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Texas A&M University
Mathematics

Events for 01/28/2025 from all calendars

Combinatorial Algebraic Geometry

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Time: 12:45PM - 1:35PM

Location: Bloc 302

Speaker: Frank Sottile, Texas A&M University

Title: Computing Schubert Problems I

Abstract: This is the first in a series of informal discussions about Grassmannians and flag varieties, with the goal of describing both how to represent Schubert problems on a computer and some goals of this computational study.

An outline of the discussions as they are given is at https://franksottile.github.io/talks/25/SchubertProblems.txt


Nonlinear Partial Differential Equations

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Time: 2:00PM - 3:00PM

Location: BLOCKER 302

Speaker: Thomas Chen, University of Texas Austin

Title: Explicit construction of global minimizers and the interpretability problem in Deep Learning

Abstract: Deep Learning (DL) as a core subfield of Machine Learning and Artificial Intelligence is at the center of extraordinary technological progress. However, despite of remarkable advances in applications and theoretical analysis, the conceptual and rigorous reasons for the functioning of DL networks are at present not clearly understood (the problem of “interpretability"). In this talk, we present some recent results aimed at the rigorous mathematical understanding of how and why supervised learning works. For underparametrized DL networks, we explicitly construct global, zero loss cost minimizers for sufficiently clustered data. In addition, we derive effective equations governing the cumulative biases and weights, and show that gradient descent corresponds to a dynamical process in the input layer, whereby clusters of data are progressively reduced in complexity ("truncated") at an exponential rate that increases with the number of data points that have already been truncated. For overparametrized DL networks, we prove that the gradient descent flow is homotopy equivalent to a geometrically adapted flow that induces a (constrained) Euclidean gradient flow in output space. If a certain rank condition holds, the latter is, upon reparametrization of the time variable, equivalent to simple linear interpolation. This in turn implies zero loss minimization and the phenomenon known as "neural collapse”. A majority of this work is joint with Patricia Munoz Ewald (UT Austin).


Nonlinear Partial Differential Equations

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Time: 3:00PM - 4:00PM

Location: Blocker 302

Speaker: Abed ElRahman Hammoud, Princeton University

Title: Artificial Intelligence for Downscaling: Application to Uncertain Chaotic Systems

Abstract: Reliable high-resolution state estimates for forecasts and reanalyses are pivotal in environmental applications, particularly in ocean and atmospheric sciences. These are typically achieved by integrating observational data into dynamical models through processes such as data assimilation (DA), when enhancing the reliability of forecasts and reanalysis, or downscaling when bridging the gap between coarse-scale observations and fine-scale information. Current DA and downscaling techniques rely on limiting assumptions and tend to be computationally demanding, especially in the presence of observational and model uncertainties. Artificial intelligence (AI) emerges as a powerful avenue for developing efficient data-driven tools that enhance reliability and alleviate computational demands of conventional DA and downscaling algorithms. This talk aims to present recent developments in AI tools that address challenges pertaining to downscaling with application to chaotic dynamical systems, and within an uncertain framework. The state-of-the-art dynamical downscaling algorithm, Continuous data assimilation (CDA), and its discrete-in-time counterpart (DDA) are first explored in the setting involving observational errors. Since CDA relies on an abstract lifting function called the determining form map, a physics-informed deep neural network (PI-DNN) named CDAnet is proposed to approximate this intractable mapping. CDAnet is then evaluated under observational and model uncertainties in application to the Rayleigh-Benard convection problem, validating and further extending upon the knowledge from theory.


Departmental Colloquia

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Time: 4:00PM - 5:00PM

Location: Bloc 117

Speaker: Anya Katsevich

Title: Anya Katsevich