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Colloquium. Academic Year 20-21

Thursdays 4:10pm on Zoom, unless otherwise noted

Colloquium Chair (2020-2021): Doug Hardin



September 17, 2020 (Thursday), 4:10 pm

Quantitative measure equivalence

Romain Tessera, Institut de Mathématiques de Jussieu-Paris
Virtual Talk via Zoom
Zoom Meeting ID: 998 6775 5871
Email mathcolloquium@vanderbilt.edu to request pass code

Measure equivalence is an equivalence relation between countable groups that has been introduced by Gromov. A fundamental instance are lattices in a same locally compact group. According to a famous result of Ornstein Weiss, all countable amenable groups are measure equivalent, meaning that geometry is completely rubbed out by this equivalence relation. Recently a more restrictive notion has been investigated called integrable measure equivalence, where the associated cocycles are assumed to be integrable. By contrast, a lot of surprising rigidity results have been proved: for instance Bowen has shown that the volume growth is invariant under integrable measure equivalence, and Austin proved that nilpotent groups that are integrable measure equivalent have bi-Lipschitz asymptotic cones. I will present a work whose goal is to understand more systematically how the geometry survives through measure equivalence when some (possibly very weak) integrability condition is imposed on the cocycles. We shall put the emphasis on amenable groups, for which we will present new rigidity results, and the first flexibility results known in this context. (Contact Person: Dietmar Bisch)

September 24, 2020 (Thursday), 4:10 pm

Equivariant homotopy commutativity, trees and chicken feet

Constanze Roitheim, University of Kent
Zoom Meeting ID: 998 6775 5871
Email mathcolloquium@vanderbilt.edu to request pass code

Commutativity up to homotopy can be daunting, and it becomes even more difficult to track when group actions get introduced. In the case of a finite group, however, the options for equivariant homotopy commutativity can be encoded using simple combinatorics, and we will show some examples. (Contact Person: Jocelyne Ishak)

*Talk will be offered live at 11:10a and replayed at 4:10pm*

October 1, 2020 (Thursday), 4:10 pm

Ramanujan: A Century Of Inspiration

Bruce C. Berndt, University of Illinois
Zoom Meeting ID: 998 6775 5871
Email mathcolloquium@vanderbilt.edu to request pass code

Srinivasa Ramanujan is perhaps the most enigmatic mathematician in the history of our subject. First, an account of Ramanujan’s life will be given. Second, the history of Ramanujan’s (earlier) notebooks and “lost” notebook will be provided. Third, the speaker will describe how he became fascinated with Ramanujan’s work, beginning with proving a few claims from his notebooks in February, 1974, and then since May, 1977, devoting all of his research efforts to proving the claims in Ramanujan’s earlier notebooks, lost notebook, and published papers. Fourth, some examples from Ramanujan’s notebooks and lost notebook will be given. This lecture will be aimed at a general audience. (Contact Person: Larry Rolen)

October 8, 2020 (Thursday), 4:10 pm

The Navier-Stokes, Euler and Other Related Equations

Edriss S. Titi, Texas A&M University, University of Cambridge, Weizmann Institute of Science
Zoom Meeting ID: 998 6775 5871
Email mathcolloquium@vanderbilt.edu to request pass code

In this talk I will present the most recent advances concerning the questions of global regularity
of solutions to the three-dimensional Navier-Stokes and Euler equations of incompressible fluids. Furthermore, I will also present recent global regularity (and finite time blow-up) results concerning certain three-dimensional geophysical flows, including the three-dimensional viscous (non-viscous) “primitive equations” of oceanic and atmospheric dynamics. (Contact Person: Gieri Simonett)

November 12, 2020 (Thursday), 4:10 pm

There are 160,839 + 160,650 3-planes in a 7-dimensional cubic hypersurface

Kirsten Wickelgren, Duke University
Zoom Meeting ID: 998 6775 5871
Zoom Meeting link: https://vanderbilt.zoom.us/j/99867755871
Email mathcolloquium@vanderbilt.edu to request pass code

Given a generic choice of polynomials with complex coefficients, one can compute the dimension of the straight lines, planes, or d-dimensional planes contained in the common zeros of the polynomials. When this dimension is 0, there is some finite number of d-planes. For example, there are 321,489 3-dimensional planes in the zero locus of a degree 3 homogeneous polynomial in 9 variables over the complex numbers. This number can be identified with the topological Euler number of a certain vector bundle. However, it only corresponds to the count of d-planes over an algebraically closed field like the complex numbers. We can get information over other fields like the real numbers, the rational numbers, or finite fields, by using an Euler number from A1-homotopy theory instead. This Euler number is no longer an integer; instead it is a bilinear form, and invariants of bilinear forms record information about the arithmetic and geometry of the planes. In this talk, we will introduce these enumerative problems and A1-Euler numbers. We establish integrality results for the A1-Euler class, and use this to compute the Euler numbers associated to arithmetic counts of d-planes on complete intersections in terms of topological Euler numbers over the real and complex numbers. The example in the title then follows from work of Finashin–Kharlamov. The new work in this talk is joint with Tom Bachmann.

(Contact Person: Anna Marie Bohmann)

December 3, 2020 (Thursday), 4:10 pm

Open problems within the N-body problem

Richard Montgomery, University of California, Santa Cruz
Zoom Meeting ID: 998 6775 5871
Zoom Meeting link: https://vanderbilt.zoom.us/j/99867755871
Email mathcolloquium@vanderbilt.edu to request pass code

The 333 year old N-body problem is alive and well. We survey four open problems within the problem  and recent progress on them, beginning with a survey of some solution curves.

(Contact Person: Marcelo Disconzi)

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