows to study the infinitesimal behavior of solutions of PDEs by reducing it to a classification problem. More concretely, a monotonicity formula is an identity implying that a certain quantity related to the problem at hand is monotone, or conserved. I will try to convey the gist of this idea that has found applications in many areas at the intersection of geometry and analysis, e.g. harmonic maps, minimal surfaces, free boundary problems, Yang-Mills connections to name just a few. I will try to maintain the level of analysis needed at a minimum, you only need to remember that the first derivative of a smooth function at an interior minimum is zero. I will explain the rest. 

5/2/24 Srivatsav (Sri) Kunnawalkam Elayavalli - Sequential commutation

I will discuss a new conceptual framework called sequential commutation that has applications to von Neumann algebra theory. These focus on joint works by the speaker and others including Patchell, Gao and Tan. 

4/18/24 David Jekel  - Infinite-dimensional, non-commutative probability spaces and their symmetries

There is a deep analogy between, on the one hand, matrices and their trace, and on the other hand, random variables and their expectation.  This idea motivates "quantum" or non-commutative probability theory.  Tracial von Neumann algebras are infinite-dimensional analogs of matrix algebras and the normalized trace, and there are several ways to construct von Neumann algebras that represent suitable "limits" of matrix algebras, either through inductive limits, random matrix models, or ultraproducts.   I will give an introduction to this topic and discuss the ultraproduct of matrix algebras and its automorphisms or symmetries.  This study incorporates ideas from model theory as well as probability and optimal transport theory.

4/4/24 Chris Gartland  - Metric Embeddings

We will survey the theory of embeddings between metric spaces. Most attention will be paid to biLipschitz embeddings between particular metric spaces of interest such as Banach spaces, Wasserstein spaces, and finitely generated groups.

3/7/24 Aranya Lahiri  Why look at p-adic groups?

Do I really do number theory? Sometimes I have no idea how I belong to the number theory group, and not say functional analysis group? Even though the only books I pretend to read are: p-adic Lie groups, non-archimidean functional analysis and Lecture notes on formal and rigid geometry? But then I realize I really don't know any functional analysis for that matter. In this talk, in very broad and crude strokes I will try to convince myself that I do number theory. Come burst my bubble.

2/22/24 Marco Carfagnini - Small Fluctuations, Spectral Theory, and Random Geometry

The goal of this talk is to discuss new developments of random geometry.  We will focus on small fluctuations (small balls) for degenerate diffusions and their connection to sub-Riemannian geometry. In particular, such diffusions can be used to describe spectral properties of their (hypoelliptic) generators, where the lack of ellipticity makes the analytic approach more challenging. Moreover, we will discuss random loops on Riemann surfaces which can be described in terms of SLEk loop measures. These are measures on the space of simple loops, and we will provide their asymptotics on small balls. Lastly, we will focus on the geometry of random Laplace eigenfunctions on the sphere and their application to physics and statistics.

2/8/24 Amit Ophir - Stable lattices in representations over $p$-adic fields

Representations of groups over $p$-adic fields arise naturally in Number Theory. Stable lattices serve as integral models for such representations. I will provide an example of these representations. I will  discuss the connection between the set of lattices and a combinatorial object called the Bruhat-Tits building. If time permits, I will discuss open problems.

1/25/24 Anthony Sanchez - Translation surfaces and renormalization dynamics.

A translation surface is a collection of polygons with edge identifications given by translations. In spite of the simplicity of the definition, the space of translation surfaces has connections to different areas of math such as the moduli space of hyperbolic surfaces. A  guiding principle centers on turning questions on a fixed translation surface into a dynamical one on the space of all translation surfaces. We consider an instance of this philosophy related to the slope gap distribution of holonomy vectors of a translation surface. We use this as a jumping off point to consider expanding translates in different spaces such as non-arithmetic hyperbolic manifolds. Aspects of this talk represent different works with L. Kumandari and J. Wang, and with K. Ohm.

1/11/24 Priyanga Ganesan - What are quantum graphs and how can we generalize classical graph theoretic ideas to their setting?

Quantum graphs are an operator generalization of classical graphs that have appeared in different branches of mathematics including operator algebras, non-commutative topology, operator systems theory and quantum information theory.  In this talk, I will present an overview of the theory of quantum graphs and discuss the connections between its different perspectives using operator algebraic methods. I will then introduce the notion of a spectrum for the quantum graph. We will then explore a combinatorial notion of coloring quantum graphs and obtain various lower bounds for the chromatic numbers of quantum graphs, using its spectrum.

11/30/23  Gongping Niu - The existence of singular isoperimetric hypersurfaces

It is well-known that isoperimetric hypersurfaces in a smooth, compact (n+1)-manifold are smooth up to a closed set of codimension at least 7. We prove that the dimension estimate of singularities is sharp. In this talk, we will explore an example of an 8-dimensional closed smooth Riemannian manifold, whose unique isoperimetric region, with half the volume of the manifold, displays two isolated singularities on its boundary. Furthermore, for n > 7, we utilize similar methods to construct singular isoperimetric hypersurfaces in higher dimensions.

11/16/23 Alec Payne (Duke) - Flexible Smooth Immersions of Cylinders in R^3

Given a smooth surface in R^3, a classical question in differential geometry asks whether the surface can be continuously deformed through a smooth, nontrivial family of isometric surfaces. If such a family exists and does not arise from rigid motions of R^3, then the surface is said to be flexible. An old conjecture asserts that flexible, smooth closed surfaces do not exist. In this talk, we survey this question and the general uniqueness problem for isometric immersions. We then present new examples of flexible, smooth immersed cylinders in R^3 which are neither minimal nor developable. We conclude with a discussion of speculative approaches to the construction of flexible, smooth closed surfaces. These results are part of upcoming work with Andrew Sageman-Furnas.

11/2/23  Christian Klevdal -  Number theory!

Come venture into number theory in this spooky post halloween talk, where I plan on talking about some objects that are (at least tangentially) related to number theory. Which objects will show up? Maybe elliptic curves, maybe p-adic numbers, maybe Lie groups. It's a bit of a mystery, so come to the talk to find out! In order to keep the talk from being too scary, I'll try to keep the prerequisite knowledge to a minimum. 

10/19/23 - Nick Treuer - The $\overline{\partial}$-Neumann Problem and the Bergman Kernel

I will give an introduction to the $\overline{\partial}$-Neumann problem and the Bergman kernel, topics that are studied in several complex variables.  I will conclude by discussing two open questions about the Bergman kernel which motivate research in several complex variables today.

05/18/23 - Job Application Panel

05/11/23 - Jon Aycock- Categorification of the Ihara Zeta Function

Zeta functions are central objects of study in number theory, and can often be found wherever there is Galois theory. In this talk, we will discuss the Ihara zeta function of an undirected graph and compare it to the Dedekind zeta function. Then we will talk about incidence algebras and use them to describe a categorification of both types of zeta functions.

05/04/23 - Pratyush Sarkar - Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

Let $\Gamma < G = \operatorname{SO}(n, 1)^\circ$ be a Zariski dense

 torsion-free discrete subgroup for $n \geq 2$. Then the frame bundle of the hyperbolic manifold $X = \Gamma \backslash \mathbb{H}^n$ is the homogeneous space $\Gamma \backslash G$ and the frame flow is given by the right translation action by a one-parameter diagonalizable subgroup of $G$. Suppose $X$ is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. Endow $\Gamma \backslash G$ with the unique probability measure of maximal entropy called the Bowen-Margulis-Sullivan measure. In a joint work with Jialun Li and Wenyu Pan, we prove that the frame flow is exponentially mixing. 

04/27/23 - Marco Carfagnini - Brownian Motion on Lie Groups and Quasi-Invariance

In this talk we will discuss how to define Brownian motions on a curved space. We will briefly discuss some definitions on Riemannian manifolds and then focus on a construction on Lie groups. If time permits, we will discuss quasi-invariance with respect to left/right-multiplication and how this is related to the geometry of the group. (p.s. a background in probability is not required) 

04/20/23 - Gwen McKinley - Entropy and Counting

In this talk, I will give an overview of some tools used in probabilistic combinatorics, and illustrate their use in two different projects. First, in joint work with Marcus Michelen and Will Perkins, we establish an asymptotic formula for the number of integer partitions of a general type: namely, partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. Second, in joint work with Lina Li and Jinyoung Park, we give enumerative and structural results for colorings of the middle layers of the Hamming cube. This talk does not assume a lot of background, and it’s definitely okay if you don’t know much probability or combinatorics!

03/09/23 - Ruth Luo - Forbidden configurations in matrices and related extremal problems for set systems  

We consider {0,1}-matrices. For matrices A and B, we say A contains B as a configuration if there is a submatrix of A that is a column and row permutation of B. For instance, if A and B are incidence matrices of graphs G and H respectively, then A contains B as a configuration if and only if G contains H as a subgraph. In this talk, we study some extremal problems for matrices and hypergraphs (set systems).

03/02/23 - Yunze Lu - Equivariant homotopy theory and the triangulation conjecture 

I will talk about Manolescu's work on the triangulation conjecture. Using equivariant homology theory, it is proved that there exists non-triangulable manifolds in high-dimensions. I will introduce equivariant stable homotopy theory and some of its applications.

02/16/23 - Sam Mattheus - Geometry over finite fields: a rich source of constructions for extremal graph theory  

Problems in extremal graph theory typically aim to maximize some graph parameter under local restrictions. In order to prove lower bounds for these kinds of problems, several techniques have been developed. The most popular one, initiated by Paul Erdős, being the probabilistic method. While this technique has enjoyed tremendous success, it does not always provide sharp lower bounds. Frequently, algebraically and geometrically defined graphs outperform random graphs. We will show how historically, geometry over finite fields has been a rich source of such graphs. I will show a broad class of graphs defined from geometry of finite fields, which has found several recent applications in extremal graph theory. Often, certain interesting families of graph had in fact already been discovered and studied, years before their value in extremal graph theory was realized. I will demonstrate some instances of this phenomenon as well, which indicates that there might still be uncharted territory to explore.

02/02/23 - Gil Goffer - Hyperbolic groups and small cancellation theory

I’ll give a short intro to hyperbolic groups and small cancellation theory, and demonstrate how this theory can be used to construct groups with desirable properties. 

01/19/23 - Kisun Lee - Rank 2 symmetric matrices, tropicalizations, and algebraic matroids 

The matrix completion problems are about completing a partially filled matrix to achieve the lowest possible rank. As they can be interpreted as an understanding of a certain algebraic variety, we consider a corresponding algebraic matroid and desire to characterize its bases. Polyhedralizing via tropical algebra may help us to figure out this characterization. We begin the talk with brief introductions on matrix completion problems, algebraic matroids, and tropical algebra. No pre-knowledge is assumed. This is based on ongoing work with May Cai, Cvetelina Hill, and Josephine Yu. 

11/03/22 - Sunny Agrawal - Using algebra to detect differential item functioning 

Differential item functioning (DIF) refers to the situation where responses to a given question on an exam (or survey or similar) differ between several groups. For several decades now, social scientists and education researchers have employed a standard battery of statistical tools to detect DIF from sample data, but essentially all of these standard tools rely on theoretical asymptotic results and presuppose sample sizes that are rarely achieved by real data sets. In this talk, we'll discuss how ideas dating back to Diaconis and Sturmfels, in which techniques from computational algebra are brought to bear in statistics, provide an alternative method to detect DIF which avoids asymptotics and is more robust with smaller sample sizes. This is joint work with Luis David Garcia-Puente, Minho Kim, and Flavia Sancier-Barbosa. 

10/20/22 - James Upton - Goss' Riemann Hypothesis for Function Fields

The Goss zeta function is a characteristic-p analogue of the Riemann zeta function for function fields. In the spirit of the Riemann hypothesis, Goss has made several conjectures concerning the distribution of its zeros. We discuss the history of these questions and some recent progress we have made in collaboration with Joe Kramer-Miller. Our main result is a comparison of the distribution of zeros between the higher-genus and genus-zero cases. As a consequence, we are able to prove Goss' conjectures in a large number of previously unknown cases.

10/06/22 - Jon Aycock  - Brauer Groups and Quadratic Reciprocity

In this talk I'll give a proof of quadratic reciprocity, which is a result about squares in finite fields, using the Alfred--Brauer--Hasse--Noether Theorem, which is a result about division algebras over local and global fields. The talk is aimed at people who might not know all of those words already; I will explain most of the steps along the way. Some prior knowledge of Hamilton's quaternions will be useful in understanding some of the arguments.

05/26/22 - Johannes Brust - Effective COVID-19 Pooling Matrix Designs

The development of vaccines for COVID-19 has enabled us to nearly return to pre-pandemic life. However, while vaccines are becoming globally widespread, high alert levels prevail. Even with vaccines, monitoring for the evolution of mutations or detecting new outbreaks calls for continued vigilance. Therefore, testing is likely to prevail to be a vital mechanism to inform decision making in the near future. In order to conserve scarce testing resources, many nations have endorsed so-called group/pooling test methods. Such methods can be expressed using linear algebra. The basic principle underlying pooling tests is the observation that to efficiently detect positive cases among a population with a very low occurrence prevalence, it can be advantageous to test groups of samples instead of testing all individual samples. We develop matrix designs, which encode all relevant information for doing pooling tests and that enable high compression rates when exactly identifying up to a certain number of positive cases.

05/19/22 - Shuang Liu - Level set simulations of cell polarity and movement

We develop an efficient and accurate level set method to study numerically a crawling eukaryotic cell using a minimal model. This model describes the cell polarity and movement using a reaction-diffusion system coupled with a sharp-interface model. 

We employ an efficient finite difference method for the reaction-diffusion equations with no-flux boundary conditions. This results in a symmetric positive definite system, which can be solved by the conjugate gradient method accelerated by preconditioners. To track the long-time dynamics, we employ techniques of the moving computational window to keep the efficiency. Our level-set simulations capture well the cell crawling, the straight line trajectory, the circular trajectory, and other features. 

Our efficient and accurate computational techniques can be extended to a broad class of biochemical descriptions of cell motility, for which problems are posed on moving domains with complex geometry and fast simulations are very important.  This is a joint work with Li-Tien Cheng and Bo Li.

05/12/22 - Yuming Paul Zhang - McKean-Vlasov equations involving hitting times: blow-ups and global solvability

We study two McKean-Vlasov equations involving hitting times. Let $(B(t); t \geq 0)$ be standard Brownian motion, and $\tau:= \inf\{t \geq 0: X(t) \leq 0\}$ be the hitting time to zero of a given process $X$. The first equation is $X(t) = X(0) + B(t) - \alpha \mathbb{P}(\tau \leq t)$.

We provide a simple condition on $\alpha$ and the distribution of $X(0)$ such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all time $t \geq 0$. We take the PDE approach and develop a new comparison principle.

The second equation is $X(t) = X(0) + \beta t + B(t) + \alpha \log \mathbb{P}(\tau \leq t)$, $t \geq 0$, whose Fokker-Planck equation is non-local. We prove that if $\beta,1/\alpha > 0$ are sufficiently large, the McKean-Vlasov dynamics is well-defined for all time $t \geq 0$. The argument is based on a relative entropy analysis. This is joint work with Erhan Bayraktar, Gaoyue Guo and Wenpin Tang.

05/05/22 - Caroline Moosmueller -  Subdivision schemes and approximation of manifold-valued data

In this talk, I will give an introduction to subdivision schemes, which are iterative refinement processes for interpolating or approximating discrete data points. Most results on subdivision schemes concern data in vector spaces and rules which are linear. I will present an adaptation of subdivision schemes to operate on manifold-valued data using the intrinsic geometry of the underlying manifold (such as the exponential map). Analysis of convergence and smoothness properties will be presented as well. Subdivision schemes find applications in computer graphics and 3D animated movies.

04/28/22 - no seminar

04/21/22 - Caroline Moosmueller -  Optimal transport in machine learning

In this talk, I will give an introduction to optimal transport, which has evolved as one of the major frameworks to meaningfully compare distributional data. The focus will mostly be on machine learning, and how optimal transport can be used efficiently for clustering and supervised learning tasks. Applications of interest include image classification as well as medical data such as gene expression profiles.

04/14/22 - Brandon Alberts - Power Savings in Number Field Counting

We will discuss some of the known power savings for the number of G-extensions of a number field with discriminant bounded above by X. We will put a focus on the existence of secondary terms in the asymptotic growth rate, and in particular will discuss a proof of the existence of some secondary terms when G is abelian.

04/07/22 - Andrew Zucker - Perspectives on the Halpern-Lauchli theorem

The aim of this talk is to introduce the audience to the Halpern-Lauchli theorem, which is a Ramsey-theoretic statement about products of trees. We will discuss several applications of the theorem and outline a number of different proofs. While the original proof was combinatorial in nature, there are now a number of proofs that interact with ideas from set-theoretic forcing. One of these proofs is new, and is joint work with Chris Lambie-Hanson.

12/02/21 - Be'eri Greenfeld - An invitation to growth: infinite dimensional algebras in real life math

Let A be an (associative/Lie/etc.) algebra, finite dimensional as a vector space over a field; its most important invariant is its vector space dimension. Suppose that A  is infinite dimensional, but having a suitable finiteness condition, e. g. finitely generated or admits a locally finite grading. Analyzing the asymptotic growth of the 'finite pieces' of A is a common and useful way to measure its infinitude. Our aim in this talk is to show how the growth of an algebra reflects its structure, and illuminate appearances and applications of this theory in various mathematical fields, including group theory, algebraic geometry, symbolic dynamics and arithmetic geometry. If time permits, we will mention some of our novel results, both in the backend (how do algebras grow?) and frontend (why do we care how algebras grow?) of the theory.

11/18/21 - Angus Chung - Analogy between Integers and Polynomials over Finite Fields

It is often mentioned that "Function Fields" look like rational numbers. In this talk, we will attempt to show some similarities between integers and polynomials over finite fields, from the number theory perspective. We will study analogues of "prime numbers" in function fields and ask questions about them. One tool that we use a lot in the study of function fields is called Drinfeld modules. We will introduce the most basic example, the Carlitz module, and briefly mention how we can use it to answer some questions.

11/04/21 - Ming Zhang - Aspects of Moduli Spaces

Roughly speaking, a moduli space is the 'space' of objects that satisfies given criteria—for example, the space of all functions satisfying a differential equation. In this talk, I will discuss some examples of moduli spaces in algebraic geometry and explain (some of) the motivations behind the 'fancy' concepts like stacks, deformation-obstruction theory, and derived algebraic geometry.

10/28/21 - Cain Edie-Michell - Four realisations of the Fibonacci category

Quantum algebra is a relatively new branch of mathematics that lives at the intersection of topology, operator algebras, representation theory, physics, and category theory. Using the concrete example of the Fibonacci fusion category, I will explain the connection between these seemingly distinct research areas.

10/21/21 - Brandon Alberts - Meromorphic Continuations of Euler Products

 Different time this week: 11 am - 12 pm!

The rate of growth of an arithmetic function $f:\mathbb{N} \to \mathbb{C}$ can be studied by determining the region of convergence of the Dirichlet series $F(s) = \sum f(n) n^{-s}$, and yet more information can be extracted when there exists an analytic (or meromorphic) function on a larger region that coincides with $F(s)$ on the region of convergence. In this expository talk, I will introduce some of the foundational ideas in this method when $f$ is a multiplicative function (that is, $f(xy) = f(x) f(y)$ if $x$ and $y$ are coprime) and how representations theory, complex analysis, and L-functions come together to construct meromorphic continuations. If we have time, I will discuss the impact of these ideas on my research in number field counting.

10/14/21 - David Jekel - Random matrix theory and transport equations

We examine the large N behavior of random d-tuples of N x N self-adjoint matrices with a probability density of the form (const) e^{-N^2 V(A_1,...,A_d)}, where V is some scalar-valued function given by taking traces of non-commutative polynomials in the matrices A_1, ..., A_d.  The large-N limit is described by a non-commutative probability gadget known as a free Gibbs law, or by a corresponding von Neumann algebra.  Certain differential equations can be used to construct a function F^{(N)}, from d-tuples of sel-adjoint matrices to itself, such that F^{(N)}(X^{(N)}) is a standard Gaussian family of self-adjoint matrices.  We showed that such an F^{(N)} will have good asymptotic behavior as N --> infinity, provided that V is sufficiently nice.  We will explain this result and its consequences for the corresponding von Neumann algebra.

04/22/21 - Shuang Liu - A  parallel cut-cell  algorithm for the free-boundary Grad-Shafranov problem 

A parallel cut-cell algorithm is described to solve the free boundary problem of the Grad-Shafranov equation.  The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a  searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the  boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's acceleration for the resulting nonlinear problem and a Cartesian grid embedded boundary method to handle the complex geometry. The algorithm is implemented in parallel using a standard domain-decomposition approach and a good parallel scaling is observed. Numerical results verify the accuracy and efficiency of the free-boundary Grad-Shafranov solver.

04/08/21 - Gwen McKinley - Random random graphs 

02/25/21 - Kisun Lee - Finding and certifying numerical roots of systems of equations? 

02/04/21 - Be'eri Greenfeld - How do algebras grow? 

01/21/21 - Martin Licht - Perspectives in structure-preserving numerical methods 

12/03/20 - Yuming Paul Zhang - Some free boundary problems

11/19/20 - Nattalie Tamam - Diophantine approximations and dynamics 

11/12/20 - Brandon Alberts - Arithmetic Statistics 

11/05/20 - Matthew Wiersma: Banach algebras over groups 

10/29/20 - Ruth Luo - An introduction to extremal graph and hypergraph theory 

10/22/20 - Andy Zucker: An introduction to big Ramsey degrees.

10/15/20 - David Jekel: An introduction to non-commutative probability 

03/05/20 - Martin Licht: Artificial Neural Networks for Dummies 

02/20/20 - Caroline Moosmüller: Three geometric data science methods for analyzing gene expression data 

02/06/20 - David Stapleton: Riemann surfaces, Riemann-Hurwitz, and Riemann-Roch 

01/23/20 - Kristin DeVleming: Moduli spaces in algebraic geometry 

11/26/19 - Paul Zhang: Front propagation of reaction-diffusion equations

11/12/19 - Josh Swanson: A brief introduction to covariant algebras

10/26/19 - Alexandr Knop: Resolution Proof System and GraphComplexity