# Math Calendar

### Upcoming Events

## Dualizable Algebras in Residually Small Congruence Modular Varieties

Speaker: Keith Kearnes, University of Colorado Boulder

Location: Stevenson 1312

Call an algebra A strongly homogeneous if for any subalgebra B it is the case that any two homomorphisms of B into A differ by an automorphism of A. I will discuss the dualizability of strongly homogeneous finite algebras in residually small congruence modular varieties.

## Hyperbole and a Half-Plane

Levi Sledd, Vanderbilt University

Location: Stevenson 1206

For thousands of years, Euclid’s unwieldy fifth axiom, known as the parallel postulate, annoyed the mathematicians of the world. Countless tried, and failed, to prove the fifth axiom from the other four. Then in the nineteenth century, geometers János Bolyai and Nikolai Lobachevsky showed that, in some settings, it’s not true. This undergraduate seminar, we’ll talk about geometries in which the parallel postulate fails, with a special emphasis on the exceedingly strange and beautiful world of hyperbolic geometry. We’ll meet the hyperbolic disk and upper half-plane, and see how they—and your old friend from calculus class, the hyperboloid—are somehow one and the same. Come this Tuesday to witness the best undergraduate seminar talk ever! Well, maybe that’s just a bit of hyperbole.

## Graded Representations of Simple Lie Algebras

Yuri Bahturin, Memorial University of Newfoundland

Location: Stevenson 1308

We try to determine when and if an irreducible module V over a simple Lie algebra L over an algebraically closed field F of characteristic zero can be given a grading by an abelian group G, which is compatible with the grading of L by G. At this time, the classification of abelian group gradings on classical simple Lie algebras is complete, so switching to the representations looks like a logical next step.

## Classical and Homotopy Group Actions

Alejandro Adem, University of British Columbia

Location: Stevenson 5211

Understanding the symmetries of a topological space is a classical problem in mathematics. In this talk we will discuss how methods from algebraic topology can be used to approach this. After reviewing well-known results about transformation groups, we will consider the notion of group actions up to homotopy. This leads to interesting interactions between topology, group theory and representation theory. A number of examples will be provided to illustrate this. Tea at 3:33 pm in SC 1425. (Contact Person: Anna Marie Bohmann)

## A Paradifferential Approach for Well-Posedness and Global Solutions for the Muskat Problem

Huy Nguyen, Brown University

Location: Stevenson 1307

We study the Muskat problem in its full generality: one fluid or two fluids, with or without viscosity jump, with or without bottoms, and in arbitrary space dimension d of the interface. The Muskat problem is scaling invariant in the Sobolev space Hsc(Rd) where sc=1+d/2. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces Hsc(Rd), s>sc. Moreover, in the infinite-depth case, if the initial interface is small in Hsc(Rd), s>sc, we prove that the obtained solutions are global in time. The starting point of these results is a reformulation written solely in terms of the Drichlet-Neumann operator. The key elements of proofs are new paralinearization results for the Drichlet-Neumann operator in rough domains. This is joint work with B. Pausader (Brown U).

## Talk Title TBA

Ionut Chifan, University of Iowa

Location: Stevenson 1432

## Posner-Robinson for Hyperjumps of Turing Degrees, Part 2

Hayden Jananthan, Vanderbilt University

Location: Stevenson 1312

One version of the Posner-Robinson Theorem states that for any A Turing above 0′ and non-recursive Z Turing reducible to A, there exists B such that A is Turing equivalent to B+Z is Turing equivalent to B’. In this way, any non-recursive Z is (relative to some B) a Turing jump. Here we give an unpublished proof (due to Slaman, and also proven independently by Woodin) of the hyperjump version, namely that for any A Turing above Kleene’s O and non-hyperarithmetical Z Turing reducible to A, there exists B such that A Turing equivalent to B+Z is Turing equivalent to O^B, so that any non-hyperarithmetical Z is (relative to some B) a hyperjump. In this first talk of two, I will cover some of the foundational background to make sense of this result. In the second talk, I will prove the theorem using Kumabe-Slaman forcing.

## Talk Title TBA

Michael Montgomery, Vanderbilt University

Location: Stevenson 1206

## Talk Title TBA

Mihaela Ignatova, Temple University

Location: Stevenson 1307

## Flavors of Bicycle Mathematics

Sergei Tabachnikov, Pennsylvania State University

Location: Stevenson 5211

This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems. Here is a sampler of problems that I hope to address: 1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This circle mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. 2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence, which defines a discrete time dynamical system on the space of curves. This system is completely integrable and it is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation. 3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam’s problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? It turns out that the known solutions are solitons of the planar filament equation. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Mark Sapir)

## Talk Title TBA

Jeremy LeCrone, University of Richmond

Location: Stevenson 1307

## Talk Title TBA

Lara Ismert, University of Nebraska, Lincoln

Location: Stevenson 1432

## Talk Title TBA

Yuanzhen Shao, Georgia Southern University

Location: Stevenson 1307

## Shanks Workshop on Homotopy Theory 2019, April 13-14

Locations: Stevenson Center 1 (Math Building)

For more information, visit the workshop website.

## Talk Title TBA

Vladimir Sverak, University of Minnesota

Location: Stevenson 5211

Tea at 3:33 pm in Stevenson 1425. (Contact Person: Gieri SImonett)

## Talk Title TBA

Lauren Ruth, Vanderbilt University

Location: Stevenson 1432

## On the Linear Forced Euler and Navier-Stokes Equations: Damping and Modified Scattering

Christian Zillinger, University of Southern California

Location: Stevenson 1307

We study the long-time asymptotic behavior of the linearized Euler and nonlinear Navier-Stokes equations close to Couette flow. As a main result we show that suitable forcing breaks asymptotic stability results at the level of the vorticity, but that solutions never the less exhibit convergence of the velocity field. Thus, here linear inviscid damping persists despite instability of the vorticity equations.

## Spring 2019 Semester

Location: Vanderbilt University

Undergraduate classes end on April 22, 2019. For more information, Visit the Office of the University Registrar online.

## Talk Title TBA

Gerard Misiolek, University of Notre Dame

Location: Stevenson 1307

## NCGOA Spring Institute 2019 and 34th Annual Shanks Lecture, May 3-9

Location: Stevenson Center 4309

The topic of the Seventeenth Annual Spring Institute on Noncommutative Geometry and Operator Algebras is “Algebra and Geometry Quantized and Quantified.” The conference will focus on common themes and recent developments in topology, quantum algebra, topological condensed matter physics, subfactor theory, and quantum information theory. NCGOA 2019 will be held in conjunction with the 34th Shanks Lecture, delivered by Fields Medalist Michael Freedman (Microsoft Research). More information is available on the conference website.

## Approximation Theory 16, May 19-22, 2019

Location: Vanderbilt University

This meeting will be the sixteenth in a series of international conferences on Approximation Theory held every three years at various locations in the U.S. For more information, please visit the conference website.