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Computational and Applied Mathematics: Jaeyong Lee, Chung-Ang University, Real-Time Solutions to PDEs with Neural Operators in SciML

SAS 4201

Recent advancements in deep learning have led to a surge in research focused on solving scientific problems under the "AI for Science." Among these efforts, Scientific Machine Learning (SciML) aims to address domain-specific data challenges and extract insights from scientific datasets through innovative methodological solutions. A particularly active area within SciML involves using neural operators…

Geometry and Topology Seminar: Saman Habibi Esfahani, Duke, Gauge theory beyond low dimensions: progress on compactness conjectures

SAS 4201

(Based on joint work with Yang Li). I will present an overview of Donaldson's program to extend the methods of gauge theory and Floer homology from 3- and 4-manifolds to higher dimensions, with a focus in this talk on Calabi-Yau 3-folds. After discussing the background material in gauge theory and Calabi-Yau geometry, I will highlight…

Algebra and Combinatorics Seminar: Rafael S. González D’León, Loyola University Chicago, On Whitney numbers of the first and second kind, or is it the other way around?

SAS 4201

The Whitney numbers of the first and second kind are a pair of poset invariants that are relevant in various areas of mathematics. One of the most interesting appearances of these numbers is as the coefficients of the chromatic polynomial of a graph. They also appear as counting regions in the complement of a real…

Differential Equations and Nonlinear Analysis Seminar: Emilio Vilches Gutiérrez, Universidad de O’Higgins, Chile, Recent Developments in Moreau’s Sweeping Processes

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The sweeping process is a first-order differential inclusion involving the normal cone to a family of moving sets. It was introduced by J.J. Moreau in the early seventies to address an elastoplastic problem. Since then, it has been used to model constrained dynamical systems, nonsmooth electrical circuits, crowd motion, mechanical problems, and other applications. The…

Computational and Applied Mathematics: Shriram Srinivasan, Los Alamos National Laboratory, Hierarchical Network Partitioning for Efficient Solution of Steady-State Nonlinear Network Flow Equations

SAS 4201

Natural gas production and distribution in the US is interconnected continent-wide, and hence the simulation of fluid flow in pipeline networks is a problem of scientific interest. While the problem of steady, unidirectional flow of fluid in a single pipeline is simple, it ceases to be so when we consider fluid flow in a large…

Colloquium: Stefan Steinerberger, University of Washington, Growth Models in the Plane

SAS 4201

We'll study the growth of (two-dimensional) things. Think about lichen growing on a tree (tends to be sort of round). Another fun example is electricity propagating through wood (tends to be sort of fractal). A famous and still very mysterious model is called DLA: it forms the most beautiful fractal patterns (pictures will be provided).…

Geometry and Topology Seminar: Thomas Weighill, The University of North Carolina at Greensboro, Coarse Embeddings into Hilbert Space for Spaces of Point Clouds

SAS 4201

Given a space X, one may want to know if it can be embedded into a vector space in a controlled way. Interestingly, this question is of interest in both abstract mathematics (for example, it has implications for the Novikov Conjecture) and in data science, where such embeddings are a necessary preprocessing step to traditional…

Algebra and Combinatorics Seminar: Erik Bates, NC State, A new combinatorial interpretation of the (sum of (generalized)) Fibonacci numbers

SAS 4201

The sum of Fibonacci numbers, i.e. the sequence 2, 4, 7, 12, 20, 33, 54, 88, ... has many combinatorial interpretations.  For instance, the n-th term in this sequence is the number of length-n binary strings that avoid 001.  In this talk, I will describe a related (but to my knowledge, new) interpretation: given a length-3 binary…

Differential Equations and Nonlinear Analysis Seminar: Stefania Patrizi, The University of Texas at Austin, The Discrete Dislocation Dynamics of Multiple Dislocation Loops

SAS 4201

We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls–Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in $\mathbb R^n$, $n\geq 2$. After suitably rescaling the equation with a small phase parameter $\epsilon>0$, the rescaled solution solves a fractional Allen–Cahn equation. We show that,…

Teaching and Learning Seminar:Megan Ryals, University of Virginia,Integrating Research and Practice: The Development of Technology-Based Learning Activities in Multivariable Calculus

SAS 4201

Students often perceive Multivariable Calculus as a collection of disconnected ideas and approach problem solving in the course formulaically.  Students need help from instructors in developing spatial reasoning and making connections between symbolic computations and graphical representations. In this talk I will share how I integrated results of research on students’ learning of differential calculus…

Computational and Applied Mathematics: Anuj Abhishek, Case Western Reserve University, Operator Learning for Inverse Problems

SAS 4201

Neural operators such as Deep Operator Networks (DeepONet) and Convolutional Neural Operators (CNO) have been shown to be fairly useful in approximating an operator between two function spaces. In this talk, we will briefly review two inverse problems that arise in Medical Imaging, namely EIT and QPAT. We will also describe the relevant operator learning architectures.…

Stochastics/Discrete Analysis Seminar: Lechao Xiao, Google DeepMind, Harmonic Analysis and Theory of Deep Learning

SAS 4201

The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles underlying the success of deep learning, the core technology behind these breakthroughs, is still far from well-understood. In this presentation, I will share some interesting…