The Colloquium is held on Mondays at 4pm on Zoom with meeting number 938 9042 5892.

Spring Quarter, 2021

• April 5
  Film Screening : Navajo Math Circles, a film by George Csicsery
• May 3, Lillian Pierce (Duke)
  Counting problems: open questions in number theory, from the perspective of moments
  Abstract: Many questions in number theory can be phrased as counting problems. How many number fields are there? How many elliptic curves are there? How many integral solutions to this system of Diophantine equations are there? If the answer is “infinitely many,” we want to understand the order of growth for the number of objects we are counting in the “family.” But in many settings we are also interested in finer-grained questions, like: how many number fields are there, with fixed degree and fixed discriminant? We know the answer is “finitely many,” but it would have important consequences if we could show the answer is always “very few indeed.” In this accessible talk, we will describe a very general way that these finer-grained questions can be related to the bigger infinite-family questions. Then we will use this perspective to survey interconnections between several big open conjectures in number theory, related in particular to class groups and number fields.
• May 24, Andy Putman (Notre Dame)
  The mapping class group of a surface
  Abstract: The mapping class group of a surface is a remarkable group with connections to low-dimensional topology, algebraic geometry, dynamics, and many other areas. I will give an elementary survey of some aspects of this topic with a focus on finiteness properties.

Winter Quarter, 2021

• January 11, Carina Curto (Penn State)
  Graphs, network motifs, and threshold-linear algebra in the brain
  Abstract: Threshold-linear networks (TLNs) are commonly-used rate models for modeling neural networks in the brain. Although the nonlinearity is quite simple, it leads to rich dynamics that can capture a variety of phenomena observed in neural activity: persistent activity, multistability, sequences, oscillations, etc. Here we study competitive threshold-linear networks, which exhibit both static and dynamic attractors. These networks have corresponding hyperplane arrangements whose oriented matroids encode important features of the dynamics. We will show how the graph associated to such a network yields constraints on the set of (stable and unstable) fixed points, and how these constraints affect the dynamics. In the special case of combinatorial threshold-linear networks (CTLNs), we find an even stronger set of “graph rules” that allow us to predict emergent sequences and to engineer networks with prescribed dynamic attractors.
• February 25
  Thursday, 4-5pm, Zoom meeting number 938 9042 5892
  Film Screening: Secrets of the Surface: the mathematical vision of Maryam Mirzakhani

Fall Quarter, 2020

• October 5, Mark Goresky (Institute for Advanced Study)
  Pseudo-random numbers and sequences
  Abstract: I will discuss three or four striking historical applications of pseudo-random numbers and some of the mathematics behind their synthesis and analysis.
• October 26, Louis Billera (Cornell University)
  On the real linear algebra of vectors of zeros and ones
  Abstract: We discuss two related properties of systems of finite sets that arose, respectively, in the study of economic equilibria and in quantum physics. These lead to combinatorial questions concerning certain arrangements of hyperplanes in real vector spaces. Resolving these questions completely will require further understanding of the real linear algebra of vectors of zeros and ones. Recent progress along these lines will be surveyed, some indicating new connections to classical combinatorial objects such as set partitions, others to ideas in number theory and topology.
• November 9, Daniel Isaksen (Wayne State University)
  Stable homotopy groups of spheres
  Abstract: A fundamental problem in topology is to determine the groups of homotopy classes of maps between spheres of different dimensions. This is a wildly difficult problem, but the Freudenthal Suspension Theorem is a useful structural result that explains how these groups are related to each other under certain dimension conditions. The stable homotopy groups are the ones that satisfy these conditions.

  The stable homotopy groups have been studied since the middle of the twentieth century, with major contributions by Hopf, Serre, Adams, May, Mahowald, and others. Until recently, explicit knowledge of the stable homotopy groups reached dimension 61. Deformations of stable homotopy theory have allowed us to extend this range to dimension 90 and possibly beyond. These ideas were inspired by motivic homotopy theory, which is a homotopy theory for algebraic varieties.

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