Colloquia: Difference between revisions
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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. | 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. | ||
<div id="Lutz">'''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. | |||
== Past Colloquia == | == Past Colloquia == | ||
Revision as of 04:12, 11 February 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 | |||
| Mar 1 | Per-Gunnar Martinsson (UT-Austin) | TBA | Li |
| Mar 8 | Anton Izosimov (U of Arizona) | Gloria Mari-Beffa | |
| Mar 15 | Peter Humphries (Virginia) | Marshall | |
| Mar 20 | Wanlin Li (Washington U St Louis) | Dymarz, GmMaW | |
| Mar 29 | Spring break | ||
| Apr 5 | Ovidiu Savin (Columbia) | Tran | |
| Apr 12 | Mikayla Kelley (U Chicago Philosophy) | Math And... seminar, title TBA | Ellenberg, Marshall |
| Apr 19 | Yanyan Li (Rutgers) | Tran | |
| Apr 26 | Chris Leininger (Rice) | TBA | Uyanik |
Abstracts
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.
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.
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.