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


=Fall 2021=


== September 17, 2021, Social Sciences 5208 + [http://128.104.155.144/ClassroomStreams/socsci5208_stream.html Live Stream], [https://markshus.wixsite.com/math Mark Shusterman] (Harvard) ==
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==
(host: Lyu)


(hosted by Gurevich)
The Gromov-Hausdorff distance between spheres.


'''Finitely Presented Groups in Arithmetic Geometry'''
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.


I will report on recent works, in part joint with Esnault—Srinivas, and with Jarden, on the finite presentability of several (profinite) groups arising in algebraic geometry and in number theory. These results build on a cohomological criterion of Lubotzky involving Euler characteristics. I will try to explain the analogy, rooted in arithmetic topology, between these results and classical facts about fundamental groups of three-dimensional manifolds.
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.


== September 24, 2021, B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [https://math.wisc.edu/staff/paul-sean/ Sean Paul] (UW-Madison) ==
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.
'''The Tian-Yau-Donaldson conjecture for general polarized manifolds'''


According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle L on a compact complex manifold X should be equivalent to an algebro-geometric "stability condition" satisfied by the pair (X,L).  The cscK metrics are the critical points of Mabuchi's K-energy functional M, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff M satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stabilityThe speaker will give a non-technical general account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry​in the spirit of Arakelov, Faltings, and Bismut-Gillet- Soule.
== February 24, 2023, Cancelled/available ==
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==


== October 1, 2021, B239 + [http://go.wisc.edu/wuas48 Live stream], [https://people.math.wisc.edu/~andreic/ Andrei Caldararu] (UW-Madison) ==
Title: How curved is a combinatorial graph?
'''Yet another Moonshine'''


The j-function, introduced by Felix Klein in 1879, is an essential ingredient in the study of elliptic curves. It is Z-periodic on the complex upper half-plane, so it admits a Fourier expansion. The original Monstrous Moonshine conjecture, due to McKay and Conway/Norton in the 1980s, relates the Fourier coefficients of the j-function around the cusp to dimensions of irreducible representations of the Monster simple group. It was proved by Borcherds in 1992.
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.


In my talk I will try to give a rudimentary introduction to modular forms, explain Monstrous Moonshine, and discuss a new version of it obtained in joint work with Yunfan He and Shengyuan Huang.  Our version involves studying the j-function around CM points (so-called Landau-Ginzburg points in the physics literature) and expanding with respect to a coordinate which arises naturally in string theory.
(hosts: Shaoming Guo, Andreas Seeger)


== October 8, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom] + live video on the 9th floor, [https://www.maths.ox.ac.uk/people/jon.chapman Jon Chapman] (University of Oxford) ==
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==


('''Wasow lecture'''; hosted by Thiffeault)
'''''Distinguished lectures'''''


'''Asymptotics beyond all orders: the devil's invention?'''
Title: Surfaces and foliations in hyperbolic 3-manifolds


"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel.
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.


The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes' phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems. Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, and Hele-Shaw flow.
(host: Kent)


== October 11, 13, 15, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom], '''[Mon, Wed, Fri 4-5pm]''', [https://www.maths.usyd.edu.au/u/geordie/ Geordie Williamson] (University of Sydney) ==
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky]  (Yale University) ==


('''Distinguished Lecture Series'''; hosted by Gurevich)
'''''Distinguished lectures'''''


'''Geometric representation theory and modular representations'''
Title: End-periodic maps, via fibered 3-manifolds


Representation theory is the study of linear symmetry. We are interested in all ways in which a group can arise as the symmetries of a vector space. Representation theory is a remarkably rich subject, with deep connections to number theory, combinatorics, algebraic geometry, differential geometry, theoretical physics and beyond. This lecture series will focus on modular representations, i.e. those representations where our vector spaces are over a field of characteristic p. I will try to highlight some of the main questions in the field and why we are interested in answering them. It is remarkable how much is still unknown and how hard some of these questions are. I will explain the role played by geometric representation theory in our attempts to understand these questions. A fascinating blend of algebra, algebraic geometry, category theory and algebraic topology is informing our understanding of basic questions. Much remains to be understood!
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.


== October 22, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom],  [https://math.berkeley.edu/people/faculty/vera-serganova Vera Serganova] (UC Berkeley) ==
(host: Kent)


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


'''Supersymmetry and tensor categories'''
'''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.


I will explain how representation theory of supergroups and
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==
supergeometry are related to general theory of tensor categories,
'''Title:''' Random plane geometry -- a gentle introduction
present old and new results and open questions
in the field. We will see how universal tensor categories can be
constructed using supergroups and discuss analogy between super
representation theory and representation theory over the fields of
positive characteristic.


== October 29, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom], [https://web.math.princeton.edu/~aionescu/ Alexandru Ionescu] (Princeton University) ==
'''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.


(hosted by Wainger)
(host: Valko)


'''Polynomial averages and pointwise ergodic theorems on nilpotent groups'''
== April 7, 2023, Friday at 4pm  [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==


I will talk about some recent work on pointwise almost
'''''Wasow lecture'''''
everywhere convergence for ergodic averages along polynomial sequences
in nilpotent groups of step two. Our proof is based on
almost-orthogonality techniques that go far beyond Fourier transform
tools, which are not available in the non-commutative nilpotent
setting. In particular we develop what we call a nilpotent circle
method}, which allows us to adapt some the ideas of the classical
circle method to the setting of nilpotent groups.


== November 5, 2021,  B239 + [http://go.wisc.edu/wuas48 Live stream], [https://faculty.washington.edu/jathreya/ Jayadev S. Athreya] (University of Washington) ==
Title: Mathematics: A key to climate research


(hosted by Uyanik)
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:


'''Surfaces and Point Processes'''
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.


We'll give several concrete examples of how to go from the geometry of surfaces to the study of point processes, following work of Siegel, Veech, Masur, Eskin, Mirzakhani, Wright, and others. We'll discuss how this "probabilistic" perspective helps inform both the direction of questions one asks, as well as providing ideas of how to prove things. We'll discuss some pieces of joint work with Cheung-Masur, Margulis, and Arana-Herrera.
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.


== November 12, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom], [https://sites.tufts.edu/kasso/ Kasso Okoudjou] (Tufts University) ==
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.


(hosted by Stovall)
(hosts: Smith, Stechmann)


'''An exploration in analysis on fractals '''
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg]  (Indiana University) ==


Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support.  In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.
(hosts: Feldman, Tran)


== November 19, 2021 , B239 + [http://go.wisc.edu/wuas48 Live stream],  [https://math.wisc.edu/staff/ai-albert/  Albert Ai](UW-Madison) ==
Title: A family of toy problems modeling liquid crystals exhibiting large disparity in the elastic coefficients.


(reserved by the hiring committee)
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.


''' Low regularity solution for quasilinear PDEs'''


In this talk, we will consider the low regularity well-posedness problem for a pair of quasilinear dispersive PDEs: the nonlinear wave equation, and the water waves equations. Two classical methods, energy estimates and Strichartz estimates, have historically yielded substantial but partial results toward advancing the low regularity theory. We will see how, using a special structure of the equations known as a normal form structure, combined with tools from harmonic and microlocal analysis, we can refine these classical methods to drastically improve the known results for low regularity well-posedness.
== April 28, 2023, Friday at 4pm [https://nqle.pages.iu.edu/ Nam Q. Le]  (Indiana University) ==
Title: Hessian eigenvalues and hyperbolic polynomials


== December 1, 2021, Wednesday at 4pm in B239  + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [https://www.math.ucla.edu/~brianrl/ Brian Lawrence] (UCLA) ==
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.


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


'''Integral points on moduli spaces'''
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. 


Mordell's conjecture, now a theorem of Faltings, states that an algebraic curve of genus at least two has only finitely many rational points.  Recent work with Venkatesh gives a new proof of Mordell's conjecture; the method gives some hope of proving finiteness results for any variety (even of higher dimension) that can be realized as a moduli space.  I'll discuss some recent results in this direction.
== Future Colloquia ==


== December 3, 2021, Friday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 ZOOM] + live video in B239, [https://people.wgtn.ac.nz/martino.lupini Martino Lupini] (Victoria University of Wellington) ==
[[Colloquia/Fall2023|Fall 2023]]


(reserved by the hiring committee)
== Past Colloquia ==
 
'''Borel-definable Algebraic Topology'''
 
In this talk, I will explain how ideas and methods from logic can be used to obtain refinements of classical invariants from homological algebra and algebraic topology. I will then present some applications to classification problems in topology. This is joint work with Jeffrey Bergfalk and Aristotelis Panagiotopoulos.
 
== December 6, 2021, Monday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 ZOOM], [https://sites.google.com/site/michaellipnowski/ Michael Lipnowski]  (McGill) ==
 
(reserved by the hiring committee)
 
'''Story about a dodecahedron'''
 
The Seifert-Weber dodecahedral space is a famous closed hyperbolic 3-manifold, one of the first to be discovered.  I'll describe some computations that I've done, together with Francesco Lin, on the dodecahedral space and some questions about small eigenvalues on hyperbolic manifolds which motivated them in the first place.  I'll also raise a question about (unlikely) intersections of geodesics on hyperbolic manifolds inspired by these computations.
 
== December 8, 2021, Wednesday at 4pm in B239, [https://padmask.github.io/ Padmavathi Srinivasan] (University of Georgia) ==
 
(reserved by the hiring committee)
 
'''Degenerations of curves, rational points, and arithmetic topology'''
 
Number theory has a rich history of long standing open problems that are fairly easy to state, but are notoriously difficult to answer. The most famous among these is Fermat's Last Theorem, whose solution spurred the development of many technical tools in use today. The quest to find explicit methods to solve other Diophantine equations continues.
 
A recent method that has had spectacular success in finding rational points on curves that were previously out of reach is the "Quadratic Chabauty" method. The explicit implementation of the Quadratic Chabauty method is a formidable computational challenge. This talk will feature a simplification of the  Quadratic Chabauty method using geometric ideas, developed jointly with Besser and Mueller. Using ideas inspired by topology, we will outline new results (joint with Li, Litt and Salter) that show that most curves have no rational solutions at all, guided by Grothendieck's Section Conjecture. The key is to study degenerations in families of curves. The talk will close with various natural ways of measuring degenerations in families of curves (such as the conductor and the discriminant) and their interrelationship.
 
== December 10, 2021, Friday at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [http://www-personal.umich.edu/~apisa/ Paul Apisa] (University of Michigan) ==
 
(reserved by the hiring committee)
 
'''Billiards, dynamics, and the moduli space of Riemann surfaces'''


The Hodge bundle is the space whose points correspond to a Riemann surface equipped with a holomorphic 1-form. This space admits a GL(2, R) action whose dynamics govern the geometry of the moduli space of Riemann surfaces, an object of central importance in geometry, algebra, and physics. I will describe work, joint with Alex Wright, that classifies roughly half of all GL(2, R) orbit closures. I will also describe applications to deceptively simple sounding problems about billiards in polygons. Along the way I will highlight connections to algebraic geometry, homogeneous dynamics, and more.
[[Colloquia/Fall2022|Fall 2022]]


== December 13, 2021, Monday at 4pm in B239, [https://sites.google.com/view/nicole-looper/home?authuser=0 Nicole Looper] (Brown) ==
[[Spring 2022 Colloquiums|Spring 2022]]


(reserved by the hiring committee)
[[Colloquia/Fall2021|Fall 2021]]
 
== December 15, 2021, Wednesday at 4pm in B239, [https://people.seas.harvard.edu/~chr/ Chris Rycroft] (Harvard) ==
 
(reserved by the hiring committee)
 
'''Uncovering the rules of crumpling with a data-driven approach'''
 
When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].
 
This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3].
 
[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018).
[2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021).
[3] J. Hoffmann et al., Sci. Advances 5, eaau6792 (2019).
 
== December 17, 2021, Friday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 ZOOM],  [http://www.pdmi.ras.ru/~dchelkak/index_en.html Dmitry Chelkak] (ENS Paris) ==
 
(reserved by the hiring committee)
 
== December 20, 2021, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream], [https://sites.google.com/view/mnovackmath/home Matthew Novack] (IAS) ==
 
(reserved by the hiring committee)
 
'''Turbulent Weak Solutions of the 3D Euler Equations'''
 
The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory of turbulence.
 
== Future ==
 
[[Colloquia/Spring2022|Spring 2022]]
 
== Past Colloquia ==


[[Colloquia/Spring2021|Spring 2021]]
[[Colloquia/Spring2021|Spring 2021]]

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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