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

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.

• November 23, 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.

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