Holly Miller, Temple University

This talk is the second in a series of seminar talks discussing Galois Cohomology. The cohomology groups of a profinite group (with coefficients in a module over said group) will be introduced twice, first via inhomogeneous cochains and then through homogeneous cochains. The first method will be used to give interpretations for the cohomology groups of low dimension. The second will be used to show that these groups are limits of the cohomology groups of finite quotients of the profinite group in question. Lastly, the Tate cohomology -- which extends the usual cohomology -- will be discussed.

Abstract: We describe a new construction of invariant train tracks for pseudo-Anosov homeomorphisms with irreducible transition matrix. This fills a gap in the literature concerning the existence of such train tracks. The construction uses invariant train tracks associated to the veering triangulation of the mapping torus of the homeomorphism to eliminate branches which obstruct irreducibility.

This week's meeting of the Temple Math Club will feature a talk by Professor Charles Osborne on a proof the Sylvester-Schur Theorem. And, of course, there will be free pizza!

Abstract: In this talk, we will go over an elementary proof of the Sylvester-Schur Theorem, which states that if k and x are positive integers, and x>k, then at least one of the numbers x, x+1, x+2, x+3, … , x+(k-1) admits a prime divisor which is greater than k.  The result was first discovered by James J. Sylvester in 1892 and rediscovered by Issai Schur in 1929.  The proof we consider here is due to Paul Erdos, and has few prerequisites besides some basic properties of binomial coefficients, especially central binomial coefficients.  This theorem may be viewed as a generalization of Bertrand’s Postulate.

We describe how one can obtain the pair correlation function of the Sine-beta process for beta=2n using the random operator framework. Our method recovers the classical formulas for beta=2 and 4. For general beta=2n, we identify the pair correlation function in terms of an ODE system, and also in terms of a matrix-valued generalization of hypergeometric functions.

Joint with Yahui Qu (UW–Madison).

A fundamental feature of turbulence, first predicted by Richardson, is the explosive dispersion of advected particles: at sufficiently late times, the separation of initially close particles grows independently of their initial separation. This implies that turbulent flows exhibit spontaneous stochasticity, or the non-uniqueness of Lagrangian trajectories in the zero-noise limit. Spontaneous stochasticity is mathematically equivalent to another well-documented phenomenon in turbulence: the anomalous dissipation of passive scalars. However, a rigorous mathematical demonstration of these effects in fluid models remains elusive. To investigate anomalous dissipation in a more tractable setting, one seeks to construct incompressible vector fields that explicitly exhibit this behaviour. Only recently have explicit deterministic vector fields with anomalous dissipation been constructed.

In this talk, I will provide an overview of spontaneous stochasticity/anomalous dissipation and discuss my recent work with Keefer Rowan, where we construct a universal total anomalous dissipator—a vector field that completely dissipates any initial data in unit time in the vanishing noise limit. Specifically, we construct a vector field such that the laws of the associated SDEs remain diffuse even as the noise vanishes. In fact, as the noise approaches zero, the laws converge to the uniform distribution on the torus.

Milen Yakimov, Northeastern University

One of the major approaches to representation theory is via support theories. They come in different flavors and have different origins. The talk will be a gentle introduction to noncommutative tensor triangular geometry, which is designed as a universal approach to support varieties. No algebraic or categorical background will be assumed. The talk is based on joint works with Dan Nakano (Univ Georgia) and Kent Vashaw (UCLA).

Violet Nguyen, Temple University

This talk is the first in a series of seminar talks discussing Galois Cohomology. We will introduce profinite spaces, profinite groups, and discrete modules over profinite groups, which will be necessary in order to define their cohomology groups. We will end with the statement of Pontryagin duality, which naturally associates to each compact abelian group (and hence each profinite group) a discrete abelian group.