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<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b>
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm in Van Vleck B239 unless otherwise noted.</b>


<!--- in Van Vleck B239, '''unless otherwise indicated'''. --->
Contacts for the colloquium are Simon Marshall and Dallas Albritton.


           


== January 10, 2022, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://www.stat.berkeley.edu/~gheissari/ Reza Gheissari] (UC Berkeley) ==
==Spring 2024==
{| cellpadding="8"
! align="left" |date
! align="left" |speaker
! align="left" |title
! align="left" | host(s)
|-
|<b>Monday Jan 22 at 4pm in B239</b>
|[https://www.mathematik.tu-darmstadt.de/fb/personal/details/yingkun_li.en.jsp Yingkun Li] (Darmstadt Tech U, Germany)
|[[#Li|Arithmetic of real-analytic modular forms]]
|Yang
|-
|'''Thursday Jan 25 at 4pm in VV911'''
|[https://chimeraki.weebly.com/scientificresearch.html Sanjukta Krishnagopal] (UCLA/UC Berkeley)
|Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes
|Smith
|-
|Jan 26
|[https://www.math.ucla.edu/~jacob/ Jacob Bedrossian] (UCLA)
|Lyapunov exponents in stochastic systems
|Tran
|-
|Feb 2
|[https://www.williamyunchen.com/ William Chen]
|[[#Chen|Orbit problems and the mod p properties of Markoff numbers]]
|Arinkin
|-
|Feb 9
|No colloquium
|
|
|-
|Feb 16
|[https://jacklutz.com/ Jack Lutz] (Iowa State)
|Algorithmic Fractal Dimensions
|Guo
|-
|Feb 23
|No colloquium
|
|
|-
|Mar 1
|[https://users.oden.utexas.edu/~pgm/ Per-Gunnar Martinsson] (UT-Austin)
|Randomized algorithms for linear algebraic computations
|Li
|-
|Mar 8
|[https://www.math.arizona.edu/~izosimov/ Anton Izosimov] (U of Arizona)
|Incidences and dimers
|Gloria Mari-Beffa
|-
|Mar 15
|[https://sites.google.com/view/peterhumphries/ Peter Humphries] (Virginia)
|[[#Humphries|Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity]]
|Marshall
|-
|'''Monday Mar 18 at 4pm in B239'''
|[https://colegraham.net/ Cole Graham] (Brown)
|Invasion in general domains
|Albritton, Smith, Tran
|-
|'''Wednesday Mar 20 at 4 pm in B239'''
|[https://www.math.wustl.edu/~wanlin/index.html Wanlin Li] (Washington U St Louis)
|Diophantine problem and rational points on curves
|Dymarz, GmMaW
|-
|Mar 29
|Spring break
|
|
|-
|Apr 5
|[https://www.math.columbia.edu/~savin/ Ovidiu Savin] (Columbia)
|
|Tran
|-
|Apr 12
|[https://www.mikaylakelley.com/about Mikayla Kelley] (U Chicago Philosophy)
|[[#Kelley|Math And... seminar: Accuracy and the Patterns of Rational Credence]]
|Ellenberg, Marshall
|-
|Apr 19
|[https://sites.math.rutgers.edu/~yyli/ Yanyan Li] (Rutgers)
|
|Tran
|-
|Apr 26
|[https://sites.google.com/view/chris-leiningers-webpage/home Chris Leininger] (Rice)
|TBA
|Uyanik
|-
|May 3
|[https://pages.cs.wisc.edu/~jyc/ Jin-Yi Cai] (UW-Madison)
|Shor's Quantum Algorithm Does Not Factor Large Integers in the Presence of Noise
|Yang
|}


(reserved by the hiring committee)
== Abstracts ==


'''Surface phenomena in the 2D and 3D Ising model'''
<div id="Li">'''Monday, January 22.  Yingkun Li'''  


Since its introduction in 1920, the Ising model has been one of the most studied models of phase transitions in statistical physics. In its low-temperature regime, the model has two thermodynamically stable phases, which, when in contact with each other, form an interface: a random curve in 2D and a random surface in 3D. In this talk, I will survey the rich phenomenology of this interface in 2D and 3D, and describe recent progress in understanding its geometry in various parameter regimes where different surface phenomena and universality classes emerge.
'''Arithmetic of real-analytic modular forms'''


== January 17, 2022, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://sites.google.com/view/lovingmath/home Marissa Loving] (Georgia Tech) ==
Modular form is a classical mathematical object dating back to the 19th century. Because of its connections to and appearances in many different areas of math and physics, it remains a popular subject today. Since the work of Hans Maass in 1949, real-analytic modular form has found important applications in arithmetic geometry and number theory. In this talk, I will discuss the amazing works in this area over the past 20 years, and give a glimpse of its fascinating future directions.      


(reserved by the hiring committee)
'''Thursday, January 25. Sanjukta Krishnagopal'''


'''Symmetries of surfaces: big and small'''
'''Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes'''


We will introduce both finite and infinite-type surfaces and study their collections of symmetries, known as mapping class groups. The study of the mapping class group of finite-type surfaces has played a central role in low-dimensional topology stretching back a hundred years to work of Max Dehn and Jakob Nielsen, and gaining momentum and significance through the celebrated work of Bill Thurston on the geometry of 3-manifolds. In comparison, the study of the mapping class group of infinite-type surfaces has exploded only within the past few years. Nevertheless, infinite-type surfaces appear quite regularly in the wilds of mathematics with connections to dynamics, the topology of 3-manifolds, and even descriptive set theory -- there is a great deal of rich mathematics to be gained in their study! In this talk, we will discuss the way that the study of surfaces intersects and interacts with geometry, algebra, and number theory, as well as some of my own contributions to this vibrant area of study.
In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application.  


== January 21, 2022, Friday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://web.math.princeton.edu/~nfm2/ Nicholas Marshall]  (Princeton) ==
First, I will discuss some recent advances in mathematical machine learning for prediction on graphs. Machine learning is often a black box. Here I will present some exact theoretical results on the dynamics of weights while training graph neural networks using graphons - a graph limit or a graph with infinitely many nodes. I will use these ideas to present a new method for predictive and personalized medicine applications with remarkable success in prediction of Parkinson's subtype five years in advance.


(reserved by the hiring committee)
Then, I will discuss some work on higher-order models of graphs: simplicial complexes - that can capture simultaneous many-body interactions. I will present some recent results on spectral theory of simplicial complexes, as well as introduce a mathematical framework for studying the topology and dynamics of ''multilayer'' simplicial complexes using Hodge theory, and discuss applications of such interdisciplinary methods to studying bias in society, opinion dynamics, and hate speech in social media.


'''Laplacian quadratic forms, function regularity, graphs, and optimal transport'''


In this talk, I will discuss two different applications of harmonic analysis to
problems motivated by data science. Both problems involve using Laplacian
quadratic forms to measure the regularity of functions. In both cases the key
idea is to understand how to modify these quadratic forms to achieve a specific
goal. First, in the graph setting, we suppose that a collection of m graphs
G_1 = (V,E_1),...,G_m=(V,E_m) on a common set of vertices V is given,
and consider the problem of finding the 'smoothest' function f : V -> R with
respect to all graphs simultaneously, where the notion of smoothness is defined
using graph Laplacian quadratic forms. Second, on the unit square [0,1]^2, we
consider the problem of efficiently computing linearizations of 2-Wasserstein
distance; here, the solution involves quadratic forms of a Witten Laplacian.


== January 24, 2022, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://sites.google.com/view/skippermath Rachel Skipper] (Ohio State) ==
'''Friday, January 26. Jacob Bedrossian'''


(reserved by the hiring committee)
'''Lyapunov exponents in stochastic systems'''


'''From simple groups to symmetries of surfaces'''
In this overview talk we discuss several results regarding positive Lyapunov exponents in stochastic systems. First we discuss proving "Lagrangian chaos" in stochastic fluid mechanics, that is, demonstrating a positive Lyapunov exponent for the motion of a particle in the velocity field arising from the stochastic Navier-Stokes equations. We describe how this chaos can be used to deduce qualitatively optimal almost-sure exponential mixing of passive scalars. Next we describe more recently developed methods for obtaining strictly positive lower bounds and some quantitative estimates on the top Lyapunov exponent of weakly-damped stochastic differential equations, such as Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (called "Eulerian chaos" in fluid mechanics). Further applications of the ideas to the chaotic motion of charged particles in fluctuating magnetic fields and the non-uniqueness of stationary measures for Lorenz 96 in degenerate forcing situations will be discussed if time permits. All of the work except for the charged particles (joint with Chi-Hao Wu) is joint with Alex Blumenthal and Sam Punshon-Smith.


We will take a tour through some families of groups of historic importance in geometric group theory, including self-similar groups and Thompson’s groups. We will discuss the rich, continually developing theory of these groups which act as symmetries of the Cantor space, and how they can be used to understand the variety of infinite simple groups. Finally, we will discuss how these groups are serving an important role in the newly developing field of big mapping class groups which are used to describe symmetries of surfaces.
<div id="Chen">'''Friday, February 2. William Chen'''


== February 11, 2022, at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://people.math.wisc.edu/~msoskova/ Mariya Soskova] (UW-Madison) ==
'''Orbit problems and the mod p properties of Markoff numbers'''


'''The e-verse'''
Markoff numbers are positive integers which encode how resistant certain irrational numbers are to being approximated by rationals. In 1913, Frobenius asked for a description of all congruence conditions satisfied by Markoff numbers modulo primes p. In 1991 and 2016, Baragar, Bourgain, Gamburd, and Sarnak conjectured a refinement of Frobenius’s question, which amounts to showing that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation”; that is to say: they conjecture that its integral points surject onto its mod p points for every prime p. In this talk we will show how to prove this conjecture for all but finitely many primes p, thus reducing the conjecture to a finite computation. A key step is to understand this problem in the context of describing the orbits of certain group actions. Primarily, we will consider the action of the mapping class group of a topological surface S on (a) the set of G-covers of S, where G is a finite group, and (b) on the character variety of local systems on S. Questions of this type have been related to many classical problems, from proving that the moduli space of curves of a given genus is connected, to Grothendieck’s ambitious plan to understand the structure of the absolute Galois group of the rationals by studying its action on “dessins d’enfant”. We will explain some of this history and why such problems can be surprisingly difficult.


Computability theory studies the relative algorithmic complexity of sets of natural numbers and other mathematical objects. Turing reducibility and the induced partial order of the Turing degrees serve as the well-established model of relative computability. Enumeration reducibility captures another natural relationship between sets of natural numbers in which positive information about the first set is used to produce positive information about the second set. The induced structure of the enumeration degrees can be viewed as an extension of the Turing degrees, as there is a natural way to embed the second partial order in the first. In certain cases, the enumeration degrees can be used to capture the algorithmic content of mathematical objects, while the Turing degrees fail. Certain open problems in degree theory present as more approachable in the extended context of the enumeration degrees, e.g. first order definability. We have been working to develop a richer “e-verse”: a system of classes of enumeration degrees with interesting properties and relationships, in order to better understand the enumeration degrees. I will outline several research directions in this context.


== February 18, 2022, at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Video over Zoom], [https://people.math.wisc.edu/~seeger/ Andreas Seeger] (UW-Madison) ==
<div id="Lutz">'''Friday, February 16. Jack Lutz'''


'''Spherical maximal functions and fractal dimensions of dilation sets'''
'''Algorithmic Fractal Dimensions '''


We survey old and new problems and results on spherical means, regarding pointwise convergence, $L^p$ improving and consequences for sparse domination.
Algorithmic fractal dimensions are computability theoretic versions of Hausdorff dimension and other fractal dimensions. This talk will introduce algorithmic fractal dimensions with particular focus on the Point-to-Set Principle. This principle has enabled several recent proofs of new theorems in geometric measure theory. These theorems, some solving long-standing open problems, are classical (meaning that their statements do not involve computability or logic), even though computability has played a central in their proofs.


== February 25, 2022, at 4pm in B239 + [http://go.wisc.edu/wuas48 Live Stream], [https://sites.google.com/view/rohini-ramadas/home Rohini Ramadas] (Warwick) ==


(hosted by WIMAW)
<div id="Martinsson">'''Friday, March 1. Per-Gunnar Martinsson'''


'''Dynamics on the moduli space of point-configurations on the Riemann sphere'''
'''Randomized algorithms for linear algebraic computations '''


A degree-$d$ rational function $f(z)$ in one variable with complex coefficients defines a holomorphic self-map of the Riemann sphere. A rational function is called post-critically finite (PCF) if every critical point is (pre)-periodic. PCF rational functions have been central in complex dynamics, due to their special dynamical behavior, and their special distribution within the parameter space of all rational maps.  
The talk will describe how randomized algorithms can effectively, accurately, and reliably solve linear algebraic problems that are omnipresent in scientific computing and in data analysis. We will focus on techniques for low rank approximation, since these methods are particularly simple and powerful, and are well understood mathematically. The talk will also briefly survey a number of other randomized algorithms for tasks such as solving linear systems, estimating matrix norms, and computing full matrix factorizations.


By work of Koch building on a result of Thurston, every PCF map arises as an isolated fixed point of an algebraic dynamical system on the moduli space $M_{0,n}$ of point-configurations on the Riemann sphere. I will introduce PCF maps and $M_{0,n}$. I will then present results characterizing the ensuing dynamics on $M_{0,n}$.


This talk includes joint work with Nguyen-Bac Dang, Sarah Koch, David Speyer, and Rob Silversmith.
<div id="Izosimov">'''Friday, March 8. Anton Izosimov'''


== March 1, 2 and 4, 2022 (Tuesday, Wednesday and Friday),  [http://www.math.stonybrook.edu/~roblaz/ Robert Lazarsfeld] (Stony Brook) ==
'''Incidences and dimers '''
(''Departmental Distinguished Lecture series'')


'''Public Lecture: Pythagorean triples and parametrized curves'''
Incidence theorems are statements about points, lines, and possibly higher-dimensional subspaces and their incidences. Examples include classical theorems of Desargues and Pappus. In this talk, we'll discuss a connection between incidence geometry and an archetypal model of statistical physics - the dimer model. The talk will be based on the work of many people, including my ongoing work with Pavlo Pylyavskyy (Minnesota).


''Tuesday, March 1, 4:00pm (Humanities 3650 + [http://go.wisc.edu/n6986j Live Stream]). Note unusual time and location!''


In this lecture, aimed at advanced undergraduate and beginning graduate students, I will discuss the question of when a curve in the plane admits a parameterization by polynomials or rational functions.  
<div id="Humphries">'''Friday, March 15. Peter Humphries'''


'''Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity'''


'''Colloquium: How irrational is an irrational variety?'''
A fundamental conjecture in number theory is the Riemann hypothesis, which implies the prime number theorem with an optimally strong error term. While a proof remains elusive, many results in number theory can nonetheless be proved using weaker inputs. I will discuss how one such weaker input, subconvexity, can be used to prove strong results on the equidistribution of geometric objects such as lattice points on the sphere. I will also discuss how various proofs of subconvexity reduce to understanding period integrals of automorphic forms.


''Wednesday, March 2, 4:00pm (VV B239 + [http://go.wisc.edu/wuas48 Live Stream]).''


Recall that an algebraic variety is said to be rational if it has a Zariski open subset that is isomorphic to an open subset of projective space. There has been a great deal of recent activity and progress on questions of rationality, but most varieties aren't rational. I will survey a body of work concerned with measuring and controlling “how irrational” a given variety might be.
'''Monday, March 18. Cole Graham'''


'''Invasion in general domains'''


'''Seminar: Measures of association for algebraic varieties'''
The sciences teem with examples of invasion, in which one steady state spatially invades another. Mathematically, we can express this phenomenon through reaction-diffusion equations. These are well understood in the free space, but applications call for more complex geometries. In this talk, I will discuss reaction-diffusion invasion in multiple dimensions and general domains.


''Friday, March 4, 4:00pm (VV B239 )''
'''Wednesday, March 20. Wanlin Li'''


I will discuss some recent work with Olivier Martin that attempts to quantify how far two varieties are from being birationally isomorphic. Besides presenting a few results, I will discuss many open problems and avenues for further investigation.
'''Diophantine problem and rational points on curves'''


== March 11, 2022, [https://people.math.wisc.edu/~anderson/ David Anderson] (UW-Madison) ==
Diophantine problem asks for integral/rational solutions to polynomial equations. These solutions correspond to rational points on algebraic varieties. The study of Diophantine problems led to many essential developments of modern number theory and arithmetic geometry. Today I will briefly discuss the history of Diophantine problems and introduce various tools developed to study these problems. I will also introduce my joint work with Litt, Salter and Srinivasan on constructing cohomology classes which provide obstruction to the existence of rational points on curves.


'''Stochastic models of reaction networks and the Chemical Recurrence Conjecture'''


Cellular, chemical, and population processes are all often represented via networks that describe the interactions between the different population types (typically called the ''species'').   
<div id="Kelley">'''Friday, April 12.  Mikayla Kelley'''


If the counts of the species are low, then these systems are most often modeled as continuous-time Markov chains on $Z^d$ (with d being the number of species), with rates determined by stochastic mass-action kinetics.  A natural (broad) question is:  how do the qualitative properties of the dynamical system relate to the properties of the network?  One specific conjecture, called the Chemical Recurrence Conjecture, and that has been open for decades, is the following: if each connected component of the network is strongly connected, then the associated stochastic model is positive recurrent (meaning the model is quite stable). 
'''Accuracy and the Patterns of Rational Credence'''


I will give a general introduction to this class of models and will present the latest work towards a proof of the Chemical Recurrence Conjecture.   I will make this talk accessible to graduate students, regardless of their field of study.  Some of the new results presented are joint with Daniele Cappelletti, Andrea Agazzi, and Jonathan Mattingly.
A credence is a belief-like attitude that encodes one's degree of confidence in some way the world could be. For example, you might be 60% confident that the Democrats will win the presidential election. Some patterns of credence are irrational. Being 90% confident that Goldbach's conjecture is true and 90% confident that Goldbach's conjecture is false seems irrational. This is because it violates the following plausible pattern of rational credence: your credences in p and not p sum to 100%. How do we identify the patterns of rational credence? According to accuracy-first epistemology, we do so by identifying which patterns promote accuracy, where accuracy is represented formally as a real-valued function. In this talk, I will introduce the basics of accuracy-first epistemology and discuss my own work on using accuracy to study the patterns of rational credence when one has infinitely many credences.


== March 25, 2022, Friday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom]. [http://www.math.lsa.umich.edu/~canary/ Richard Canary] (Michigan) ==


(hosted by Zimmer)
'''Friday, May 3. Jin-Yi Cai'''


'''Hitchin representations of Fuchsian groups'''
'''Shor's Quantum Algorithm Does Not Factor Large Integers in the Presence of Noise'''


Abstract:  The Teichm&uuml;ller space $\mathcal T(S)$ of all hyperbolic structures on a fixed closed surface $S$ is a central object in geometry, topology and dynamics. It may be viewed  as the orbifold universal cover of the moduli space of algebraic curves of fixed genus and also as a component of the space of (conjugacy classes of) representations of $\pi_1(S)$ into $\mathsf{PSL}(2,\mathbb R)$ which is topologically a cell.
Shor's quantum factoring algorithm is the raison d'être for the field of quantum computing. The security of encryption systems such as RSA depends on the (conjectured) infeasibility of factoring in (classical) polynomial time, but Shor's algorithm can do so in Bounded-error Quantum Polynomial time (BQP). The key ingredient of this algorithm is the so-called Quantum Fourier Transform (QFT). BQP (in particular QFT) assumes infinite precision quantum rotation gates are available. This talk presents the [https://arxiv.org/abs/2306.10072 first proof] that, if the rotation gates have a vanishingly small level of noise, Shor's algorithm does not factor integers of the form n = pq for a positive density of primes p and q. It also fails with probability 1 - o(1) for random primes p and q. This proof applies to any algorithm that uses QFT. If time permits, I will also discuss my (speculative) view on the suitability of BQP replacing P or BPP in the strong Church-Turing thesis.
Hitchin discovered a component $\mathcal H_d(S)$ of the space of (conjugacy classes of) representations of $\pi_1(S)$ into $\mathsf{PSL}(d,\mathbb R)$
== Future Colloquia ==
which is topologically a cell. Subsequently, many striking analogies between the Hitchin component $\mathcal H_d(S)$ and Teichm&uuml;ller space $\mathcal T(S)$ were found. For example, Labourie showed that all representations in $\mathcal H_d(S)$ are discrete, faithful quasi-isometric embeddings.
[[Colloquia/Spring 2025|Spring 2025]]


In this talk, we will begin by gently reviewing the parallel theories of Teichm&uuml;ller space and the Hitchin component. We will finish by reviewing a long term project to develop a geometric theory of the augmented Hitchin component which parallels the classical theory of the augmented Hitchin component (which one may view as the "orbifold universal cover" of the Deligne-Mumford compactification of Teichm&uuml;ller space). This program includes joint work with Harry Bray, Nyima Kao, Giuseppe Martone, Tengren Zhang and Andy Zimmer.
[[Colloquia/Fall 2024|Fall 2024]]


== April 1, 2022, Friday at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom broadcast], [https://www.patelp.com/ Priyam Patel] (Utah) ==
== Past Colloquia ==


(hosted by WIMAW)
[[Colloquia/Spring2024|Spring 2024]]


'''Infinite-type surfaces'''
[[Colloquia/Fall 2023|Fall 2023]]


Surfaces fall into two categories: finite-type and infinite-type. The theory of infinite-type surfaces has been historically less developed than that of finite-type surfaces, but in the last few years, there has been a surge of interest in surfaces of infinite type and their mapping class groups (informally thought of as the groups of topological symmetries of these surfaces). In this talk, I will survey some of the biggest open problems in this quickly growing subfield of geometric group theory and topology, and discuss some of my own recent joint work towards resolving them.
[[Colloquia/Spring2023|Spring 2023]]
 
== April 8, 2022, Friday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream], [https://math.temple.edu/~tuf27009/index.html Matthew Stover] (Temple University) ==
 
(hosted by Zimmer)
 
'''A geometric characterization of arithmeticity'''
 
An old, fundamental problem is classifying closed n-manifolds admitting a metric of constant curvature. The most mysterious case is constant curvature -1, that is, hyperbolic manifolds, and these divide further into "arithmetic" and "nonarithmetic" manifolds. However, it is not at all evident from the definitions that this distinction has anything to do with the differential geometry of the manifold. Uri Bader, David Fisher, Nicholas Miller and I gave a geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen. I will give an overview, assuming only basic differential topology, of how (non)arithmeticity and totally geodesic submanifolds are connected, then describe how this allows us to import tools from ergodic theory and homogeneous dynamics originating in groundbreaking work of Margulis to prove our characterization. Given time, I will mention some more recent developments and open questions.
 
== April 15, 2022, Friday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream], [https://www.qatar.tamu.edu/programs/science/faculty-and-staff/berhand-lamel Bernhard Lamel], (Texas A&M University at Qatar) ==
 
(hosted by Gong)
 
'''Convergence and Divergence of Formal Power Series Maps'''
 
Consider two real-analytic hypersurfaces (i.e. defined by convergent power series) in complex spaces. A formal holomorphic map is said to take one into the other if the composition of the power series defining the target with the map (which is just another formal power series) is a (formal) multiple of the defining power series of the source. In this talk, we are going to be interested in conditions for formal holomorphic maps to necessarily be convergent. Now, a formal holomorphic map taking the real line to itself is just a formal power series with real coefficients; this example also gives rise to real hypersurfaces in higher dimensional complex spaces having divergent formal self-maps. On the other hand, a formal map taking the unit sphere in higher dimensional complex space to itself is necessarily a rational map with poles outside of the sphere, in particular, the formal power series defining it converges. The convergence theory for formal self-maps of real hypersurfaces has been developed in the late 1990s and early 2000s. For formal embeddings, “ideal" conditions had been long conjectured. I’m going to give an introduction to this problem and talk about some joint work from 2018 with Nordine Mir giving a basically complete answer to the question when a formal map taking a real-analytic hypersurface in complex space into another one is necessarily convergent.
 
== April 25-26-27 (Monday [VV B239], Tuesday [Chamberlin 2241], Wednesday [VV B239]) 4 pm  [https://math.mit.edu/directory/profile.php?pid=1461 Larry Guth] (MIT) ==
 
(''Departmental Distinguished Lecture series'')
 
'''Reflections on decoupling and Vinogradov's mean value problem.'''
 
Decoupling is a recent development in Fourier analysis that has solved several longstanding problems.  The goal of the lectures is to describe this development to a general mathematical audience.
 
We will focus on one particular application of decoupling: Vinogradov's mean value problem from analytic number theory.  This problem is about the number of solutions of a certain system of diophantine equations.  It was raised in the 1930s and resolved in the last decade.
 
We will give some context about this problem, but the main goal of the lectures is to explore the ideas that go into the proof.  The method of decoupling came as a big surprise to me, and I think to other people working in the field.
The main idea in the proof of decoupling is to combine estimates from many different scales.  We will describe this process and reflect on why it is helpful.
 
'''Lecture 1:'''Introduction to decoupling and Vinogradov's mean value problem.
 
In this lecture, we introduce Vinogradov's problem and give an overview of the proof.
 
'''Lecture 2:''' Features of the proof of decoupling.
 
In this lecture, we look more closely at some features of the proof of decoupling.  The first feature we examine is the exact form of writing the inequality, which is especially suited for doing induction and connecting information from different scales.  The second feature we examine is called the wave packet decomposition.  This structure has roots in quantum physics and in information theory.
 
'''Lecture 3:''' Open problems.
 
In this lecture, we discuss some open problems in number theory that look superficially similar to Vinogradov mean value conjecture, such as Hardy and Littlewood's Hypothesis K*.  In this lecture, we probe the limitations of decoupling by exploring why the techniques from the first two lectures don't work on these open problems.  Hopefully this will give a sense of some of the issues and difficulties involved in these problems.
 
== May 10+12, 2022, Tuesday+Thursday, 12pm on [https://go.wisc.edu/d456cn Zoom]. [http://www.ma.huji.ac.il/~kalai/ Gil Kalai] (Hebrew University) ==
 
(''Hilldale Lectures / Special colloquium'')
 
'''The argument against quantum computers'''
 
In the first lecture I will introduce quantum computers and present an argument for why
quantum computers are impossible.  From my analysis I will derive general principles for the
behavior of noisy quantum systems and will also briefly discuss the recent announcements
concerning "quantum computational supremacy,” which conflict with my theory.
 
In the second lecture I will discuss the connection between the possibility of quantum computers,
the predictability of complex quantum systems in nature, and the issue of free will.
 
Both lectures are self-contained, intended for a wide audience and assume no background
on quantum computers or philosophy. 
 
'''Lecture I: The Argument Against Quantum Computers'''
 
A quantum computer is a new type of computer based on quantum physics.
When it comes to certain computational objectives, the computational ability of quantum
computers is tens, and even hundreds of orders of magnitude faster than that of the familiar digital computers, and their construction will enable us to factor large
numbers and to break most of the current cryptosystems.
We will describe a computational complexity argument against the feasibility of quantum
computers. We identify a very low complexity class of probability distributions described by
noisy intermediate-scale quantum computers (NISQ computers), and explain why it will
allow neither good-quality quantum error-correction nor a demonstration of "quantum
supremacy."
 
The analysis also shows that for a wide range of noise rates NISQ computers are inherently
chaotic in the strong sense that their output cannot be predicted even probabilistically. 
Some general principles governing the behavior of noisy quantum systems in a "world
devoid of quantum computers" will be derived.
 
I will briefly discuss the recent announcements regarding "quantum computational
supremacy" by scientists from Google ("Sycamore") and from USTC, which conflict with my
theory.
 
The lecture is going to be self-contained, it is intended for a wide audience, and we assume
no prior knowledge of quantum computers.
 
Relevant papers are:
https://arxiv.org/abs/1908.02499
https://arxiv.org/abs/2008.05188
https://arxiv.org/abs/1409.3093
https://arxiv.org/abs/2008.05177
 
'''Lecture II: Quantum Computers, Predictability and Free Will'''
 
We will discuss the connection between the possibility of quantum computers, the
predictability of complex quantum systems in nature, and the issue of free will.
The argument regarding the impossibility of quantum computers implies that the future of
complex quantum systems in nature cannot be predicted. A more involved argument shows
that the impossibility of quantum computation supports the view whereby the laws of nature
do not in fact contradict free will. For this philosophical journey, we discuss in parallel the
Google “Sycamore” quantum computer of 12 computational units (qubits), and the human-
being Alice, whose free will we attempt to analyze.
 
At the center of the argument is the ambiguity inherent in the way the future is determined
by the past; ambiguity that is not expressed in terms of the mathematical laws of physics
(which are fully deterministic) but rather in terms of the physical description of the objects
we refer to.
 
The lecture will be self-contained and we will not assume prior background regarding
quantum computers or philosophy. (It will also not rely on the first lecture.)
 
A relevant paper is:
https://arxiv.org/abs/2204.02768
 
== Future Colloquia ==


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


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


== Past Colloquia ==
[[Colloquia/Fall2021|Fall 2021]]
[[Colloquia/Fall2021|Fall 2021]]



Latest revision as of 00:37, 19 March 2024


UW Madison mathematics Colloquium is on Fridays at 4:00 pm in Van Vleck B239 unless otherwise noted.

Contacts for the colloquium are Simon Marshall and Dallas Albritton.


Spring 2024

date speaker title host(s)
Monday Jan 22 at 4pm in B239 Yingkun Li (Darmstadt Tech U, Germany) Arithmetic of real-analytic modular forms Yang
Thursday Jan 25 at 4pm in VV911 Sanjukta Krishnagopal (UCLA/UC Berkeley) Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes Smith
Jan 26 Jacob Bedrossian (UCLA) Lyapunov exponents in stochastic systems Tran
Feb 2 William Chen Orbit problems and the mod p properties of Markoff numbers Arinkin
Feb 9 No colloquium
Feb 16 Jack Lutz (Iowa State) Algorithmic Fractal Dimensions Guo
Feb 23 No colloquium
Mar 1 Per-Gunnar Martinsson (UT-Austin) Randomized algorithms for linear algebraic computations Li
Mar 8 Anton Izosimov (U of Arizona) Incidences and dimers Gloria Mari-Beffa
Mar 15 Peter Humphries (Virginia) Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity Marshall
Monday Mar 18 at 4pm in B239 Cole Graham (Brown) Invasion in general domains Albritton, Smith, Tran
Wednesday Mar 20 at 4 pm in B239 Wanlin Li (Washington U St Louis) Diophantine problem and rational points on curves Dymarz, GmMaW
Mar 29 Spring break
Apr 5 Ovidiu Savin (Columbia) Tran
Apr 12 Mikayla Kelley (U Chicago Philosophy) Math And... seminar: Accuracy and the Patterns of Rational Credence Ellenberg, Marshall
Apr 19 Yanyan Li (Rutgers) Tran
Apr 26 Chris Leininger (Rice) TBA Uyanik
May 3 Jin-Yi Cai (UW-Madison) Shor's Quantum Algorithm Does Not Factor Large Integers in the Presence of Noise Yang

Abstracts

Monday, January 22. Yingkun Li

Arithmetic of real-analytic modular forms

Modular form is a classical mathematical object dating back to the 19th century. Because of its connections to and appearances in many different areas of math and physics, it remains a popular subject today. Since the work of Hans Maass in 1949, real-analytic modular form has found important applications in arithmetic geometry and number theory. In this talk, I will discuss the amazing works in this area over the past 20 years, and give a glimpse of its fascinating future directions.

Thursday, January 25. Sanjukta Krishnagopal

Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes

In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application.

First, I will discuss some recent advances in mathematical machine learning for prediction on graphs. Machine learning is often a black box. Here I will present some exact theoretical results on the dynamics of weights while training graph neural networks using graphons - a graph limit or a graph with infinitely many nodes. I will use these ideas to present a new method for predictive and personalized medicine applications with remarkable success in prediction of Parkinson's subtype five years in advance.

Then, I will discuss some work on higher-order models of graphs: simplicial complexes - that can capture simultaneous many-body interactions. I will present some recent results on spectral theory of simplicial complexes, as well as introduce a mathematical framework for studying the topology and dynamics of multilayer simplicial complexes using Hodge theory, and discuss applications of such interdisciplinary methods to studying bias in society, opinion dynamics, and hate speech in social media.


Friday, January 26. Jacob Bedrossian

Lyapunov exponents in stochastic systems

In this overview talk we discuss several results regarding positive Lyapunov exponents in stochastic systems. First we discuss proving "Lagrangian chaos" in stochastic fluid mechanics, that is, demonstrating a positive Lyapunov exponent for the motion of a particle in the velocity field arising from the stochastic Navier-Stokes equations. We describe how this chaos can be used to deduce qualitatively optimal almost-sure exponential mixing of passive scalars. Next we describe more recently developed methods for obtaining strictly positive lower bounds and some quantitative estimates on the top Lyapunov exponent of weakly-damped stochastic differential equations, such as Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (called "Eulerian chaos" in fluid mechanics). Further applications of the ideas to the chaotic motion of charged particles in fluctuating magnetic fields and the non-uniqueness of stationary measures for Lorenz 96 in degenerate forcing situations will be discussed if time permits. All of the work except for the charged particles (joint with Chi-Hao Wu) is joint with Alex Blumenthal and Sam Punshon-Smith.

Friday, February 2. William Chen

Orbit problems and the mod p properties of Markoff numbers

Markoff numbers are positive integers which encode how resistant certain irrational numbers are to being approximated by rationals. In 1913, Frobenius asked for a description of all congruence conditions satisfied by Markoff numbers modulo primes p. In 1991 and 2016, Baragar, Bourgain, Gamburd, and Sarnak conjectured a refinement of Frobenius’s question, which amounts to showing that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation”; that is to say: they conjecture that its integral points surject onto its mod p points for every prime p. In this talk we will show how to prove this conjecture for all but finitely many primes p, thus reducing the conjecture to a finite computation. A key step is to understand this problem in the context of describing the orbits of certain group actions. Primarily, we will consider the action of the mapping class group of a topological surface S on (a) the set of G-covers of S, where G is a finite group, and (b) on the character variety of local systems on S. Questions of this type have been related to many classical problems, from proving that the moduli space of curves of a given genus is connected, to Grothendieck’s ambitious plan to understand the structure of the absolute Galois group of the rationals by studying its action on “dessins d’enfant”. We will explain some of this history and why such problems can be surprisingly difficult.


Friday, February 16. Jack Lutz

Algorithmic Fractal Dimensions

Algorithmic fractal dimensions are computability theoretic versions of Hausdorff dimension and other fractal dimensions. This talk will introduce algorithmic fractal dimensions with particular focus on the Point-to-Set Principle. This principle has enabled several recent proofs of new theorems in geometric measure theory. These theorems, some solving long-standing open problems, are classical (meaning that their statements do not involve computability or logic), even though computability has played a central in their proofs.


Friday, March 1. Per-Gunnar Martinsson

Randomized algorithms for linear algebraic computations

The talk will describe how randomized algorithms can effectively, accurately, and reliably solve linear algebraic problems that are omnipresent in scientific computing and in data analysis. We will focus on techniques for low rank approximation, since these methods are particularly simple and powerful, and are well understood mathematically. The talk will also briefly survey a number of other randomized algorithms for tasks such as solving linear systems, estimating matrix norms, and computing full matrix factorizations.


Friday, March 8. Anton Izosimov

Incidences and dimers

Incidence theorems are statements about points, lines, and possibly higher-dimensional subspaces and their incidences. Examples include classical theorems of Desargues and Pappus. In this talk, we'll discuss a connection between incidence geometry and an archetypal model of statistical physics - the dimer model. The talk will be based on the work of many people, including my ongoing work with Pavlo Pylyavskyy (Minnesota).


Friday, March 15. Peter Humphries

Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity

A fundamental conjecture in number theory is the Riemann hypothesis, which implies the prime number theorem with an optimally strong error term. While a proof remains elusive, many results in number theory can nonetheless be proved using weaker inputs. I will discuss how one such weaker input, subconvexity, can be used to prove strong results on the equidistribution of geometric objects such as lattice points on the sphere. I will also discuss how various proofs of subconvexity reduce to understanding period integrals of automorphic forms.


Monday, March 18. Cole Graham

Invasion in general domains

The sciences teem with examples of invasion, in which one steady state spatially invades another. Mathematically, we can express this phenomenon through reaction-diffusion equations. These are well understood in the free space, but applications call for more complex geometries. In this talk, I will discuss reaction-diffusion invasion in multiple dimensions and general domains.

Wednesday, March 20. Wanlin Li

Diophantine problem and rational points on curves

Diophantine problem asks for integral/rational solutions to polynomial equations. These solutions correspond to rational points on algebraic varieties. The study of Diophantine problems led to many essential developments of modern number theory and arithmetic geometry. Today I will briefly discuss the history of Diophantine problems and introduce various tools developed to study these problems. I will also introduce my joint work with Litt, Salter and Srinivasan on constructing cohomology classes which provide obstruction to the existence of rational points on curves.


Friday, April 12. Mikayla Kelley

Accuracy and the Patterns of Rational Credence

A credence is a belief-like attitude that encodes one's degree of confidence in some way the world could be. For example, you might be 60% confident that the Democrats will win the presidential election. Some patterns of credence are irrational. Being 90% confident that Goldbach's conjecture is true and 90% confident that Goldbach's conjecture is false seems irrational. This is because it violates the following plausible pattern of rational credence: your credences in p and not p sum to 100%. How do we identify the patterns of rational credence? According to accuracy-first epistemology, we do so by identifying which patterns promote accuracy, where accuracy is represented formally as a real-valued function. In this talk, I will introduce the basics of accuracy-first epistemology and discuss my own work on using accuracy to study the patterns of rational credence when one has infinitely many credences.


Friday, May 3. Jin-Yi Cai

Shor's Quantum Algorithm Does Not Factor Large Integers in the Presence of Noise

Shor's quantum factoring algorithm is the raison d'être for the field of quantum computing. The security of encryption systems such as RSA depends on the (conjectured) infeasibility of factoring in (classical) polynomial time, but Shor's algorithm can do so in Bounded-error Quantum Polynomial time (BQP). The key ingredient of this algorithm is the so-called Quantum Fourier Transform (QFT). BQP (in particular QFT) assumes infinite precision quantum rotation gates are available. This talk presents the first proof that, if the rotation gates have a vanishingly small level of noise, Shor's algorithm does not factor integers of the form n = pq for a positive density of primes p and q. It also fails with probability 1 - o(1) for random primes p and q. This proof applies to any algorithm that uses QFT. If time permits, I will also discuss my (speculative) view on the suitability of BQP replacing P or BPP in the strong Church-Turing thesis.

Future Colloquia

Spring 2025

Fall 2024

Past Colloquia

Spring 2024

Fall 2023

Spring 2023

Fall 2022

Spring 2022

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012

WIMAW