The purpose of the Bridge to Research Seminar is to introduce students (primarily students early on in the doctoral program at Purdue) to the exceptionally wide variety of topics which are studied in the math department at Purdue. Each week, a professor gives a one-hour talk on their research. I organized this seminar with Liam Ashton from Fall 2024 to Spring 2025.
January 13 - NO SEMINAR
January 20 - NO SEMINAR (MLK DAY)
January 27 - Lvzhou Chen
Title: Residual properties of groups via geometry and topology
Abstract: The topology of manifolds in low dimensions are closely related to their geometry and fundamental groups. In this talk, we will discuss some classical theorems of Hall and Scott about residually finite (RF) and locally extended residually finite (LERF) properties of free groups and surface groups. In particular, we will see how these group-theoretic properties are related to the existence of certain finite covering spaces of graphs/surfaces, and how to use the geometric and topological point of view to prove such group-theoretic results.
February 3 - Alisa Sedunova
Title: A gentle introduction to analytic number theory via the Dirichlet divisor problem and the Gauss circle problem
Abstract: Gauss circle problem asks how many points with integer coordinates lie within a circle of radius x. Given that the area inside the circle is pi*x^2 , we expect this to be a rough estimate for the number of such points. The natural follow-up question is how accurate is this approximation? Similarly, the Dirichlet divisor problem asks for the best approximation to the number of points with integer coordinates under the hyperbola xy=c. While both of these problems remain unsolved, they received a lot attention of analytic number theorists and represent somewhat model problems for various modern techniques in the field. With this in mind we will introduce these techniques and aim at demonstrating at how they can be applied to these particular problems.
February 10 - Uli Walther
Title: Bernstein-Sato Polynomials
Abstract: The power rule, (x^s)'=s*x^{s-1}, is the simplest of a class of functional equations known as "Bernstein--Sato identities". In the talk, I explain them, and several aspects that are related to it. This shows a mix of analysis, topology, and algebra.
February 17 - NO SEMINAR (CANCELLED)
February 24 - Ilya Shkredov
Title: On some problems of additive combinatorics
Abstract: Additive combinatorics is a rapidly developing relatively new field of modern mathematics, lying between number theory and combinatorics. A variety of tools are used such as (in addition to number theory and combinatorics) dynamical systems, computer science, probability, geometry, algebra etc. Roughly speaking, additive combinatorics is the area that studies combinatorial problems that can be expressed through a group operation.
In our survey, we plan to present the fundamental results of the area and describe some connections of additive combinatorics with number theory, combinatorics, ergodic theory, graph theory, Fourier analysis, geometry, and other branches of mathematics.
March 3 - NO SPEAKER
March 10 - Samy Tindel
Title: Introducing rough paths theory
Abstract: This talk focuses on an important problem in stochastic calculus: how to define an integral with respect to a random forcing term, when this forcing term is not differentiable. I will briefly describe 3 settings: (1) Itô calculus for Brownian motion. (2) Young integration. (3) Rough paths setting. The third setting is the most general, and allows stochastic integration with a minimal amount of probability. I will try to advertise analytic and algebraic aspects of this beautiful theory, and will keep technical considerations to a minimum.
March 17 - NO SEMINAR (SPRING BREAK)
March 24 - NO SEMINAR
March 31 - Steven Dong
Title: Machine Learning Methods with Spectral-Like Accuracy
Abstract: In this talk I will provide a light introduction to a class of scientific machine learning methods based on randomized neural networks for numerical PDE problems, which demonstrate remarkable accuracy and computational efficiency.
April 7 - Xiangxiong Zhang
Title: Sobolev gradient flows for the ground state of Gross-Pitaevskii eigenvalue problem
Abstract: For the Schrodinger equation with a cubic nonlinearity modelling Bose-Einstein Condensation (BEC), the separation of variables gives the Gross-Pitaevskii eigenvalue problem. For computing the ground state of Gross-Pitaevskii eigenvalue problem, it is also the minimizer of the Gross-Pitaevskii energy, for which there exist a few simple and efficient numerical algorithms including Sobolev gradient flow induced by H1 norm and other norms. The Sobolev gradient flow methods can also be regarded as Riemannian gradient descent methods on unit sphere under different metrics. I will comment on provable convergence results for some of these algorithms and show how to efficiently implement a large problem like discretization on a 1000^3 grid for 3D problems on Nvidia GPU cards with easy coding in Matlab and Python.
April 14 - NO SEMINAR
April 21 - Manuel Rivera
August 26 - Alexandria Volkening
Title: Mathematics of cell behavior in biological patterns
Abstract: Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on two examples of such emergent behavior in biology, specifically cell interactions during pattern formation in zebrafish skin and tissue development in ferns. Different modeling approaches provide complementary insights into these systems and face different challenges. For example, vertex-based models describe cell shape, while more efficient agent-based models treat cells as particles. PDE models, which track the evolution of cell densities, are more amenable to analysis, but it is often difficult to relate their few parameters to specific cell interactions. In this talk, I will overview our models of cell behavior in biological patterns and discuss our ongoing work on quantitatively relating different types of models using topological data analysis and data-driven techniques. I’ll also highlight example thesis projects in data-driven mathematical biology.
September 2 - NO SEMINAR (LABOR DAY)
September 9 - Eric Samperton
Title: Tricolorings and dihedral colorings of knots and links
Abstract: I'll introduce some of the simplest invariants of knots and links that come from the dihedral group of order 6. But what does "simple" mean? The invariants are "simple" to define and (with a little algebraic topology) it is "simple" to prove they are invariants. However, only some of them are clearly "simple" to compute; I expect the rest of them are NP-hard to compute. I'll explain what this means, state the precise conjecture I've been thinking about, and ramble about my current struggles proving it.
September 16 - Margaret Thomas
Title: Definability and o-minimality
Abstract: Mathematical Logic is the study of the foundations of mathematics, and gives us formalized ways of considering which statements are true or provable about certain mathematical objects, as well as tools to exploit this perspective for new mathematical discoveries. One example is to consider the sets `definable' in a given mathematical structure, and try to understand their geometric/combinatorical/algebraic nature using information about the logical complexity of their definitions. This understanding can then be applied to mathematical problems where such sets naturally arise. In this talk, I will try to give some of the flavour of this perspective, focusing in particular on a class of structures that I work with whose definable sets are geometrically very well-behaved (i.e. non-pathological), called `o-minimal' structures. Time permitting, I'll try to discuss how this idea of definability has interacted with many other areas, such as diophantine geometry, combinatorics, classical topology, dynamical systems, machine learning and economic preference theory, several within my own research.
September 23 - Plamen Stefanov
Title: Inverse Problems: Between Pure and Applied Math
Abstract: I will introduce the notion of an inverse problem, which roughly speaking can be formulated as follows: If I give you the answer, can you tell me what the question was? More prosaically, inverse problems are reduced to solving certain equations where even the equation may not be explicit. I will present some classical examples, starting with the Radon Transform which is in the heart of Computed tomography (CT), performed more than 80 million times in the US per year. In 1979, Cormack and Hounsfield (but not Radon!) got a Nobel prize for the development of CT. In higher dimensions, CT scan is modeled by the inversion of the X-ray transform (in 2D, they are the same). I will discuss the Radon/X-ray Transforms with full and partial data, and its generalization to Riemannian geometry motivated by propagation of waves/rays in non-homogeneous media. Then I will discuss the non-linear Traveltime Tomography to recover, say a sound speed, from travel times of waves through inhomogeneous media. Its linearization is the geodesic X-ray transform. An important motivation to study this problem comes from geophysics – we want to recover the inner structure of Earth from times of travel of seismic waves measured on its surface. If time permits, I will discuss application to cloaking and cosmology. While those problems do not exhaust all possible inverse problems by far, my intention is to emphasize on so-called tomography problems as an introduction to the field and to connect them to problems in medical imaging and physics.
September 30 - Matt Novack
Title: The interplay of advection and diffusion (or why we stir cream into coffee)
Abstract: When we pour cream into a cup of coffee, why do we actively stir the mixture instead of waiting for the cream to naturally diffuse throughout the cup? The goal of this talk is to gain some understanding of how stirring and diffusion interact. We will do so via an analogy with a toy version of this stirring/diffusion problem, which may be formulated in the context of ordinary differential equations. Understanding the toy problem requires only basic linear algebra; in particular, no background knowledge of PDEs will be assumed.
October 7 - NO SEMINAR (FALL BREAK)
October 14 - Yuan Gao
Title: Macroscopic Dynamics for Nonequilibrium Chemical Reactions from a Hamiltonian Perspective
Abstract: Most biochemical reactions in living cells are an open system interacting with the environment through chemostats. At a mesoscopic scale, the molecular number of each species in those biochemical reactions can be modeled by the random time-changed Poisson processes. To characterize the macroscopic behaviors in the large volume limit, the law of large numbers in path space determines a mean-field limit nonlinear ODE. At the same time, the WKB expansion yields a Hamilton-Jacobi equation and the corresponding Lagrangian gives the good rate function in the large deviation principle (LDP). Rigorous proof can be done by recasting Varadhan's discrete nonlinear semigroup as a monotone scheme which approximates the limiting first-order Hamiltonian-Jacobi equations(HJE). The convergence of Varadhan's discrete nonlinear semigroup (the monotone scheme) to the continuous Lax-Oleinik semigroup yields the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered. Moreover, the LDP for invariant measures can be used to construct the global energy landscape for non-equilibrium reactions. This enables the dissipative-conservative decomposition for the Hamilton dynamics of chemical reactions, which also facilitates the understanding of thermodynamics. This global energy landscape is also proved to be a selected weak KAM solution to the corresponding stationary HJE. Optimal control method for rare events simulation in biochemistry will also be briefly reviewed.
October 21 - Daniel Le
Title: Pythagorean triples and reciprocity
Abstract: Reciprocity has been a central motif of modern algebraic number theory. We will give a brief overview of reciprocity over the last 400 years motivated by some concrete considerations with Pythagorean triples.
October 28 - Deepam Patel
Title: What is ... monodromy?
Abstract: I will begin with a discussion of monodromy as a natural obstruction to analytic continuation of complex analytic functions, and then discuss a more geometric/topological approach via sheaves which is then applicable in the setting of geometry objects in char. p. If there is time, I will explain how understanding of monodromy in geometric settings leads to applications in number theory and arithmetic geometry.
November 4 - Jianlin Xia
Title: Matrix Computations for Data Science
Abstract: The rapid development of the data science field provides significant new opportunities for numerical computations. Although there have been many successful data analysis methods, the studies of relevant matrices and the development of relevant fast and reliable matrix computations are typically overlooked.
In fact, data analysis techniques (especially neural network and machine learning methods) provide highly interesting new opportunities to perform matrix analysis and design new matrix algorithms. Some classical examples of scenarios where large and challenging matrices arise include the following.
1. In neural network approximations of functions, large mass matrices and Hessian matrices may be constructed from activation functions such as ReLU functions as basis functions.
2. Structured matrices have been often used in the design of effective neural networks, efficient training algorithms, and fast dimension reduction strategies.
3. In optimization and training algorithms such as natural gradient descent and batch normalization, the underlying matrices are often closely related to certain preconditioners.
In this talk, we will show some connections between several data science subjects and matrix computations and discuss how the two fields can interact and mutually benefit each other. This can potentially lead to highly interesting research subjects in both fields.
November 11 - Sasha Tsymbaliuk
Title: Shuffle Algebras
Abstract: Shuffle algebras have been used since 90s to study quantum groups. In the recent decade, this approach has been generalized to many more types of quantum groups of loop type, with a broad range of applications to: geometric representation theory, integrable systems, symmetric functions, and topological invariants. I will present an overview of some of the key aspects of the theory.
November 18 - Ben McReynolds
Title: Voyeuristic Mathematics - Group Act and We Watch
Abstract: For many, groups are unnatural algebraic objects to study and their utility is often not fully appreciated. Unlike many other objects in mathematics, groups are best studied through their actions on spaces. I will discuss some important examples of group actions in algebra, topology/geometry, algebraic geometry, and analysis. We will see that groups can be used to make important spaces, solve equations, and can be used to produce invariants of spaces. The focus will be largely on two seemingly unrelated topics in topology (covering space theory) and algebra (Galois theory).
November 25 - Tong Liu
Title: Fermat's Last Theorem and p-adic Galois representation
Abstract: In this talk, I will briefly review the history of Fermat's Last Theorem, particularly how FLT stimulated the development of algebraic number theory and arithmetic geometry. Ultimately, I will briefly discuss Wiles' strategy to prove FLT: using p-adic Galois representation to connect modular forms.
December 2 - Ning Wei
Title: The Impact of Ephaptic Coupling and Electrodiffusion on Arrhythmogenesis in the Heart
Abstract: Cardiac myocytes synchronize through electrical signaling to contract heart muscles, facilitated by gap junctions (GJs) in the intercalated disc (ID). GJs provide low-resistance pathways for electrical impulse propagation between myocytes, serving as the primary mechanism for electrical communication in the heart. However, research indicates that conduction can persist without GJs. For instance, GJ knockout mice still exhibit slow, discontinuous electrical propagation, suggesting alternative communication mechanisms. Ephaptic coupling (EpC) serves as an alternative way for cell communication, relying on electrical fields within narrow clefts between neighboring myocytes. Studies show that EpC can enhance conduction velocity (CV) and reduce conduction block (CB), especially when GJs are compromised. Reduced GJs and significant electrochemical gradients are prevalent in various heart diseases. However, existing models often fail to capture their combined influence on cardiac conduction, which limits our understanding of both the physiological and pathological aspects of the heart. Our study aims to address this gap by developing a two-dimensional (2D) multidomain electrodiffusion model that incorporates EpC. This is the first model to capture the dynamics of all ions across multiple domains, enabling us to reveal the impact of EpC in the heart. In particular, we investigated the interplay between ionic electrodiffusion and EpC on action potential propagation, morphology, electrochemical properties and arrhythmogenesis in both healthy and ischemic hearts. Our findings indicate that ionic electrodiffusion enhances CV and reduces CB under strong EpC. Specifically, the electrodiffusion of Ca$^{2+}$ and K$^+$ intensifies the effects of EpC on action potential morphology, whereas Na$^+$ diffusion mitigates these effects. Ionic electrodiffusion also facilitates action potential propagation into ischemic regions when EpC is substantial. Moreover, strong EpC can effectively terminate reentry, prevent its initiation, and lower the maximum dominant frequency (max DF), irrespective of GJ functionality. However, weak EpC may help counteract proarrhythmic effects when GJ coupling is slightly to moderately reduced, contributing to the stabilization of conduction patterns. Additionally, strong EpC notably alters ionic concentrations in the cleft, significantly increasing [K$^+$] and nearly depleting [Ca$^{2+}$], while causing moderate changes in [Na$^+$]. This multidomain electrodiffusion model sheds light on the mechanisms of EpC in the heart.