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<b>UW Madison mathematics Colloquium is ONLINE 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.


=Spring 2021=
           


== January 27, 2021 '''[Wed 4-5pm]''', [https://sites.google.com/view/morganeaustern/home Morgane Austern] (Microsoft Research) ==
==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 Roch)
== Abstracts ==


'''Asymptotics of learning on dependent and structured random objects'''
<div id="Li">'''Monday, January 22.  Yingkun Li'''  


Classical statistical inference relies on numerous tools from probability theory to study
'''Arithmetic of real-analytic modular forms'''
the properties of estimators. However, these same tools are often inadequate to study
modern machine problems that frequently involve structured data (e.g networks) or
complicated dependence structures (e.g dependent random matrices). In this talk, we
extend universal limit theorems beyond the classical setting.


Firstly, we consider distributionally “structured” and dependent random object–i.e
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.      
random objects whose distribution are invariant under the action of an amenable group.
We show, under mild moment and mixing conditions, a series of universal second and
third order limit theorems: central-limit theorems, concentration inequalities, Wigner
semi-circular law and Berry-Esseen bounds. The utility of these will be illustrated by
a series of examples in machine learning, network and information theory. Secondly
by building on these results, we establish the asymptotic distribution of the cross-
validated risk with the number of folds allowed to grow at an arbitrary rate. Using
this, we study the statistical speed-up of cross validation compared to a train-test split
procedure, which reveals surprising results even when used on simple estimators.


== January 29, 2021, [https://sites.google.com/site/isaacpurduemath/ Isaac Harris] (Purdue) ==
'''Thursday, January 25. Sanjukta Krishnagopal'''


(Hosted by Smith)
'''Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes'''


'''Direct Sampling Algorithms for Inverse Scattering'''
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 this talk, we will discuss a recent qualitative imaging method referred to as the Direct Sampling Method for inverse scattering. This method allows one to recover a scattering object by evaluating an imaging functional that is the inner-product of the far-field data and a known function. It can be shown that the imaging functional is strictly positive in the scatterer and decays as the sampling point moves away from the scatterer. The analysis uses the factorization of the far-field operator and the Funke-Hecke formula. This method can also be shown to be stable with respect to perturbations in the scattering data. We will discuss the inverse scattering problem for both acoustic and electromagnetic waves. This is joint work with A. Kleefeld and D.-L. Nguyen.
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.


== February 1, 2021 '''[Mon 4-5pm]''', [https://services.math.duke.edu/~nwu/index.htm Nan Wu] (Duke) ==
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.


(Hosted by Roch)


'''From Manifold Learning to Gaussian Process Regression on Manifolds'''


In this talk, I will review the concepts in manifold learning and discuss a famous manifold learning algorithm, the Diffusion Map. I will talk about my recent research results which theoretically justify that the Diffusion Map reveals the underlying topological structure of the dataset sampled from a manifold in a high dimensional space. Moreover, I will show the application of these theoretical results in solving the regression problems on manifolds and ecological problems in real life.
'''Friday, January 26. Jacob Bedrossian'''


== February 5, 2021, [https://hanbaeklyu.com/ Hanbaek Lyu] (UCLA) ==
'''Lyapunov exponents in stochastic systems'''


(Hosted by Roch)
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.


'''Dictionary Learning from dependent data samples and networks'''
<div id="Chen">'''Friday, February 2. William Chen'''


Analyzing group behavior of systems of interacting variables is a ubiquitous problem in many fields including probability, combinatorics, and dynamical systems. This problem also naturally arises when one tries to learn essential features (dictionary atoms) from large and structured data such as networks. For instance, independently sampling some number of nodes in a sparse network hardly detects any edges between adjacent nodes. Instead, we may perform a random walk on the space of connected subgraphs, which will produce more meaningful but correlated samples. As classical results in probability were first developed for independent variables and then gradually generalized for dependent variables, many algorithms in machine learning first developed for independent data samples now need to be extended to correlated data samples. In this talk, we discuss some new results that accomplish this including some for online nonnegative matrix and tensor factorization for Markovian data. A unifying technique for handling dependence in data samples we develop is to condition on the distant past, rather than the recent history. As an application, we present a new approach for learning "basis subgraphs" from network data, that can be used for network denoising and edge inference tasks. We illustrate our method using several synthetic network models as well as Facebook, arXiv, and protein-protein interaction networks, that achieve state-of-the-art performance for such network tasks when compared to several recent methods.
'''Orbit problems and the mod p properties of Markoff numbers'''


== February 8, 2021 '''[Mon 4-5pm]''', [https://sites.google.com/view/mndaoud/home Mohamed Ndaoud] (USC) ==
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.


(Hosted by Roch)


'''SCALED MINIMAX OPTIMALITY IN HIGH-DIMENSIONAL LINEAR REGRESSION: A NON-CONVEX ALGORITHMIC REGULARIZATION APPROACH'''
<div id="Lutz">'''Friday, February 16. Jack Lutz'''


The question of fast convergence in the classical problem of high dimensional linear regression has been extensively studied. Arguably, one of the fastest procedures in practice is Iterative Hard Thresholding (IHT). Still, IHT relies strongly on the knowledge of the true sparsity parameter s. In this paper, we present a novel fast procedure for estimation in the high dimensional linear regression. Taking advantage of the interplay between estimation, support recovery and optimization we achieve both optimal statistical accuracy and fast convergence. The main advantage of our procedure is that it is fully adaptive, making it more practical than state of the art IHT methods. Our procedure achieves optimal statistical accuracy faster than, for instance, classical algorithms for the Lasso. Moreover, we establish sharp optimal results for both estimation and support recovery. As a consequence, we present a new iterative hard thresholding algorithm for high dimensional linear regression that is scaled minimax optimal (achieves the estimation error of the oracle that knows the sparsity pattern if possible), fast and adaptive.
'''Algorithmic Fractal Dimensions '''


== February 12, 2021, [https://sites.math.washington.edu/~blwilson/ Bobby Wilson] (University of Washington) ==
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.


(Hosted by Smith)


== February 17, 2021 '''[Wed 9-10am]''', [https://www.math.ias.edu/~visu/ Visu Makam] (IAS)==
<div id="Martinsson">'''Friday, March 1. Per-Gunnar Martinsson'''


(Hosted by Roch)
'''Randomized algorithms for linear algebraic computations '''


== February 19, 2021, [http://www.mauricefabien.com/ Maurice Fabien] (Brown)==
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.


(Hosted by Smith)


== February 26, 2021, [https://www.math.ias.edu/avi/home Avi Wigderson] (Princeton IAS) ==
<div id="Izosimov">'''Friday, March 8. Anton Izosimov'''


(Hosted by Gurevitch)
'''Incidences and dimers '''


== March 12, 2021, [] ==
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).


(Hosted by )


== March 26, 2021, [] ==
<div id="Humphries">'''Friday, March 15. Peter Humphries'''


(Hosted by )
'''Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity'''


== April 9, 2021, '''8pm''' [] ==
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.


(Hosted by )


== April 23, 2021, [] ==
'''Monday, March 18. Cole Graham'''


(Hosted by )
'''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.
<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/Fall2020|Fall 2020]]
[[Colloquia/Fall2020|Fall 2020]]

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