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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.


=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) ==
==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
|}


(hosted by Gurevich)
== Abstracts ==


'''Finitely Presented Groups in Arithmetic Geometry'''
<div id="Li">'''Monday, January 22.  Yingkun Li'''  


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.
'''Arithmetic of real-analytic modular forms'''


== 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) ==
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.      
'''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 stability.  The 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.
'''Thursday, January 25. Sanjukta Krishnagopal'''  


== October 1, 2021, B239 + [http://go.wisc.edu/wuas48 Live stream], [https://people.math.wisc.edu/~andreic/ Andrei Caldararu] (UW-Madison) ==
'''Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes'''
'''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.
In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application.  


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.
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.


== 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) ==
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.


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


'''Asymptotics beyond all orders: the devil's invention?'''


"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel.
'''Friday, January 26. Jacob Bedrossian'''


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.
'''Lyapunov exponents in stochastic systems'''


== 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) ==
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.


('''Distinguished Lecture Series'''; hosted by Gurevich)
<div id="Chen">'''Friday, February 2. William Chen'''


'''Geometric representation theory and modular representations'''
'''Orbit problems and the mod p properties of Markoff numbers'''


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!
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.


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


(hosted by Gurevich/Gorin)
<div id="Lutz">'''Friday, February 16. Jack Lutz'''


'''Supersymmetry and tensor categories'''
'''Algorithmic Fractal Dimensions '''


I will explain how representation theory of supergroups and
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.
supergeometry are related to general theory of tensor categories,
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) ==


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


'''Polynomial averages and pointwise ergodic theorems on nilpotent groups'''
'''Randomized algorithms for linear algebraic computations '''


I will talk about some recent work on pointwise almost
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.
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) ==


(hosted by Uyanik)
<div id="Izosimov">'''Friday, March 8. Anton Izosimov'''


'''Surfaces and Point Processes'''
'''Incidences and dimers '''


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.
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).


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


(hosted by Stovall)
<div id="Humphries">'''Friday, March 15. Peter Humphries'''


'''An exploration in analysis on fractals '''
'''Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity'''


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.
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.


== November 19, 2021 , [https://math.wisc.edu/ TBA] (TBA) ==


(reserved by the hiring committee)
'''Monday, March 18. Cole Graham'''
Title: Low regularity solution for quasilinear PDEs


Abstract: 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.
'''Invasion in general domains'''


== December 3, 2021 , [https://math.wisc.edu/ TBA] (TBA) ==
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.


(reserved by the hiring committee)
'''Wednesday, March 20. Wanlin Li'''


== December 10, 2021 , [https://math.wisc.edu/ TBA] (TBA) ==
'''Diophantine problem and rational points on curves'''


(reserved by the hiring committee)
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.


== Future ==


[[Colloquia/Spring2022|Spring 2022]]
<div id="Kelley">'''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 [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.
== Future Colloquia ==
[[Colloquia/Spring 2025|Spring 2025]]
 
[[Colloquia/Fall 2024|Fall 2024]]


== Past Colloquia ==
== Past Colloquia ==
[[Colloquia/Spring2024|Spring 2024]]
[[Colloquia/Fall 2023|Fall 2023]]
[[Colloquia/Spring2023|Spring 2023]]
[[Colloquia/Fall2022|Fall 2022]]
[[Spring 2022 Colloquiums|Spring 2022]]
[[Colloquia/Fall2021|Fall 2021]]


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

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