Colloquium Archives for Fall 2023 to Spring 2024
Predictive Science and Deep Learning - A Bright Future or an Odd Couple?
When: Wed, September 20, 2023 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Wolfgang Dahmen (Aachen, University of South Carolina) - https://sc.edu/study/colleges_schools/artsandsciences/mathematics/our_people/directory/dahmen_wolfgang.php
Abstract: Modern machine learning methodologies appear to exert a
transformative impact on society and science, especially in “Big Data”
application scenarios. While these applications are typically error-tolerant
this talk focuses on a rigorous accuracy quantification when trying to to use
machine learning tools to recover “physical states of interest” from several
sources of incomplete information, e.g. in terms of observational data and
governing possibly deficient physical laws. The latter “background model” is
typically given as a parameter dependent family of PDEs. Efficient “forward
exploration” of corresponding solution manifolds as well as related inverse
tasks, like state or parameter estimation, hinge on “learning” efficient and
certifiable surrogates for the parameter-to-solution map. We highlight
intrinsic challenges and discuss conceptual pathways exploiting the role of
stable variational formulations of the governing PDEs as well as tailored
optimization strategies for training on variationally correct residual loss
functions.
The optimal paper Moebius band
When: Fri, September 29, 2023 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Richard Schwartz (Brown University) - https://www.math.brown.edu/reschwar/
Title: The optimal paper Moebius band
Abstract: Ever since the last ice age, when children
wandered out of their frozen caves and made Moebius
bands from strips of paper, humans have wondered
how short a strip of paper they could use to make such
Moebius bands. In this talk I will give a hands-on and
elementary account of my recent solution of the
optimal paper Moebius band conjecture of B. Halpern and
C. Weaver from 1977. My result is that a unit width strip
of paper needs to be more than sqrt(3) units long in order
for you to be able to twist it up into a paper Moebius band,
and the bound is sharp.
Riehl (TBA)
When: Fri, October 6, 2023 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Emily Riehl (Johns Hopkins University) - https://math.jhu.edu/~eriehl/
Abstract: TBA
Categorification and geometry
When: Fri, October 13, 2023 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Lars Hesselholt (Nagoya University) - https://www.math.nagoya-u.ac.jp/~larsh/
Abstract: The key principle in Grothendieck's algebraic geometry is that every commutative ring be considered as the ring of functions on some geometric object. Clausen and Scholze have introduced a categorification of algebraic and analytic geometry, where the key principle is that every stable dualizably symmetric monoidal infinity-category be considered as the infinity-category of quasi-coherent modules on some geometric object. In this talk, I will explain this shift in paradigm as well as Clausen's philosophy that *every* cohomology theory should arise from this picture, complete with a six-functor formalism of categories of coefficients. The Hahn-Raksit-Wilson even filtration and Efimov continuity are key players in this picture.
Mathematics Around the Heisenberg Group
When: Thu, October 26, 2023 - 3:45pmWhere: Kirwan Hall 3206
Speaker: Roger Howe (Yale University) - https://www.norbertwiener.umd.edu/fft/2023/Speakers/Roger_Howe.html
Abstract: The Heisenberg group is the group-theoretic embodiment of the Canonical Commutation Relations (CCR) of quantum mechanics, formulated by Werner Heisenberg just under a century ago. In the intervening years, the CCR and the Heisenberg group have come to be seen as a central nexus in mathematics, connecting and unifying many seemingly disparate phenomena, in harmonic analysis, partial differential equations, operator theory, mathematical physics, Lie theory, representation theory, classical invariant theory, number theory, and more. This talk will survey topics where the Heisenberg group plays an important role, with an emphasis on the connections. This talk is part of the FFT 2023 conference, and it is given by the Norbert Wiener Center Distinguished Lecturer.
Decoding Time's Mysteries for Better Predictions
When: Thu, October 26, 2023 - 6:45pmWhere: Kirwan Hall 3206
Speaker: James Howard (Johns Hopkins University) - https://www.norbertwiener.umd.edu/fft/2023/Speakers/James_Howard.html
Abstract: In our data-driven world, making sense of complex information is paramount. Data comes in various shapes and forms, from healthcare to finance, but perhaps none as intricate as time-series data. How can we unravel the underlying stories that this sort of data tells us? In this talk, we will journey through the cross-disciplinary avenues of harmonic analysis, survival models, and machine learning to answer this question. By taking a closer look at signature methods and rough paths, we will explore how mathematics not only dissects the intricacies of time-series data but also enhances our understanding of stochastic processes. These advancements have unprecedented applications in predicting vital healthcare outcomes and beyond. Drawing upon the latest research, including my own, I will illustrate how this mathematical framework provides a robust and efficient way to revolutionize predictive models. The aim is to bring together insights from various fields to show how an enriched mathematical understanding can lead to practical, real-world applications that can potentially save lives. This Keynote talk is part of the FFT 2023 conference.
A tale of two invariants
When: Wed, November 15, 2023 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Paul Feehan (Rutgers) - https://sites.math.rutgers.edu/~feehan/
Abstract:
In 1994, Edward Witten published his celebrated formula that expressed the countably many Donaldson invariants of a smooth 4-manifold in terms of its finitely many integer Seiberg-Witten invariants. Witten used ideas from theoretical physics to predict his formula, subsequently generalized by him and Gregory Moore (1997). Later in 1994, Victor Pidstrigatch and Andrei Tyurin proposed a mathematical program to give a rigorous proof of Witten’s formula, entirely within the realm of classical Yang-Mills gauge theory. This program was developed by Thomas Leness and the speaker, leading to a complete proof of Witten’s formula based on moduli spaces of non-Abelian monopoles to connect moduli spaces of anti-self-dual Yang-Mills connections, which define Donaldson invariants, and moduli spaces of Seiberg-Witten monopoles, which define Seiberg-Witten invariants.
In this talk, we will review this paradigm and also explain how an analogous idea may provide a similar geometric explanation of the mysterious Gopakumar-Vafa formula for Gromov-Witten invariants of symplectic 6-manifolds and the integrality and finiteness of the BPS states, results that were recently proved by Ionel & Parker (2018) and by Doan, Ionel, & Walpuski (2021).
Using logic to study homeomorphism groups
When: Wed, November 29, 2023 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Thomas Koberda (University of Virginia) - https://sites.google.com/view/koberdat
Abstract: I will describe some recent results on the first order rigidity of homeomorphism groups of compact manifolds, and their applications to dynamics of group actions on manifolds. I will also describe how to find "syntactic" invariants of manifolds, and how these can be used to give a conjectural model-theoretic characterization of the genus of a surface.
Generative Models for Implicit Distribution Estimation: a Statistical Perspective
When: Thu, January 25, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Yun Yang (University of Illinois Urbana-Champaign) - https://sites.google.com/site/yunyangstat/
Abstract: The estimation of distributions of complex objects from high-dimensional data with low-dimensional structures is an important topic in statistics and machine learning. Modern generative modeling techniques accomplish this by encoding and decoding data to generate new, realistic synthetic data objects, including images and texts. A key aspect of these models is the extraction of low-dimensional latent features, assuming the data lies on a low-dimensional manifold. Our study develops a minimax framework for distribution estimation on unknown submanifolds, incorporating smoothness assumptions on both the target distribution and the manifold. Through the perspective of minimax rates, we examine some existing popular generative models, such as variational autoencoders, generative adversarial networks, and score-based generative models. By analyzing their theoretical properties, we characterize their statistical capabilities in implicit distribution estimation and identify certain limitations that could lead to potential improvements.
Video Imputation and Prediction Methods with Applications in Space Weather
When: Tue, January 30, 2024 - 4:00pmWhere: Kirwan Hall 3206
Speaker: Yang Chen (University of Michigan) - https://yangchenfunstatistics.github.io/yangchen.github.io/
Abstract:
The total electron content (TEC) maps can be used to estimate the signal delay of GPS due to the ionospheric electron content between a receiver and a satellite. This delay can result in a GPS positioning error. Thus, it is important to monitor and forecast the TEC maps. However, the observed TEC maps have big patches of missingness in the ocean and scattered small areas on the land. Thus, precise imputation and prediction of the TEC maps are crucial in space weather forecasting.
In this talk, I first present several extensions of existing matrix completion algorithms to achieve TEC map reconstruction, accounting for spatial smoothness and temporal consistency while preserving essential structures of the TEC maps. We show that our proposed method achieves better reconstructed TEC maps than existing methods in the literature. I will also briefly describe the use of our large-scale complete TEC database. Then, I present a new model for forecasting time series data distributed on a matrix-shaped spatial grid, using the historical spatiotemporal data and auxiliary vector-valued time series data. Large sample asymptotics of the estimators for both finite and high dimensional settings are established. Performances of the model are validated with extensive simulation studies and an application to forecast the global TEC distributions.
Arboreal Galois groups: an introduction
When: Wed, February 7, 2024 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Robert Benedetto (Amherst College) - https://rlbenedetto.people.amherst.edu/
Abstract: Let \(K\) be a field, over the field of rational numbers. Let \(f(z)\in
K[z]\) be a polynomial of degree \(d\geq 2\) with coefficients in \(K\), and let
\(x_0\in K\).
The roots of \(f^n(x)-x_0\) are the iterated preimages of \(x_0\) under
\(f\), and together they have the natural structure of a \(d\)-ary
rooted tree \(T\).
Thus, the Galois groups of the equations \(f^n(z)=x_0\) are known as arboreal
Galois groups, because they act as automorphisms of this tree. Our focus in
this talk will be on cases when the arboreal Galois groups are strictly smaller
than the full automorphism group of \(T\), which can happen when the dynamical
orbits of the critical points of \(f\) have certain special properties.
Higher theta series
When: Wed, February 28, 2024 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Zhiwei Yun (MIT) - https://math.mit.edu/~zyun/
Abstract: Theta series play an important role in the classical
theory of modular forms. In the modern language of automorphic
representations, they are constructed from a pair of groups G and
H (one orthogonal and one symplectic, or both unitary groups) and
the remarkable Weil representation of G×H. Kudla introduced
an analogue of theta series in arithmetic geometry, by forming a
generating series of algebraic cycles on Shimura varieties. The
arithmetic theta series has since become a very active program.
In joint work with Tony Feng and Wei Zhang, we consider analogues of
arithmetic theta series over function fields, and try to go further
than what was done over number fields. Our work concentrated on
unitary groups. We defined a generating series of algebraic cycles on
the moduli stack of unitary Drinfeld Shtukas (called the higher theta
series). We made the Modularity Conjecture: the higher theta series
is an automorphic form valued in a certain Chow group. This is a
function field analogue of the special cycles generating series
defined by Kudla and Rapoport, but with an extra degree of freedom
namely the number of legs of the Shtukas.
One concrete formula we proved was a higher derivative version of the
Siegel-Weil formula. It is an equality between degrees of
0-dimensional special cycles on the moduli of unitary Shtukas and
higher derivatives of the Siegel-Eisenstein series of another unitary
group. More recently, we have obtained a proof of a weaker version of
the Modularity Conjecture, confirming that the cycle class of the
higher theta series (valued in the cohomology of the generic fiber) is
automorphic.
The series of talks will feature a colloquium-style introduction to
some representation-theoretic and geometric background (the second talk), the other
two being more technical talks in which I will explain some ingredients in
the proofs of the higher Siegel-Weil formula and the weak Modularity
Conjecture.
Random lattices and their applications in number theory, geometry and statistical mechanics
When: Fri, March 1, 2024 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Jens Marklof (School of Mathematics, University of Bristol) - https://www.bristol.ac.uk/people/person/Jens-Marklof-6eb63e14-a018-4833-9cf8-b95272b5a09e/
Abstract: Lattices are fundamental objects in physics, mathematics and computer science. Starting from a cubic lattice, say, we can perturb the structure by linear transformations (shearing, stretching, rotating) to obtain a whole family of lattices. I will discuss the resulting "space of lattices", the dynamics of group actions on this space, natural probability measures, as well as some fascinating applications to long-standing problems in various areas of mathematics and mathematical physics. My plan is to tell you about kinetic transport in crystals and quasicrystals (the Lorentz gas), pseudo-random properties of simple
arithmetic sequences, knapsack problems, diameters of random Cayley graphs and (time permitting) subtle lattice point counting problems in hyperbolic geometry.
Bio: Jens Marklof is Professor of Mathematical Physics at the University of Bristol, specialising in dynamical systems and ergodic theory, quantum chaos, and the theory of automorphic forms. Marklof received his PhD in 1997 from the University of Ulm, and held research fellowships at Princeton University, Hewlett-Packard, the Isaac Newton Institute in Cambridge, the Institut des Hautes Etudes Scientifique and the Laboratoire de Physique Theorique et Modeles Statistiques near Paris. He delivered a plenary address at the
International Congress of Mathematical Physics in Prague 2009, and was an invited section speaker at the International Congress of Mathematicians in Seoul 2014. Major awards include a 2010 LMS Whitehead Prize and a five-year ERC Advanced Grant. In 2015 Marklof was elected a Fellow of the Royal Society, the UK's national academy of sciences. From November 2023 he will serve a two-year term as President of the London Mathematical Society.
TBA
When: Thu, March 14, 2024 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Svetlana Jitomirskaya (University of California, Berkeley) - https://math.berkeley.edu/people/faculty/svetlana-jitomirskaya
Instantaneous everywhere-blowup of parabolic stochastic PDEs
When: Wed, April 3, 2024 - 3:15pmWhere: Kirwan Hall 3206
Speaker: Davar Khoshnevisan (University of Utah) - http://www.math.utah.edu/~davar/
Abstract: We consider a one-dimensional heat equation of the form $\partial_t u = \partial^2_x u + b(u) + \sigma(u)\dot{W}$, where the forcing term $\dot{W}$ is space-time white noise. We survey aspects of a long history of the study of blowup for this family of heat equations, and conclude with recent joint work with Mohammud Foondun and Eulalia Nualart on instantaneous, everywhere blowup which seems to be a novel property. Time permitting, we might mention also aspects of the proof of the latter results, especially as they relate to new developments in the ergodic theory of parabolic stochastic partial differential equations, developed by Le Chen, the speaker, David Nualart, and Fei Pu (2021, 2022).
This is based on joint work with Mohammud Foondun and Eulalia Nualart.