Colloquium Archives for Fall 2022 to Spring 2023

Unramified correspondance and virtual homology of mapping class groups

When: Wed, September 7, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Vlad Markovic (University of Oxford),

Abstract: I shall discuss my recent work showing that the Bogomolov-Tschinkel universality conjecture holds if and only if the mapping class groups of a punctured surface is large (which is essentially the negation of the Ivanov conjecture about the mapping class groups). I will also discuss my recent work with O. Tosic regarding the closely related Putman-Wieland conjecture.

Maps between configuration spaces and moduli spaces

When: Wed, September 28, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Lei Chen (University of Maryland) -
Abstract: In this talk, I will survey what is known or conjectured about maps between configuration spaces and moduli spaces of surfaces. Among them, we consider two categories: continuous maps and also holomorphic maps. We will also talk about some results and conjectures about maps between finite covers of those spaces. Unlike the case of the whole spaces, much less is known about their finite covers.

(Madry FFT) TBA

When: Fri, October 7, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Aleksander Madry (MIT) -
Abstract: TBA


When: Wed, October 12, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Jacob Bedrossian (University of Maryland) -

Deterministic surface growth models

When: Wed, October 19, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Takis Souganidis (The University of Chicago) -
Abstract: This talk is about the asymptotic behavior of large classes of (hyperbolically and parabolically) scaled deterministic surface growth models that are monotone and equivariant under translations by constants, The limits are solutions of degenerate elliptic partial differential equations which typically are discontinuous in some gradient directions consistent with Finsler metrics, such as the crystalline infinity Laplacian.

Finite quotients of 3-manifold groups

When: Wed, November 2, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Melanie Wood (Harvard University) -
Abstract: It is well-known that for any finite group G, there exists a closed 3-manifold M with G as a quotient of the fundamental group of M. However, we can ask more detailed questions about the possible finite quotients of 3-manifold groups, e.g. for G and H_1,...,H_n finite groups, does there exist a 3-manifold group with G as a quotient but no H_i as a quotient? We answer all such questions. To prove non-existence, we prove new parity properties of the fundamental groups of 3-manifolds. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the fundamental group of a random 3-manifold, in the sense of Dunfield-Thurston. This is joint work with Will Sawin.

Inference and Uncertainty Quantification for Low-Rank Models

When: Wed, November 30, 2022 - 11:30am
Where: Kirwan Hall 3206
Speaker: Yuling Yan (Princeton University) -
Abstract: Many high-dimensional problems involve reconstruction of a low-rank matrix from highly incomplete and noisy observations. Despite substantial progress in designing efficient estimation algorithms, it remains largely unclear how to assess the uncertainty of the obtained low-rank estimates, and how to construct valid yet short confidence intervals for the unknown low-rank matrix.  

In this talk, I will discuss how to perform inference and uncertainty quantification for two widely encountered low-rank models: (1) noisy matrix completion, and (2) heteroskedastic PCA with missing data. For both problems, we identify statistically efficient estimators that admit non-asymptotic distributional characterizations, which in turn enable optimal construction of confidence intervals for, say, the unseen entries of the low-rank matrix of interest.  Our inferential procedures do not rely on sample splitting, thus avoiding unnecessary loss of data efficiency. All this is accomplished by a powerful leave-one-out analysis framework that originated from probability and random matrix theory. 

This is based on joint work with Yuxin Chen, Jianqing Fan and Cong Ma.

On the boundary regularity of holomorphic mappings of positive codimension

When: Wed, November 30, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Nordine Mir (Texas A&M University at Qatar) -
Abstract: We shall start by discussing the known theory (as well as open questions) regarding the boundary regularity of biholomorphic mappings between smoothly
bounded domains in multidimensional complex euclidean spaces. We shall then
shift to the same problem for proper holomorphic mappings between domains in
complex spaces of different dimension, or the localized version for germs of CR
maps, where much less is understood. In this regard, we shall describe recent
results (with B. Lamel) showing how the existence of irregular CR maps between
smooth CR manifolds impacts the CR geometry of the target manifold. Our new
techniques open the way to prove general (boundary) regularity results for map-
pings between domains with degenerate boundaries, and also generalize results
from the literature for non degenerate boundaries.

Revisiting latent variable models from a deep learning perspective

When: Thu, December 1, 2022 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Xiao Wang (Purdue University ) -
Abstract: Latent variable models are an indispensable and powerful tool for uncovering the hidden structure of observed data. The well-known latent variable models in statistics include linear mixed models and Gaussian mixture models. Many powerful generative models in machine learning also belong to latent variable models. Existing methods often suffer from two major challenges in practice: (a) a proper latent variable distribution is difficult to specify; (b) making an exact likelihood inference is formidable due to the intractable computation. We propose a new framework for the inference of latent variable models that overcomes these two limitations. This new framework allows for a fully data-driven latent variable distribution via deep neural networks, and the proposed stochastic gradient method, combined with the Langevin algorithm, is efficient and suitable for complex models and big data. We provide theoretical results for the Langevin algorithm and establish the convergence analysis of the optimization method. This framework has demonstrated superior practical performance through numerical studies. Some other related research topics will also be presented.

Essential dimension and prismatic cohomology

When: Wed, December 7, 2022 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Mark Kisin (Harvard University) -
Abstract: The smallest number of parameters needed to define an algebraic covering space is called its essential dimension. Questions about this invariant go back to Klein, Kronecker and Hilbert and are related to Hilbert's 13th problem. In this talk, I will give a little history, and then explain a new approach which relies on recent developments in p-adic Hodge theory.

Adaptive variational Bayes: Optimality, computation and applications

When: Thu, December 8, 2022 - 3:30pm
Where: Physics 1204
Speaker: Lizhen Lin (The University of Notre Dame) -
Abstract: In this talk, I will discuss adaptive statistical inference based on variational Bayes. Although a number of studies have been conducted to analyze theoretical properties such as posterior contraction properties of variational posteriors, there is still a lack of general and computationally tractable variational Bayes methods that can achieve adaptive inference. To fill this gap, we propose a novel adaptive variational Bayes framework, which can operate on a collection of models. The proposed framework first computes a variational posterior over each individual model separately and then combines them with certain weights to produce a variational posterior over the entire model. It turns out that this combined variational posterior is the closest member to the posterior over the entire model in a predefined family of approximating distributions. We show that the proposed variational posterior achieves optimal contraction rates adaptively under very general conditions and attains model selection consistency when the true model structure exists. We apply the general results obtained for the adaptive variational Bayes to a large class of statistical models including deep learning models and derive some new and adaptive inference results.

Statistical Methods for Observational Data on Infectious Diseases

When: Mon, December 12, 2022 - 3:30pm
Where: Physics 1204
Speaker: Fan Bu (UCLA) -
Abstract: Emerging modern datasets in public health call for development of innovative statistical methods that can leverage complex real-world data settings. We first discuss a stochastic epidemic model that incorporates contact tracing data to make inference about transmission dynamics on an adaptive contact network. An efficient data-augmented inference scheme is designed to accommodate partially epidemic observations. This networked epidemic model allows flexible extensions to account for individual heterogeneity, disease latency and social interventions to help bring new epidemiological insights. We then discuss a collaborative work with the US FDA to improve post-market vaccine safety surveillance procedures. We propose a Bayesian statistical framework to tackle the challenge of sequentially analyzing observational healthcare data to detect vaccine adverse events. This new framework is substantially more flexible as it does not require a pre-specified analysis schedule in the standard procedure. It also adaptively corrects for bias in observational healthcare data to improve the quality of data-driven decision-making by reducing decision error.

Recognizing groups and fields in Erdős geometry and model theory

When: Fri, January 27, 2023 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Artem Chernikov (UCLA) -
Abstract:  Erdős-style geometry is concerned with combinatorial questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be typically viewed as asking for the possible number of intersections of a given (semi-)algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. It turns out that techniques from model theory are very useful in recognizing these algebraic structures, and allow us to obtain higher arity and dimension generalizations of the Elekes-Szabó theorem. I will overview this active area and present some recent results from joint work with (subsets of) Abdul Basit, Kobi Peterzil, Sergei Starchenko, Terence Tao and Chieu-Minh Tran.

Compact moduli and degenerations

When: Mon, January 30, 2023 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Dori Bejleri (Harvard University) -
Abstract: It has been said that working with non-compact spaces is like trying to hold change in your pocket with a hole in it. One of the central examples of non-compact spaces in algebraic geometry are moduli spaces. Broadly speaking, the points of a moduli space represent equivalence classes of algebraic varieties of a given type, and its geometry reflects the ways these varieties deform in algebraic families. The classification of algebraic varieties of a given type is tantamount to understanding the geometry of the corresponding moduli space. The goal of this talk is to discuss recent progress on compactifying moduli spaces of higher dimensional varieties, focusing on the interplay between compactifications of moduli spaces and singular degenerations of the objects they classify.

Leveraging Quantum Computers to Solve Partial-Differential Equations

When: Wed, March 1, 2023 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Ryan Vogt (Department of Defense) -
Abstract: We first set the stage by presenting a brief history of quantum mechanics and quantum computing. For the remainder of the colloquium, we focus on solving partial-differential equations (PDEs) using a quantum algorithm. The strategy we employ is to leverage the semi-discretization approach. In this approach, we first choose to discretize the PDE in space, leading to  a system of ordinary differential equations (ODEs) in time. We then discuss our quantum algorithm for solving the ODE system in time. Afterwards, we explain how applying our algorithm to a PDE that admits a non-smooth (weak) solution will yield a quadratic time computational speed up versus classical approaches; and briefly discuss the impact this would have practically on state of the art PDE simulations. In this talk, for simplicity, we discuss solving the 2-D heat equation, using finite-difference, spectral, and finite-element spatial discretizations to demonstrate that our quantum algorithm recovers the numerical solution. We then summarize our work, and briefly discuss other areas of applied mathematics we have developed quantum algorithms for e.g. stochastic ODE/PDE, optimal control, and supervised/unsupervised machine learning.

Are Black Holes Real? A Mathematical Approach to an Astrophysical Question

When: Thu, March 9, 2023 - 3:15pm
Where: Kirwan 3206
Speaker: Sergiu Klainerman (Princeton University) -
Abstract: The question whether black holes are real can be approached mathematically by addressing basic issues concerning their rigidity, stability and how they form in the first place. I will review these and focus on recent results concerning the nonlinear stability of slowly rotating Kerr black holes.

Nonlinear stability of Kerr Black Holes for Small Angular Momentum

When: Fri, March 10, 2023 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Sergiu Klainerman (Princeton University) -
Abstract: I will describe the main ideas behind  my recent results with J. Szeftel and with E. Giorgi and J. Szeftel  concerning  the nonlinear stability of slowly rotating black holes.

Higher rank Teichmüller spaces

When: Wed, March 15, 2023 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Anna Wienhard (Max Planck Institute ) -

  Abstract: Classical Teichmüller space describes the space of conformal structures on a given topological surface. It plays an important role in several areas of mathematics as well as in theoretical physics. Due to the uniformization theorem, Teichmüller space can be realized as the space of hyperbolic structures, and is closely related to discrete and faithful representations of the fundamental group of the surface into PSL(2,R), the group of isometries of the hyperbolic plane. Higher rank Teichmüller spaces generalize many aspects of this classical theory when PSL(2,R) is replaced by other Lie groups of higher rank, for example the symplectic group PSp(2n, R) or the special linear group PSL(n, R). In this talk I will give an introduction to higher rank Teichmüller spaces and their properties.  I will also highlight connections to other areas in geometry, dynamics and algebra.

On equivariance in scientific machine learning

When: Wed, April 12, 2023 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Lexing Ying (Stanford University) -
Abstract: Equivariance is a key principle in the design of deep neural networks. In this talk, I will discuss a few examples where equivariance plays a key role in scientific machine learning, including how the equivariance principle helps in the design of neural network architectures for several inverse problems, how equivariance naturally leads to the attention mechanism in sequence-to-sequence models, and etc.

Geometry of Music Perception

When: Tue, April 18, 2023 - 2:00pm
Where: ATL 3100A and Virtual Via Zoom:
Speaker: Benjamin Himpel (Reutlingen University and University of Plymouth)

Abstract: Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of all chords is shown to be a Whitney stratified space. Each stratum is a Riemannian manifold which naturally yields a geodesic distance across strata. The resulting metric is compatible with voice-leading satisfying the triangle inequality. The geometric model allows for rigorous studies of psychoacoustic quantities like roughness and harmonicity as height functions. In order to show how to use the geometric framework in psychoacoustic studies, concepts for the perception of chord resolutions are introduced and analyzed.

The Mathematics of Consilience

When: Thu, April 27, 2023 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Simon Levin (Princeton University) -
Abstract: In his book, “Consilience: The Unity of Knowledge,” the late biologist E.O. Wilson lamented the “fragmentation of knowledge and resulting chaos in philosophy,” primarily between the sciences and humanities, which he attributed to “artifacts of scholarship.” His concerns and aspirations similarly could have been applied within the disciplines of science, as made clear in Philip Anderson’s essay, “More is Different,” highlighting the disparities between reductionistic and holistic views of science. Wilson goes on to write that mathematics “because of its effectiveness in the natural sciences seems to point arrowlike toward the goal of objective truth,” indeed even referring to Wigner’s famous essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” But Wilson seems unconvinced, arguing that logical positivism, the notion that the philosophical problems worth studying are just those that are subject to logical analysis, “are more commonly studied in philosophy, as dinosaur fossils are studied in paleontology laboratories, to understand the causes of extinction.” I will argue that mathematics has much to offer in providing a unification of the disciplines, drawing on concepts from the subject of complex adaptive systems to relate the reductionistic and holistic approaches, to understand scaling, emergence, pattern formation, critical transitions and, most crucially, the conflicts between the interests of individual agents and the collective good that were central to much of Wilson’s own work, and are key to the search for assuring a sustainable planet.

About the speaker: Simon A. Levin is the James S. McDonnell Distinguished University Professor in Ecology and Evolutionary Biology at Princeton University. He received his B.A. from Johns Hopkins University and his Ph.D. in Mathematics from the University of Maryland. Levin is a Fellow of the American Academy of Arts and Sciences and the American Association for the Advancement of Science, a Member of the National Academy of Sciences and the American Philosophical Society, and a Foreign Member of the Istituto Veneto and the Istituto Lombardo.

Levin is a former President of the Ecological Society of America and the Society for Mathematical Biology, Chair of the Council of IIASA, Chair of the Board of the Beijer Institute, and Chair of the Science Board of the Santa Fe Institute. He has received numerous awards including the Kyoto Prize in Basic Sciences, Heineken Prize for Environmental Sciences, Margalef Prize for Ecology, Tyler Prize for Environmental Achievement, the U.S. National Medal of Science, and most recently, the BBVA Foundation Frontiers of Knowledge Award in Ecology and Conservation Biology. He has mentored more than 150 Ph.D. students and Postdoctoral Fellows.

A Tale of Two Theorems of Thurston

When: Wed, May 3, 2023 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Dan Margalit (Georgia Institute of Technology) -
Abstract: In the 20th century, Thurston proved two classification theorems, one for surface homeomorphisms and one for branched covers of surfaces.  While the theorems have long been understood to be analogous, we will present new work with Belk and Winarski showing that the two theorems are in fact special cases of one Ubertheorem.  We will also discuss joint work with Belk, Lanier, Strenner, Taylor, Winarski, and Yurttas on further algorithmic and theoretical aspects of Thurston’s theorems.  This talk is meant to be accessible to a wide audience of mathematicians.

The games that people, lizards, and cancer cells play

When: Wed, May 10, 2023 - 3:15pm
Where: CSIC 4122
Speaker: Alex Vladimirsky (Cornell University) -
Abstract: In many applications, it is important to model continuing changes in the relative popularity of "strategies" used by a large population of "players". If these strategies are inherited (as in distinct sub-populations of animals with different phenotypes), changes in relative abundance are driven by different reproductive rates, which might also be changing with the composition of the entire population. Perhaps surprisingly, the popularity of patterns in human (non-inherited) behavior may also evolve in a similar fashion if the strategy-switching is primarily driven by (occasional) emulation of more successful players. Evolutionary Game Theory (EGT) provides a formal framework for describing the dynamics of such processes by ordinary (or stochastic) differential equations. But in some cases, passive observation is not enough: we might want to change the rules of the game dynamically, thus altering that population's trajectory and driving it to some target outcome. Mathematical control theory provides useful computational tools to accomplish this rule-changing "optimally".

We will first illustrate the EGT by attempting to answer a few classical questions: Why aren't animals fighting to death more often? How do many people manage to stay community-minded while others around them remain selfish? Are aggressive, defensive, or sneaky lizards more successful in producing offspring?

We will then illustrate the ideas of "controlled EGT" in the context of cancer modeling and optimization of adaptive drug therapies. In a joint project with Mark Gluzman and Jake Scott, we showed that the competition among sub-populations of cancer cells can be exploited to find the best time and duration for treatment, thus decreasing the total amount of drugs used and the total time to recovery. This was accomplished by solving a Hamilton-Jacobi-Bellman PDE numerically to obtain the optimal treatment policy in the feedback form. In a more recent project with MingYi Wang and Jake Scott, we extend this model to account for environmental stochasticity in cancer evolution. To deal with the possibility of failure due to random perturbations, we switch to a different optimization goal and compute treatment policies that maximize the probability of desirable outcomes (e.g., curing a patient without exceeding the prescribed time and drugs "budget"). These underlying models are certainly too simple to make them directly relevant for oncological practice. But our control-theoretic approach is general, and the qualitative insights gained from the EGT will, hopefully, inform the design of future clinical trials.

Nochetto’s 70th Birthday Conference

When: Tue, May 16, 2023 - 8:00am

Nochetto’s 70th Birthday Conference

When: Wed, May 17, 2023 - 8:00am

Nochetto’s 70th Birthday Conference

When: Thu, May 18, 2023 - 8:00am

Nochetto’s 70th Birthday Conference

When: Fri, May 19, 2023 - 8:00am