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In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated.


==September 9 , 2022, Friday at 4pm  [https://math.ou.edu/~jing/ Jing Tao] (University of Oklahoma)==
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b>
(host: Dymarz, Uyanik, WIMAW)


'''On surface homeomorphisms'''
<!--- in Van Vleck B239, '''unless otherwise indicated'''. --->


In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.
==September 23, 2022, Friday at 4pm  [https://www.pabloshmerkin.org/ Pablo Shmerkin] (University of British Columbia) ==
(host: Guo, Seeger)


'''Incidences and line counting: from the discrete to the fractal setting'''
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==
(host: Lyu)


How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.
The Gromov-Hausdorff distance between spheres.


==September 30, 2022, Friday at 4pm [https://alejandraquintos.com/ Alejandra Quintos] (University of Wisconsin-Madison, Statistics) ==
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.
(host: Stovall)


'''Dependent Stopping Times and an Application to Credit Risk Theory'''
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its ''precise'' value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.


Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.


In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.
== February 24, 2023, Cancelled/available ==
==October 7, 2022, Friday at 4pm [https://www.daniellitt.com/ Daniel Litt] (University of Toronto)==
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==
(host: Ananth Shankar)


'''The search for special symmetries'''
Title: How curved is a combinatorial graph?


What are the canonical sets of symmetries of n-dimensional space? I'll describe the history of this question, going back to Schwarz, Fuchs, Painlevé, and others, and some new answers to it, obtained jointly with Aaron Landesman. While our results rely on low-dimensional topology, Hodge theory, and the Langlands program, and we'll get a peek into how these areas come into play, no knowledge of them will be assumed.
Abstract:   Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed.  I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open.  No prior knowledge of differential geometry (or graphs) is required.


==October 14, 2022, Friday at 4pm  [https://math.sciences.ncsu.edu/people/asagema/ Andrew Sageman-Furnas] (North Carolina State)==
(hosts: Shaoming Guo, Andreas Seeger)
(host: Mari-Beffa)


'''Constructing isometric tori with the same curvatures'''
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky]  (Yale University) ==


Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid.
'''''Distinguished lectures'''''


For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.
Title:  Surfaces and foliations in hyperbolic 3-manifolds


The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.
Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.


== October 20, 2022, Thursday at 4pm, VV911  [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==
(host: Kent)
(host: Kurtz, Roch)


''Note the unusual time and room!''
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky]  (Yale University) ==


'''An introduction to counts-of-counts data'''
'''''Distinguished lectures'''''


Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size
Title: End-periodic maps, via fibered 3-manifolds


''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>''  containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.
Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning<nowiki>''</nowiki> construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.


In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.
(host: Kent)


''References''
== March 24, 2023 , Friday at 4pm  [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==
'''Title''': Boundaries, boundaries, and more boundaries


[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943
'''Abstract:''' It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups.  This is joint work with Jason Behrstock and Jacob Russell.


[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==
'''Title:''' Random plane geometry -- a gentle introduction


[3] Ewens WJ. Theoret Popul Biol, 3, 1972
'''Abstract:''' Consider Z^2, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percolation, is conjectured to have a scaling limit. Most random plane geometric models (including hidden geometries) should have the same scaling limit. I will explain the basics of the limiting geometry, the "directed landscape", and its relation to traffic jams, tetris, coffee stains and random matrices.


[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)
(host: Valko)


==October 21, 2022, Friday at 4pm  [https://web.ma.utexas.edu/users/ntran/ Ngoc Mai Tran] (Texas)==
== April 7, 2023, Friday at 4pm  [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==
(host: Rodriguez)


'''Forecast science, learn hidden networks and settle economics conjectures with combinatorics, geometry and probability.'''  
'''''Wasow lecture'''''


In many problems, one observes noisy data coming from a hidden or complex combinatorial structure. My research aims to understand and exploit such structures to arrive at an efficient and optimal solution. I will showcase a few successes, achieved with different tools, from different different fields: networks forecasting, hydrology, and auction theory. Then I will outline some open questions in each field.
Title: Mathematics: A key to climate research


==October 28, 2022, Friday at 4pm  [https://people.math.wisc.edu/~qinli/ Qin Li] (UW)==
Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper:
'''Multiscale inverse problem, from Schroedinger to Newton to Boltzmann'''


Inverse problems are ubiquitous. We probe the media with sources and measure the outputs, to infer the media information. At the scale of quantum, classical, statistical and fluid, we face inverse Schroedinger, inverse Newton’s second law, inverse Boltzmann problem, and inverse diffusion respectively. The universe, however, expects a universal mathematical description, as Hilbert proposed in 1900. In this talk, we discuss the connection between these problems. We will give arguments for justifying that these are the same problem merely represented at different scales. It is a light-hearted talk, and I will mostly focus on the story instead of the derivation. PDE background is appreciated but not necessary.
1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly.


== November 7, 2022, Monday at 4pm [https://ai.facebook.com/people/kristin-lauter/ Kristen Lauter] (Facebook) ==
2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent mathematical research offers advanced data analysis techniques that could be “game changing” in this respect.
Distinguished lectures


(host: Yang).
3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research.


'''Private AI: Machine Learning on Encrypted Data'''
(hosts: Smith, Stechmann)


As the world adopts Artificial Intelligence, the privacy risks are many. AI can improve our lives, but may leak our private data. Private AI is based on Homomorphic Encryption (HE), a new encryption paradigm which allows the cloud to operate on private data in encrypted form, without ever decrypting it, enabling private training and private prediction. Our 2016 ICML CryptoNets paper showed for the first time that it was possible to evaluate neural nets on homomorphically encrypted data, and opened new research directions combining machine learning and cryptography. The security of Homomorphic Encryption is based on hard problems in mathematics involving lattices, recently standardized by NIST for post-quantum cryptography. This talk will explain Homomorphic Encryption, Private AI, and explain HE in action.
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg]  (Indiana University) ==


== November 8, 2022, Tuesday at 4pm [https://ai.facebook.com/people/kristin-lauter/ Kristen Lauter] (Facebook) ==
(hosts: Feldman, Tran)
Distinguished lectures in VV911. ''Note: unusual room.''


(host: Yang).
Title: A family of toy problems modeling liquid crystals exhibiting large disparity in the elastic coefficients.


'''Artificial Intelligence & Cryptography: Privacy and Security in the AI era'''
Abstract: Certain classes of liquid crystals have been found to strongly favor particular types of deformations over others; for example, the cost of splay may greatly exceed the cost of bend or twist. In a series of studies with Dmitry Golovaty (Akron), Michael Novack (UT Austin) and Raghav Venkatraman (Courant), we explore the implications of assuming various asymptotic regimes for the elastic constants. Through a mixture of formal and rigorous analysis, along with computations, we identify the limiting behavior of minimizers to the associated energies. We find that a variety of singular structures emerge corresponding to jumps in the profile of these limiting minimizers that effectively save on the cost of splay, bend or twist—whichever is assumed to be most expensive.


How is Artificial Intelligence changing your life and the world?  How do you expect your data to be kept secure and private in the future?  Artificial intelligence (AI) refers to the science of utilizing data to formulate mathematical models that predict outcomes with high assurance. Such predictions can be used to make decisions automatically or give recommendations with high confidence. Cryptography is the science of protecting the privacy and security of data.  This talk will explain the dynamic relationship between cryptography and AI and how AI can be used to attack post-quantum cryptosystems.


The first talk is based on my 2019 ICIAM Plenary Lecture and the second one is based on my 2022 SIAM Block Prize Lecture.
== April 28, 2023, Friday at 4pm [https://nqle.pages.iu.edu/ Nam Q. Le]  (Indiana University) ==
Title: Hessian eigenvalues and hyperbolic polynomials


== November 11, 2022, Friday at 4pm [http://users.cms.caltech.edu/~jtropp/ Joel Tropp] (Caltech)==
Abstract: Hessian eigenvalues are natural nonlinear analogues of the classical Dirichlet eigenvalues. The Hessian eigenvalues and their corresponding eigenfunctions are expected to share many analytic and geometric properties (such as uniqueness, stability, max-min principle, global smoothness, Brunn-Minkowski inequality, convergence of numerical schemes, etc) as their Dirichlet counterparts. In this talk, I will discuss these issues and some recent progresses in various geometric settings. I will also explain the unexpected role of hyperbolic polynomials in our analysis. I will not assume any familiarity with these concepts.
This is the Annual LAA lecture. See [https://math.wisc.edu/laa-lecture/ this] for its history.


(host: Qin, Jordan)
== May 5, 2023, Friday at 4pm [https://www.math.ucdavis.edu/~gravner/ Janko Gravner]  (UC Davis) ==
Title: Long-range nucleation


'''Rocket-propelled Cholesky: Addressing the challenges of large-scale kernel computations'''
Abstract: Nucleation is a mechanism by which one equilibrium displaces another through formation of small unstoppable nuclei. Typically, nucleation is local, as the size of the nuclei is much smaller than the time scale of convergence to the new state. We will discuss a few simple models where nuclei are not small in diameter but instead are a result of lower-dimensional structures that grow and interact significantly before most of the space is affected. Analysis of such models includes a variety of combinatorial and probabilistic methods. 


Kernel methods are used for prediction and clustering in many data science and scientific computing applications, but applying kernel methods to a large number of data points N is expensive due to the high cost of manipulating the N x N kernel matrix. A basic approach for speeding up kernel computations is low-rank approximation, in which we replace the kernel matrix A with a factorized approximation that can be stored and manipulated more cheaply. When the kernel matrix A has rapidly decaying eigenvalues, mathematical existence proofs guarantee that A can be accurately approximated using a constant number of columns (without ever looking at the full matrix). Nevertheless, for a long time designing a practical and provably justified algorithm to select the appropriate columns proved challenging.
== Future Colloquia ==


Recently, we introduced RPCholesky ("randomly pivoted" or "rocket-propelled" Cholesky), a natural algorithm for approximating an N x N positive semidefinite matrix using k adaptively sampled columns. RPCholesky can be implemented with just a few lines of code; it requires only (k+1)N entry evaluations and O(k^2 N) additional arithmetic operations. In experiments, RPCholesky matches or improves on the performance of alternative algorithms for low-rank psd approximation. Moreover, RPCholesky provably achieves near-optimal approximation guarantees. The simplicity, effectiveness, and robustness of this algorithm strongly support its use for large-scale kernel computations.
[[Colloquia/Fall2023|Fall 2023]]


Joint work with Yifan Chen, Ethan Epperly, and Rob Webber. Available at arXiv:2207.06503.
== Past Colloquia ==
 
==November 18, 2022, Friday at 4pm [http://homepages.math.uic.edu/~freitag/index.html Jim Freitag] (U of Illinois-Chicago) Zoom link: https://go.wisc.edu/jimfreitag<nowiki/>==
'''Now available:''' [https://people.math.wisc.edu/logic/talks/221118-Freitag.mp4 Recording] and [https://people.math.wisc.edu/logic/talks/221118-Freitag.pdf Slides]
 
(hosts: Lempp, Andrews)
 
'''When any three solutions are independent'''
 
In this talk, we'll talk about a surprising recent result about the algebraic relations between solutions of a differential equation. The result has applications to functional transcendence, diophantine geometry, and compact complex manifolds.
 
==November 21, 2022, <span style="color: red;">Monday</span> at 4pm [https://math.mit.edu/directory/profile.html?pid=1698 Andrei Negut] (MIT) Zoom link: [https://go.wisc.edu/andreinegut https://go.wisc.edu/andreinegut]==
Hiring talk.
 
(hosts: Arinkin, Caldararu)
 
'''From gauge theory to geometric representation theory and back'''
 
We start from the celebrated construction (due to Grojnowski and Nakajima) of a Heisenberg algebra action on the cohomology groups of Hilbert schemes of points on surfaces
 
# replacing Hilbert schemes with moduli spaces of higher rank sheaves yields a computation of Nekrasov partition functions in 5d supersymmetric gauge theory, and a proof of the deformed Alday-Gaiotto-Tachikawa conjecture.
# replacing cohomology by Chow groups gives a proof of the Beauville conjecture in the hyperkahler geometry of Hilbert schemes of points on K3 surfaces (with Maulik)
# working with derived categories allows us to construct a detailed framework realizing categorical knot invariants in terms of the geometry of Hilbert schemes of points on the affine plane (with Gorsky and Rasmussen)
 
==December 2, 2022, Friday at 4pm:  Promit Ghosal (MIT)==
'''Fractal Geometry of the KPZ equation'''
 
The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to many important models like random growth processes, Burgers turbulence, interacting particles system, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an interesting story about the  (2+1)-d and (3+1)-d case (work in progress with Jaeyun Yi).
 
== December 5, 2022, Monday at 4pm:  Di Fang (Berkeley) ==
(reserved by HC. contact: Stechmann)
 
==December 9, 2022, Friday at 4pm:  Dallas Albritton (Princeton)==
(reserved by HC. contact: Stechmann)
 
== December 12, 2022, Monday at 4pm:  Laurel Ohm (Princeton) ==
(reserved by HC. contact: Stechmann)
 
== Future Colloquia ==


[[Colloquia/Fall2022|Fall 2022]]
[[Colloquia/Fall2022|Fall 2022]]


[[Colloquia/Spring2023|Spring 2023]]
== Past Colloquia ==
[[Spring 2022 Colloquiums|Spring 2022]]
[[Spring 2022 Colloquiums|Spring 2022]]



Revision as of 20:58, 16 April 2023


UW Madison mathematics Colloquium is on Fridays at 4:00 pm.


February 3, 2023, Friday at 4pm Facundo Mémoli (Ohio State University)

(host: Lyu)

The Gromov-Hausdorff distance between spheres.

The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.

Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its precise value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.

In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.

February 24, 2023, Cancelled/available

March 3, 2023, Friday at 4pm Stefan Steinerberger (University of Washington)

Title: How curved is a combinatorial graph?

Abstract:   Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed.  I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open.  No prior knowledge of differential geometry (or graphs) is required.

(hosts: Shaoming Guo, Andreas Seeger)

March 8, 2023, Wednesday at 4pm Yair Minsky (Yale University)

Distinguished lectures

Title: Surfaces and foliations in hyperbolic 3-manifolds

Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.

(host: Kent)

March 10, 2023, Friday at 4pm Yair Minsky (Yale University)

Distinguished lectures

Title: End-periodic maps, via fibered 3-manifolds

Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning'' construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.

(host: Kent)

March 24, 2023 , Friday at 4pm Carolyn Abbott (Brandeis University)

Title: Boundaries, boundaries, and more boundaries

Abstract: It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell.

March 31, 2023 , Friday at 4pm Bálint Virág (University of Toronto)

Title: Random plane geometry -- a gentle introduction

Abstract: Consider Z^2, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percolation, is conjectured to have a scaling limit. Most random plane geometric models (including hidden geometries) should have the same scaling limit. I will explain the basics of the limiting geometry, the "directed landscape", and its relation to traffic jams, tetris, coffee stains and random matrices.

(host: Valko)

April 7, 2023, Friday at 4pm Rupert Klein (FU Berlin)

Wasow lecture

Title: Mathematics: A key to climate research

Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper:

1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly.

2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent mathematical research offers advanced data analysis techniques that could be “game changing” in this respect.

3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research.

(hosts: Smith, Stechmann)

April 21, 2023, Friday at 4pm Peter Sternberg (Indiana University)

(hosts: Feldman, Tran)

Title: A family of toy problems modeling liquid crystals exhibiting large disparity in the elastic coefficients.

Abstract: Certain classes of liquid crystals have been found to strongly favor particular types of deformations over others; for example, the cost of splay may greatly exceed the cost of bend or twist. In a series of studies with Dmitry Golovaty (Akron), Michael Novack (UT Austin) and Raghav Venkatraman (Courant), we explore the implications of assuming various asymptotic regimes for the elastic constants. Through a mixture of formal and rigorous analysis, along with computations, we identify the limiting behavior of minimizers to the associated energies. We find that a variety of singular structures emerge corresponding to jumps in the profile of these limiting minimizers that effectively save on the cost of splay, bend or twist—whichever is assumed to be most expensive.


April 28, 2023, Friday at 4pm Nam Q. Le (Indiana University)

Title: Hessian eigenvalues and hyperbolic polynomials

Abstract: Hessian eigenvalues are natural nonlinear analogues of the classical Dirichlet eigenvalues. The Hessian eigenvalues and their corresponding eigenfunctions are expected to share many analytic and geometric properties (such as uniqueness, stability, max-min principle, global smoothness, Brunn-Minkowski inequality, convergence of numerical schemes, etc) as their Dirichlet counterparts. In this talk, I will discuss these issues and some recent progresses in various geometric settings. I will also explain the unexpected role of hyperbolic polynomials in our analysis. I will not assume any familiarity with these concepts.

May 5, 2023, Friday at 4pm Janko Gravner (UC Davis)

Title: Long-range nucleation

Abstract: Nucleation is a mechanism by which one equilibrium displaces another through formation of small unstoppable nuclei. Typically, nucleation is local, as the size of the nuclei is much smaller than the time scale of convergence to the new state. We will discuss a few simple models where nuclei are not small in diameter but instead are a result of lower-dimensional structures that grow and interact significantly before most of the space is affected. Analysis of such models includes a variety of combinatorial and probabilistic methods.

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