Colloquia
Colloquia for the Department of Mathematics and Statistics are normally held in University Hall 4010 on Fridays at 4:00 p.m. Any departures from this are indicated below. Light refreshments are served before the colloquia in 2040 University Hall. Driving directions, parking information and maps are available on the university website.
2024-2025 Colloquia
Below is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.
Spring Semester
Shoemaker Lecture Series April 2-4, 2025
Rui Loja Fernandes (University of Illinois at Urbana-Chicago)
Lecture I: An Invitation to Poisson Geometry
April 2, 2025, 5:00-6:00pm in Gillham Hall 5300
Abstract: Poisson geometry provides the mathematical framework underlying classical mechanical systems and plays a fundamental role in the transition to quantum mechanics. Its origins trace back to the work of 18th- and 19th-century mathematicians such as Lagrange, Legendre, Hamilton, Jacobi and Poisson. The modern development of Poisson geometry began in the 1970s and 1980s through the contributions of Arnold, Kirillov, Lichnerowicz and Weinstein.
In this lecture, I will provide a gentle introduction to modern Poisson geometry, highlighting its key concepts and exploring a few of its applications. I will also discuss some of its most significant achievements, including Kontsevich’s seminal result that every Poisson manifold admits a formal deformation quantization.
Lecture II: The Differential Geometry of Poisson Manifolds
April 3, 2025, 4:00-5:00pm in University Hall 4010
Abstract: The study of Poisson manifolds necessitates a special form of differential geometry, often referred to as contravariant geometry. This framework introduces analogs of fundamental geometric notions, such as paths, connections, differential forms and even a version of Stokes’ theorem, all adapted to the Poisson setting. In this lecture, I will provide an overview of this contravariant geometry and illustrate how it can be employed to establish new results in local Poisson geometry.
Lecture III: Global Poisson Geometry and Symplectic Groupoids
April 4, 2025
Abstract: This lecture explores the global aspects of Poisson geometry, with a particular focus on one of its most fundamental structures: symplectic groupoids. Introduced independently by Karasev, Maslov, and Weinstein in the late 1980s, symplectic groupoids provide a natural framework for understanding the global behavior of Poisson manifolds. Much like a Lie algebra integrates to a Lie group, a Poisson manifold may integrate to a symplectic groupoid. I will discuss the origins of symplectic groupoids, the conditions for their existence, and their role in establishing global results in Poisson geometry.
March 14, 2025
Fabricio Valencia Quintero (University of Sao Paulo, Brazil)
A double extension procedure for flat pseudo-Riemannian F-Lie algebras.
Abstract: The aim of this talk is to introduce the notion of flat pseudo-Riemannian F-Lie algebra (FSRFL) and construct its corresponding double extension. The latter can be achieved by applying some basic techniques from the theory of both associative and left symmetric algebras. Motivated by the notion of Frobenius manifold, we begin by introducing the sort of geometric object encompassing the simply connected Lie groups integrating FSRFLs and then we lay the groundwork for our constructions. We finish by showing some examples which exhibit the algebraic conditions underlying the double extension procedure. Based on joint work with Alexander Torres-Gomez.
January 31, 2025
Aissa Wade (Penn State University)
Jacobi fibrations and Yang-Mills fields.
Abstract: A Jacobi fibration is a fiber bundle E → B whose typical fiber F is equipped with a Jacobi structure and whose structure group G is a subgroup of the group of diffeomorphisms of F preserving its Jacobi structure. Poisson fibrations, symplectic and contact fibrations are special cases of Jacobi fibrations.
Symplectic fibrations first appeared in Sternberg’s paper entitled “Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field”, where he studied the motion of a particle minimally coupled to a gauge field. In 1978, Alan Weinstein further investigated Sternberg’s “YangMills setup” and gave a nice construction of a symplectic fibration obtained from a connection on a given principal G-bundle and a given Hamiltonian G-space. Weinstein’s construction has been extended to several geometric frameworks, including the Poisson case as well as the contact and locally conformal symplectic settings.
In this talk, I will present a way to unify all these known cases and discuss applications of Jacobi fibrations to Dirac-Jacobi geometry
January 16, 2025 10:00am
Luca Vitagliano (University of Salerno, Italy)
The Symplectic-to-Contact Dictionary
Abstract: Symplectic Geometry and Poisson Geometry (the possibly singular, contravariant version), are hot areas of contemporary Differential Geometry originally inspired by Theoretic Mechanics. Symplectic structures can only be supported by even dimensional manifolds. Contact Geometry is an odd dimensional analogue of Symplectic Geometry and its possibly singular, contravariant version is Jacobi Geometry. We will review the fundamentals of Symplectic/Poisson and Contact/Jacobi Geometry, including the basic examples. In the end of the talk, we will present a Dictionary from symplectic-related geometries to contact-related geometries explaining in which precise sense the latter are odd dimensional analogues of the former.
Fall Semester
Shoemaker Lecture Series December 4-6, 2024
Gunther Uhlmann (University of Washington)
Lecture I: Inverse Problems and Harry Potter’s Cloak
December 4, 2024, 5:00-6:00pm in Doermann Theater
Abstract: Inverse problems arise in all fields of science and technology where causes for a desired or observed effect are to be determined. By solving an inverse problem is in fact how we obtain a large part of our information about the world. An example is human vision: from the measurements of scattered light that reaches our retinas, our brains construct a detailed three-dimensional map of the world around us. In the first part of the talk, we will describe several inverse problems arising in different contexts.
In the second part of the lecture, we will discuss invisibility. Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last 20 years or so there have been several scientific proposals to achieve invisibility. We will describe in a non-technical fashion a simple and powerful proposal, the so-called transformation optics, and some of the progress that has been made in achieving invisibility.
Lecture II: Calderon’s Inverse Problem and Electrical Impedance Tomography
December 5, 2024 in HH 1600
Abstract: Calderon's inverse problem, also called Electrical Impedance Tomography or Electric Resistivity Imaging, asks whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This question arises in several areas of applications including medical imaging and geophysics. I will report on some of the progress that has been made on this problem since Calderon proposed it in 1980, including recent developments on similar problems for nonlinear equations and nonlocal operators. We will also discuss several open problems.
Lecture III: Journey to the Center of the Earth
December 6, 2024 in GH 5300
Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics, medical imaging and oceanography, among several others.
October 25, 2024
Dr. Jianping Sun (University of North Carolina)
Dynamic Treatment Effect Analysis in Crossover Designs through Repeated Measures
Abstract: This talk introduces an extended model that harnesses the power of convolution operations to represent time-varying treatment and carry-over effects in a crossover study design. Unlike the traditional model, the proposed approach unifies the treatment and carry-over effects through time-varying response functions, one for each treatment. The model is not only flexible enough to accommodate a variety of treatment plans, including multiple administrations at different doses, but also allows for the inclusion of more treatment periods. The advantages of this approach are accentuated by its ability to be generalized, to avoid prior assumptions about the carry-over effect, and to maintain consistent estimation and hypothesis testing procedures. In the talk, I will explore the details of hypothesis testing under this extended model, focusing in particular on the comparison of two response functions within specified intervals. The goal of this work is to improve the modeling of carryover effects, thereby strengthening the applicability of the model to a variety of experimental settings.
September 6, 2024
Biao Ou (University of Toledo)
A revisit to the Jordan canonical form theorem
Abstract: We present a proof of the Jordan canonical form theorem that probably resembles the original proof of Jordan. We then find the Jordan canonical form for special matrices including rank-one matrices and all 3 by 3 matrices. We then apply the Jordan canonical form theorem in two occasions. One is about an infinite sequence of complex numbers satisfying a linear recurrence relation. The other is about a square matrix with all positive number entries.
2023-2024 Colloquia
Below is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.
Spring Semester
Shoemaker Lecture Series April 11-12, 2024
Zhezhen Jin (Columbia)
Lecture I: Commonly used statistical methods for assessing agreement or concordance will be reviewed: intra-class correlation coefficient (ICC), Bland-Altman plot, Cronbach’s alpha, Cohen’s kappa, ROC curve, etc.
April 11, 2024 (Thursday), 4:00-5:00pm in UH 4010
Lecture II: A real example from Alzheimer’s disease study on the item selection and assessment. It is based in a 40-item test by the standardized University of Pennsylvania Smell Identification Test (UPSIT) of olfactory function, for screening patients for the risk of developing Alzheimer’s disease. The challenge lies on the comparison of reduced items and complete items which can be addressed by modification of the statistical methods for agreement and concordance.
April 12, 2024 (Friday), 4:00-5:00pm in UH 4010
Presentation Abstracts: Statistical methods in the analysis of biomedical data
It is very important to incorporate basic statistical principles and ideas in data analysis. It is essential to compare and identify biomarkers that are more informative to disease diagnosis and monitoring, and to evaluate various treatment procedure and plan on health outcome. After a discussion on the issues and challenges with real examples, I will review available statistical methods and present our newly developed semiparametric statistical methods that are useful for item reduction, differentiation of significant exposure factors and high dimensional data analysis.
March 15, 2024
Ryad Ghanam (Virginia Commonwealth University, Qatar)
Lie Algebras and Symmetries of Differential Equations.
Abstract: In this talk we shall focus on the origins of Lie theory and discuss several examples of differential equations to which the Lie symmetry method is applicable. Thereafter we shall trace the development of the theory of abstract Lie algebras and its importance in theoretical physics. We will also consider a special system of differential equations on Lie groups; called the geodesic equations of the canonical connection.
February 16, 2024
Youjin Zhang (Tsinghua University)
A class of generalized Frobenius manifolds and integrable systems
Abstract: The notion of Frobenius manifolds was introduced by Boris Dubrovin in the early 1990s. It is a geometric characterization of the Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity which arise in the study of 2d topological field theory, and has played important roles in the study of Gromov-Witten theory, singularity theory, integrable systems and some other research fields of mathematical physics. In this talk, I will first recall the definition of Frobenius manifolds, their basic properties, and some well-known results on their relation with integrable evolutionary PDEs of KdV type, and then I will introduce some recent results on the relation of a class of generalized Frobenius manifolds with integrable systems.
February 9, 2024
Tony Shaska (Oakland Univerity)
Machine learning in the moduli space of curves
Abstract: Studying arithmetic properties of the moduli space of algebraic curves is a classical problem with many implications in arithmetic geometry and applications to cryptography. We explore how to use methods of deep learning to better understand the rational points of the moduli space. Several examples of such methods, from the moduli space of genus two curves, will be described in detail illustrating our approach. The talk will be accessible to a general audience.
Fall Semester
Shoemaker Lecture Series October 5-6, 2023
Van Vu (Yale)
Lecture I: Matrix completionand random perturbation
October 5, 2023 (Thursday), 5:00-6:00pm in GH 5300
Abstract: A practical problem of fundamental interestis to complete a large data matrix from relatively few observed entries. A well-known example is the Netflix prize problem.
Perturbation theory provides perturbation bounds onspectral parameters of a matrix under a small perturbation. In recent works, we discovered that many classical perturbation bounds (such as Weyl theorem) can be improved significantly when the perturbation is random.
In this talk, I will going to give a brief survey on mathematical approaches to the matrix completion problems, and discuss a very simple new algorithm based on results concerning random perturbation.
Lecture II: The circular law
October 6, 2023 in GH 5300
Abstract: The circular law is the asymmetric counterpart of Wigner semi-circle law. It asserts that the (complex) eigenvalues of a typical random asymmetric matrix distribute uniformly on the unit circle (after a standard normalization).
Wigner semi-circle law was proved in the 1950s. But Circular Law was proved in full generality only in the 2000s. Why this delay?
We are going to discuss the mathematical developmentsthat lead to the solution of the Circular Law conjecture.
2022-2023 Colloquia
Below is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.
Spring Semester
April 28, 2023
Roland Roeder (Indiana University-Purdue University Indianapolis)
The Ising Model for magnets and the mysterious Lee-Yang zeros
Abstract:The Ising Model was developed by Lenz and Ising in the 1920s to describe magnetic materials. It is the most fundamental model of such materials, but there are still many open questions about it that continue to challenge present day physicists and mathematicians. In suitable variables, the Ising model can be described by a single polynomial $Z(z,t)$ of two variables $z$ and $t$ that is called the "partition function". Lee and Yang proved in 1952 that if $t \in [0,1]$, then the zeros of the partition function lie on the unit circle $|z| = 1$. Most of the physical properties of the magnet are determined by the location of these "Lee-Yang zeros".
In this talk, I will explain the Ising model, the Lee-Yang Theorem, and describe several interesting results and open questions about the locations of the Lee-Yang zeros for the classical square lattice $\mathbb{Z}^2$. I will conclude by describing two hierarchical lattices (the Cayley Tree and the Diamond Hierachical Lattice) for which dynamical systems techniques allow us to say considerably more about the limiting distribution of Lee-Yang zeros.
In the case of the Cayley Tree, this is joint work with Ivan Chio, Anthony Ji, and Caleb He and in the case of the Diamond Hierarchical Lattice it is joint work with Pavel Bleher and Mikhail Lyubich.
March 31, 2023
Pavel Etingof (Massachusetts Institute of Technology)
Lie theory in tensor categories with applications to modular representation theory
Abstract: Let $G$ be a group and $k$ an algebraically closed field of characteristic $p>0$. If $V$ is a finite dimensional representation of $G$ over $k$, then by the classical Krull-Schmidt theorem, the tensor power $V^{\otimes n}$ can be uniquely decomposed into a direct sum of indecomposable representations. But we know very little about this decomposition, even for very small groups, such as $G=(\Bbb Z/2)^3$ for $p=2$ or $G=(\Bbb Z/3)^2$ for $p=3$. For example, what can we say about the number $d_n(V)$ of such summands of dimension coprime to $p$? It is easy to show that there exists a finite limit $d(V):={\rm lim}_{n\to \infty}d_n(V)^{1/n}$, but what kind of number is it? For example, is it algebraic or transcendental? Until recently, there was no techniques to solve such questions (and in particular the same question about the sum of dimensions of these summands is still wide open). Remarkably, a new subject which may be called "Lie theory in tensor categories" gives methods to show that $d(V)$ is indeed an algebraic number, which moreover has the form $$d(V)=\sum_{1\le j\le p/2}n_j(V)[j]_q,$$ where $n_j(V)\in \Bbb N$, $q:=\exp(\pi i/p)$, and $[j]_q:=\frac{q^j-q^{-j}}{q-q^{-1}}$. Moreover, $$d(V\oplus W)=d(V)+d(W),\ d(V\otimes W)=d(V)d(W),$$ i.e., $d$ is a character of the Green ring of $G$ over $k$. Furthermore, $$d_n(V)\ge C_Vd(V)^n$$ for some $0<C_V\le 1$ and we can give lower bounds for $C_V$. In the talk I will explain what Lie theory in tensor categories is and how it can be applied to such problems. This is joint work with K. Coulembier and V. Ostrik.
March 17, 2023 1:30pm
Apoorva Khare (Indian Institute of Science; and Analysis & Probability Research Group)
Groups with norms: from word games to a PolyMath project
Abstract:Consider the following three properties of an arbitrary group $G$:
- Algebra: $G$ is abelian and torsion-free.
- Analysis: $G$ is a metric space that admits a "norm", namely, a translation-invariant metric $d(.,.)$ satisfying: $d(1,g^n) = |n| d(1,g)$ for all $g$ in $G$ and integers $n$.
- Geometry: $G$ admits a length function with "saturated" subadditivity for equal arguments: $l(g^2) = 2 l(g)$ for all $g$ in $G$. While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a "norm".
We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and finally, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.
(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)
Fall Semester
December 2, 2022
Xiaojun Chen (Sichuan University)
Calabi-Yau algebras and the noncommutative shifted symplectic structure
Abstract: The notion of Calabi-Yau algebras was introduced by Ginzburg in 2007, and has been widely studied by mathematicians in recent years. In this talk, we study several geometric structures on Calabi-Yau algebras and show that there is a noncommutative version of the “shifted" symplectic structure on Calabi-Yau algebras, which induces a shifted symplectic structure on the moduli stacks of their derived representations. If time permits we also discuss the derived noncommutative Poisson structure on Calabi-Yau algebras as well as its quantization. This talk is based on the joint works with Farkhod Eshmatov.
September 23, 2022
Jari Taskinen (University of Helsinki)
Applications of the Floquet-transform to elliptic spectral PDE problems and to operator
theory in Bergman spaces
Abstract: The Floquet-transform $\mathsf{F}$ is a variation of the Fourier-transform, which can applied in function spaces defined on periodic subdomains of the Euclidean space $\mathbb{R^N}$, whereas the Fourier-transform only works for functions defined on the entire $\mathbb{R^N}$. It is a standard tool also in the theory of the Schrödinger equation with periodic potentials, and in quantum physics and chemistry it is rather known by the name Bloch-transform.
I will review some recent results on the applications of the Floquet-transform to spectral PDE-problems on periodic domains with thin stuctures. Here, it is interesting to open gaps of the continuous or essential spectrum of the problem, which otherwise in typical self-adjoint problems would fill the entire positive real-axis. In the beginning of the talk I will give an introduction to the necessary spectral concepts.
Another topic will be a recent study of Bergman kernels in periodic domains $\Pi$ of the complex plane. Here, the Floquet-transform can be used to express the kernel of a geometrically complicated unbounded domain $\Pi$ in terms of a kernel of a simpler bounded periodic cell.
For the 2021-2022 academic year the colloquia will be held online due to COVID-19.
Here is the link: Blackboard Collaborate Colloquia Meeting Room
2021-2022 Colloquia
Below is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.
Spring Semester
April 29, 2022
Chanaka Kottegoda (The University of Toledo)
Complex Dynamics and Bifurcations of Predator-prey Systems with Generalized Holling
Type Functional Responses and Allee Effects in Prey
Zoom Link Meeting ID: 614 048 1628
Abstract: This talk is devoted to high codimension bifurcations of a predator-prey system with generalized Holling type response function and Allee effects in prey. The nilpotent cusp singularity of order 3 and degenerate Hopf bifurcation of codimension 3 are completely analyzed. Remarkably it is the first time that three limit cycles are discovered in predator-prey models with multiplicative Allee effects. Moreover, a new unfolding of nilpotent saddle of codimension 3 with a fixed invariant line is discovered and fully developed, and the existence of codimension 2 heteroclinic bifurcation is proven. Our work extends the existing results of predator-prey systems with Allee effects. The bifurcation analysis and diagram allow us to give biological interpretations of predator-prey interactions.
April 22, 2022
Gregg Musiker (University of Minnesota)
From Cluster Algebras and Snake Graphs to Teichmueller Spaces and Super Teichmueller
Spaces
Abstract: This talk will provide an introduction to cluster algebras from surfaces, and their generators, known as cluster variables. We will also describe how certain combinatorial objects known as snake graphs yield generating functions that agree with Laurent expansions of said cluster variables. From the point of view of decorated Teichmueller theory, these generating functions also provide formulas for lambda lengths, also known as Penner coordinates, with respect to a reference triangulation. We will then switch gears and describe formulas for super lambda lengths in the context of decorated super Teichmeuller space of a marked disc or an annulus. This latter work generalizes an earlier interpretation of dimers on snake graphs for ordinary lambda lengths defined with R. Schiffler and L. Williams. For the case of super lambda-lengths, this is joint work with N. Ovenhouse and S. Zhang.
April 15, 2022
Javad Mashreghi (Laval University / President of the Canadian Math Society)
A briefing on dilation theory
Abstract: This talk is mostly a historical recount of dilation theory in the complex plane. We provide a wide range of dilation-related theorems in different function spaces such as the Hardy space, Bergman space, super-harmonically weighted Dirichlet spaces, Bloch space, model spaces, de Branges-Rovnyak spaces, etc. We finish with some recent results and open questions, emerged in the last decade, to highlight the need for novel tools in dealing the remaining problems.
April 8, 2022
Anton Zeitlin (Louisiana State University)
Super-Teichmueller spaces and Penner coordinates
Abstract: The Teichmüller space parametrizes Riemann surfaces of fixed topological type and is fundamental in various contexts of mathematics and physics. It can be defined as a component of the moduli space of flat G=PSL(2,R) connections on the surface. Higher Teichmüller space extends these notions to appropriate higher rank classical Lie groups G, and N=1 super Teichmüller space likewise studies the extension to the super Lie group G=OSp(1|2). In this talk, I will discuss the solution to the problem of producing Penner-type coordinates on super-Teichmüller space and its higher analogues. I will also talk about some applications of these coordinates.
April 1, 2022 - In person in UH 4010
Mark Pengitore (University of Virginia)
Coarse Embeddings and Homological Filling Functions
Abstract: In this talk, we will relate homological filling functions with coarse embeddings. We will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we can show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher-dimensional generalizations of our main result.
March 18, 2022
Yao Li (University of Massachusetts Amherst)
The Fokker-Planck equation: theory, computation, and data-driven method
Abstract: The time evolution of the probability density function of a stochastic differential equation is described by the Fokker-Planck equation, which has lots of applications in various areas. In this talk I will discuss some recent progress about the Fokker-Planck equation. On the theoretical side, I’ll introduce some refined estimates about how the invariant probability density function concentrates at the vicinity of the attractor. On the computational side, I will discuss a few different numerical methods that solve low (up to 4) and medium (up to 20) dimensional Fokker-Planck equations. Finally, I will introduce how to use a data-driven approach to study the speed of convergence of the Fokker-Planck equation to its steady state, and (if time permits) its application to the theoretical foundation of deep learning.
January 21, 2022
Yun Kang (Arizona State University)
How Does Mathematics Save Honeybees?
Abstract: The honeybee, Apis mellifera, is not only crucial in maintaining biodiversity by pollinating 85% plant species but also is the most economically valuable pollinator of agricultural crops worldwide with value between $15 and $20 billion annually as commercial pollinators in the U.S. Unfortunately, the recent sharp declines in honeybee population have been considered a global crisis. In this talk, we will demonstrate how we develop different types of mathematical trackable models to explore how the crucial feedback mechanisms linking disease, parasitism, nutrition, and foraging behavior might be responsible for the colony growth dynamics and survival in a dynamical environment. Rigorous mathematics from those models that use nonlinear nonautonomous and/or delayed differential equations within metapopulation frameworks have been integrated with data to explore the contributing factors to the mysterious and dramatic loss of honeybees as well as provide a basis for new strategies for controlling Varroa and reducing colony losses for beekeepers, and benefit land managers.
Fall Semester
December 10, 2021
Mingji Zhang (New Mexico Institute of Mining and Technology)
Qualitative properties of ionic flows via Poisson-Nernst-Planck models with nonzero
but small permanent charges and multiple cations
Abstract: A quasi-one-dimensional Poisson-Nernst-Planck system for ionic flow through a membrane channel is studied. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes and current-voltage relations explicitly, and this allows us to examine the small permanent charge effects on ionic flows in detail.
December 3, 2021
Radakrishnan 'Kit' Nair (University of Liverpool, UK)
On Weyl’s theorem on uniform distribution of polynomials
Suppose $P(x) = \alpha _k x^k + \ldots + \alpha _1 x + \alpha$ is a polynomial with one of $\alpha _k , \ldots , \alpha _1$ irrational. Suppose $( k_l )_{l\geq 1}$ is a class of highly regular sequences of natural numbers, to be specified. Then if a function $f: [0,1) \to \Bbb{C}$ is continuous, we have $$ \lim _{l\to \infty}{1\over l}\sum _{i=1}^{l}f(P({k_l})) = \int _0^1f(t)dt. $$ In the case of $k_l=l \ (l=1,2, \ldots )$ this is a seminal theorem of Herman Weyl, that underlies much of modern analytic number theory ergodic theory. We will prove this theorem and give some arithmetic applications.
November 5, 2021
Emil Straube (Texas A&M University)
d-bar methods in complex analysis
Abstract: In this lecture, we will indicate some typical applications of $\overline{\partial}$-methods in complex analysis of one and several variables. We start with the solution of the inhomogeneous $\overline{\partial}$-equation in a planar domain, which lends itself to a simple proof of the Mittag--Leffler theorem on holomorphic functions with prescribed poles. Next, we describe how one variable methods give the solution of the inhomogeneous Cauchy-Riemann equations in $\mathbb{C}^{2}$ when the right hand side has compact support, and how this simple observation already leads to a phenomenon that is radically different from the one variable case: holomorphic functions in certain domains automatically extend holomorphically to a bigger domain. This leads to the question of domains of existence of holomorphic functions, and we indicate how these domains are characterized by the solvability of the inhomogeneous $\overline{\partial}$- equations. Finally, we add some remarks about solving the inhomogeneous equations with regularity estimates up to the boundary (time permitting).
October 29, 2021
Hoi Nguyen (The Ohio State University)
Random matrices: universality of the spectrum and cokernels
Abstract: Random Matrix Theory is a rich area with many applications. In this talk I will give a brief survey on some recent developments in the area, focusing mainly on the universality aspect of the spectrum and cokernels.
October 22, 2021
Pablo Roldan (Yeshiva University)
Continuation of relative equilibria in the n–body problem to spaces of constant curvature
Abstract: The curved n–body problem is a natural extension of the planar Newtonian n–body problem to surfaces of non-zero constant curvature. We prove that all non-degenerate relative equilibria of the planar problem can be continued to spaces of constant curvature k, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. For example, we extend Lagrange’s triangle configuration with different masses to both positive and negative curvature spaces.
October 8, 2021
Vladimir Retakh (Rutgers)
Noncommutative versions of classical and semi-classical problems
Abstract: I will talk about the Gelfand program on noncommutative versions of some classical problems based on "down-to-Earth" approach. In particular, I will discuss factorizations of polynomials, noncommutative symmetric functions, noncommutative versions of Plücker coordinates, Ptolemy identity and noncommutative Laurent phenomenon for triangulated surfaces.
October 1, 2021
Jo Nelson (Rice)
Contact Invariants and Reeb Dynamics
Abstract: Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non- integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will also explain how to generalize Morse theoretic constructions using J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.