Past Seminars
Seminar: The Cauchy-Riemann Equations on Domains in the Complex Projective Space
Mei-Chi Shaw, University of Notre Dame
April 1st, 2025
11:00AM to 12:00PM
Armitage – 124
Abstract: The Cauchy-Riemann equations play central role in one and several complex variables. The Cauchy-Riemann operator ∂ has been studied extensively on domains in the complex Euclidean space Cⁿ. Much less is known when the ambient manifold is not Cⁿ.
In this talk, we discuss the range of ∂ on domains in the complex projective space CPⁿ. We also study the ∂-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for ∂ on pseudoconcave domains in CPⁿ. In particular, we show that ∂ does not have closed range in L² for (2,1)-forms on the Hartogs triangle in CP². This is in sharp contrast to ∂ on the Hartogs triangle in C², where L² results have long been established by Hörmander.
Seminar: Continuity Equations in Fibered Wasserstein Spaces – A common framework for meanfield and graphon dynamics
Beniot Bonnet-Weill, Chargé de Recherche CNRS (Junior Researcher)
March 25th, 2025
12:45pm to 1:45pm (Free Period)
Armitage – 124
Abstract: During the past fifteen to twenty years, the the concept of meanfield approximation has become one of the leading paradigms in the mathematical analysis of large multiagent systems. This prominence can be explained by its mathematical versatility, its modelling power, and its general amenability to various families of numerical schemes.
However, meanfield limits are, by essence, confined to operating at the level of homogeneous particle systems, wherein the dynamics of each agent only depends on purely spatial quantities (e.g. its own position and that of the others). In order to provide a macroscopic description of heterogeneous multiagent systems, a more recent trend has consisted in leveraging the concept of graph limit, introduced by Lovasz and Szegedy. These are quite natural and relatively easy to manipulate, albeit a bit rigid as they lead to considering ODEs in Lebesgue spaces, coined graphon dynamics.
In this ongoing work in collaboration with Nastassia Pouradier Duteil (INRIA, Sorbonne Université), we investigate a new class of evolutions taking the form of continuity equations over spaces of Young measures endowed with an adequate “fibered” Wasserstein metric. The main interest in doing so is that the latter combine some of the desirable features of both meanfield and graphon dynamics, while providing an embedding of both in natural limit cases. In this context, I will present the basics of all three models, discuss some of the topological properties of fibered Wasserstein distances, and expose Carathéodory and Cauchy-Lipschitz well-posedness results for the underlying dynamics.
Seminar: Nonlocal Systems of Hyperbolic Equations
Mauro Garavello, University of Milano Bicocca
March 13th, 2025
12:45pm to 1:45pm (Free Period)
Armitage – 121
Abstract: We consider a multi non-linear system of hyperbolic equation in conservation form with non-local terms in the flux functions. The non-local terms considered here are convolutions with smooth kernels.
The resulting model is of macroscopic type and is able to describe different behaviors typically emerging in population dynamics. Indeed different shapes of the kernel function and of the velocity vectors may result from example in an aggregation phenomenon, with the possible formation of clusters (or opinions), or in a segregation of the various populations.
An important role is played by the support of the kernel function, which corresponds to the visual range of the individuals. In this talk we present several numerical integrations together with its well posedness and some analytic qualitative properties. Moreover, we also discuss the coupling between local and non-local equations. This interplay is of a particular interest in the context of traffic flow, due to the presence of autonomous vehicles, aware of traffic conditions at a significant distance from their locations, and of standard vehicles, which typically behaves according to the traffic conditions at their locations.
These are joint works with Rinaldo M. Colombo (University of Brescia, Brescia, Italy) and Claudia Nocita (University of Milano Bicocca, Milano, Italy).
Seminar: Why Neural Networks find simple solutions
Benoit Dherin, Google
March 10th, 2025
11:20am to 12:20pm (Free Period)
Armitage – 124
Abstract: Despite their ability to model very complicated functions and equipped with enough parameters to grossly overfit the training dataset, overparameterized neural networks seem instead to learn simpler functions that generalize well. In this talk, we present the notions Implicit Gradient Regularization (IGR) and Geometric Complexity (GC), which shed light on this perplexing phenomenon. IGR helps to guide the learning trajectory towards flatter regions in parameter space for any overparameterized differentiable model. This effect can be derived mathematically using Backward Error Analysis, a powerful and flexible method borrowed from the numerics of ODEs. For neural networks, we explain how IGR translates to a simplicity bias measured by the neural network GC. We will also show how various common training heuristics put a pressure on the GC, creating a built-in geometric Occam’s razor in deep learning.
Seminar: Transforming STEM culture
Brigitte Lahme, Omayra Ortega and Aris Winger
Thursday, December 5th, 2024
12:45 PM – 1:45 PM (Free Period)
Zoom Meeting
The TIPS: Toward Justice pathway is a two-year departmental pathway that addresses persistent marginalization and under representation of Latine students in STEM. Originally developed at Sonoma State University with the support of the National Science Foundation, The pathway comprises of:
- workshops and exploration regarding factors contributing to Latine under representation in STEM, including stereotype threat and implicit bias
- introduction to culturally responsive pedagogies
- collaborative implementation of these practices in gateway STEM courses via lesson study
- review of institutional barries and STEM students’ connections to campus resources
- implementation of High Impact Practices to increase student sense of belonging in STEM fields.
The TIPS pathway is a significant undertaking on the part of a STEM department – with the potential for exceptional reward as we support many more of our students to thrive in our classrooms, departments and disciplines.
Math Seminar: Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging and Optimal Transport
Harlin Lee, University of North Carolina at Chapel Hill
Thursday, November 14th, 2024
12:45 PM – 1:45 PM (Free Period)
Armitage Hall, Room 121
Abstract: Capturing data from dynamic processes through cross-sectional measurements is seen in many fields such as computational biology. Trajectory inference deals with the challenge of reconstructing continuous processes from such observations. In this work, we propose methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is instrinsic to the Wasserstein space. Combining subdivision schemes with optimal transport-based geodesic, our methods carry out trajectory inference at a chosen level of precision and smoothness, and can automatically handle scenarios where particles undergo division over time. We rigorously evaluate our method by providing convergence guarantees and testing it on cell data characterized by bifurcations and merges, comparing its performance against state-of-the-art trajectory inference and interpolation methods. The results not only underscore the effectiveness of our method in inferring trajectories, but also highlight the benefit of performing interpolation and approximation that respect the inherent geometric properties of the data.
Math Seminar: Emerging Applications of Graphons in Dynamical Systems
Georgi Medvedev, Drexel University
Thursday, November 7th, 2024
12:45 PM – 1:45 PM (Free Period)
Armitage Hall – Room 121
Abstract: Natural and man-made networks, ranging from neuronal networks to power grids and social networks, can be effectively modeled by interacting dynamical systems on graphs. One of the key challenges in studying such models is handling network connectivity. The theory of graphons, originally developed for problems in discrete mathematics, offers natural and effective analytical tools for integrating network connectivity into dynamical models. The application of graphons has facilitated progress in understanding the dynamics of network models that were previously inaccessible to analysis.
In this talk, we review the elements of the theory of graphons that have proven useful for the mathematical analysis of dynamical systems. This will be followed by a discussion of recent results on synchronization and bifurcations in the Kuramoto model of coupled phase oscillators.
Math Seminar: Optimization in Bures-Wasserstein space via interacting particle systems
Giacomo Borghi, Heriot-Watt University, Edinburgh, UK
Thursday, October 31st, 2024
12:45 PM – 1:45 PM (Free Period)
Armitage Hall, Room 121
Abstract: Motivated by the development of computational algorithms dealing with probabilities, we present in this talk a novel optimization method based on Gaussian particles. The particles interact via a stochastic consensus-type dynamics exploiting the Riemannian structure of the Bures-Wasserstein space. We will also present an extension to the more general case of 2-Wasserstein space and compare the method with gradient flow dynamics in benchmark problems. This is joint work with M. Herty and A. Stavitskiy.
Math Seminar: Smart Data, Smarter Models: Enhancing the Predictive Power of Mathematical Models of Cancer
Jana Gevertz, The College of New Jersey
Tuesday, October 15th, 2024
11:30 AM – 12:30 PM
Joint Health Science Center – Room 104A
Abstract: Mathematical models are powerful tools that can vastly improve our understanding of cancer dynamics and treatment response. However, to be useful, experimental or clinical data are necessary to both train and validate such predictive models, and not all data are created equal. Here I present two methodologies that improve upon model-informed experimental design and model-based predictions. First, I will introduce a multi-objective optimization algorithm to identify combination protocols that maximize synergy from the perspective of both efficacy and potency (toxicity), while simultaneously reconciling sometimes contradictory assessments made by different synergy metrics. Second, using the notion of parameter identifiability, I will address the question of what is the minimal amount of experimental data that needs to be collected, and when it should be collected, to have confidence in a model’s predictions. Real-word applications of both methodologies will be presented.
Math Seminar: On the differentiation of integrals along filter
Fausto Di Biase, University of Chieti/Pescara
Thursday, May 21, 2024
12:00 – 1:00 PM
BSB Room 117
Abstract: I will describe the background of the theory of the differentiation of integrals, a topic that played a central role in some of the main developments in harmonic analysis in the previous century, and then present some recent results, obtained in collaboration with Steven G. Krantz.
Math Seminar: On the moving sofa problem
Jineon Baek, University of Michigan
Thursday, April 11, 2024
12:45 – 1:45 PM (Free Period)
Armitage Hall, Room 123
Abstract: Modeling the situation of moving furniture around, the moving sofa problem asks for the maximum area of a planar shape that can move around the corner in an L-shaped hallway of width 1. The problem was posed by Leo Moser in 1966, and the best known lower bound of 2.2195… was proved by Gerver in 1994, by constructing a sofa whose boundary consists of 18 special curves. While it is conjectured that Gerver’s sofa attains the maximum area, the best published upper bound of 2.37 was proved by Kallus and Romik in 2018 using computer assistance.
Improving upon the computer-assisted approach of Kallus and Romik, we improve the upper bound to 2.32. Moreover, without any computer assistance, we prove a conceptually new upper bound of $1 + \pi^2/8$ = 2.2337… that is much closer to the lower bound of Gerver, on a large subset of shapes which includes Gerver’s sofa. We also discuss the possibility of making the upper bound of 2.2337… unconditional by building upon the approaches of the two results.
Math Seminar: The polynomial method in the study of zero-sum theorems
Dr. Sukumar Das Adhikari
Friday, November 17, 2023
1:30 PM – 2:30 PM
BSB 132
We consider some elementary algebraic techniques in the area of Zero-sum Combinatorics. Originating from a beautiful theorem of Erdos-Ginzberg-Ziv about sixty years ago and some other questions around the same time, it has found various ramifications and generalizations, with many interesting results and several unanswered questions. It has ushered in many algebraic as well as combinatorial techniques which have found other applications also. In this talk, we shall report on applications of some elementary algebraic techniques in addressing some questions related to a weighted generalization.
Math Seminar: Infinite-dimensional Wishart processes
Dr. Sonja Cox, Korteweg-de Vries Institute for Mathematics, University of Amsterdam
Friday, October 20, 2023
11:20 AM – 12:20 PM
BSB 132
A Wishart process is a stochastic process $(X_t)_{t\geq 0}$ taking values in the space of positive semi-definite matrices such that $X_t$ has a (generalized) Wishart distribution for every $t\geq 0$. Wishart processes were introduced in the ’90s by Bru and have become a popular choice for modeling stochastic covariance. For example, Wishart processes are used in multi-dimensional Heston models to describe the instantaneous volatility of multiple assets. Models for energy and interest rate markets involve stochastic \emph{partial} differential equations, and thus call for infinite-dimensional covariance models. In our work, we introduce and analyze infinite-dimensional Wishart processes, and discuss some of their advantages and shortcomings.
Math Seminar: Central limit theorems in deterministic dynamical systems
Dr. Dalibor Volny
Monday, October 2, 2023
11:20 AM – 12:20 PM
BSB 108
We deal with central limit theorems in dynamical systems, i.e. for strictly stationary sequences of random variables. Methods that have been used work in dynamical systems of positive entropy only. By Burton and Denker, later by Volny (functional CLT) it has been proved that in any ergodic and aperioadic dynamical system, a CLT with convergence to a normal law (Brownian motion, in the functional case) exists. Here we deal with convergence towards stable laws (stable processes). The results are due to Zemer Kosloff and Dalibor Volny.
Math Seminar: The Cauchy-Riemann Equations on Hartogs Triangles
Dr. Mei-Chi Shaw, University of Notre Dame
Friday, April 28, 2023
11AM – 12PM
BSB 132
Also available on Zoom: https://tinyurl.com/9nrnveur
The Hartogs triangle in the complex Euclidean space is an important example in several complex variables. It is a bounded pseudoconvex domain with non-Lipschitz boundary. In this talk, we discuss the extendability of Sobolev spaces on the Hartogs triangle and show that the weak and strong maximal extensions of the Cauchy-Riemann operator agree (joint work with A. Burchard, J. Flynn and G. Lu). These results are related to the Dolbeault cohomology groups with Sobolev coefficients on the complement of the Hartogs triangle. We will also discuss some recent progress for the Cauchy-Riemann equations on Hartogs triangles in the complex projective space (joint work with C. Laurent- Thiébaut).
This seminar is held at Rutgers-Camden, as part of a joint seminar with the Complex Analysis and Geometry seminar at Rutgers-New Brunswick.
Math Seminar: Metropolized Hamiltonian Monte Carlo on highdimensional Gaussian Targets
Dr. Stefan Oberdörster, University of Bonn, Germany
Friday, March 24, 2023
12-1 PM in BSB132
In this talk, we will discuss recent developments regarding the convergence of Metropolis-adjusted Hamiltonian Monte Carlo on Gaussian target distributions. These targets stand out amongst the strongly log-concave distributions due to their analytical feasibility. Based on coupling techniques, we show contractivity in a specially designed Wasserstein distance. This yields upper mixing time bounds with respect to total variation that are dimensionally tight for cold start by a conductance argument.
Joint work with Nawaf Bou-Rabee (Rutgers) and Andreas Eberle (Bonn).
Math Seminar: Principled Mathematical Models for the Spotted Lanternfly Invasion
Dr. Benjamin Seibold, Temple University
Friday, February 17, 2023
12-1 PM in BSB132
Also available on Zoom: https://tinyurl.com/9nrnveur
The spotted lanternfly is an invasive species that is spreading in the Eastern United States. Introduced in 2014 to Eastern Pennsylvania, it has since spread within PA and to several adjacent states. Due to its ability to severely compromise lumber, grape, and crop production, it has been called “the worst invasive species to establish in the US in a century.” In this presentation we showcase our team’s efforts to produce principled models for the lanternfly life cycle and its dependence on climatic conditions, with the goal to generate quantitatively accurate predictions of the pest’s establishment potential across the country. In addition to intriguing mathematical models and challenging cross-disciplinary efforts on properly calibrating the models, this research also induces an intriguing need for specialized moving mesh methods. We showcase how biological properties like diapause manifest in a characteristic rank-1 structure of the population evolution operator, and highlight predictions on the pest’s future establishment, including how humans facilitate its spread.
Math Seminar: Weighted L² -estimates for ∂ and its applications
Dr. Song-Ying Li, University of California, Irvine
Friday, October 28th, 2022
11:30 AM – BSB116
Also available on Zoom: https://tinyurl.com/9nrnveur
Abstract: In this talk, I will introduce the Hörmander’s weighted L² estimates for Cauchy-Riemann operator and then present some applications which includes sharp pointwise estimate and uniform estimate for the canonical solution for Cauchy-Riemann equation ∂u = f on a classical bounded symmetric domain in Cn and productive domains. The second application is my recent work on applying the weighted L2 estimates to study the Corona problem in several complex variables.
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