Bogdan Chornomaz, Vanderbilt University
Location: SC 1312

A natural question about finite lattices is how large they can be with respect to the size of the underlying set of join-irreducible elements. An obvious exponential bound is reachable simply by considering a boolean lattice. Yet, the question remains about what “causes” a lattice to be exponentially big. In this talk we will argue that the only reason for that is the growth of its Vapnik-Chervonekis (VC) dimention. Having restricted this dimension, we will prove a polynomial bound on size, show that this bound is tight and characterize the class of “extremal” lattices, reaching this bound. Given time, we will also show that generalized extremality yield a combinatorial characterization of convex geometries.

Glenn Webb, Vanderbilt University
Location: SC 1206

Deterministic models are developed for the spatial spread of epidemic diseases in geographical settings. The models are focused on outbreaks that arise from a small number of infected hosts imported into sub-regions of the geographical settings. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The models consist of systems of partial differential equations with diffusion terms describing the spatial spread of the underlying microbial infectious agents. Applications are given to seasonal influenza epidemics.

Ilya Kapovich, CUNY Hunter College
Location: Stevenson 1308

For automorphisms of the free group \(F_r\), being “fully irreducible” is the main analog of the property of being a pseudo-Anosov element of the mapping class group. It has been known, because of general results about random walks on groups acting on Gromov-hyperbolic spaces, that a “random” (in the sense of being generated by a long random walk) element \(\phi\) of \(\mathrm{Out}(F_r)\) is fully irreducible and atoroidal. But finer structural properties of such random fully irreducibles \(\phi\in \mathrm{Out}(F_r)\) have not been understood. We prove that for a “random” \(\phi\in \mathrm{Out}(F_r)\) (where \(r\ge 3\)), the attracting and repelling \(\mathbb R\)-trees of \(\phi\) are trivalent, that is all of their branch points have valency three, and that these trees are non-geometric (and thus have index \(<2r-2\)). The talk is based on a joint paper with Joseph Maher, Samuel Taylor and Catherine Pfaff.

Tetiana Stepaniuk, Technische Universität Graz
Location: SC 1310

In the first part of talk we find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of generalized Poisson integrals. This problem leads to the problem of finding the asymptotic equality of Lp-norm of the remainder of trigonometric series. In the second part of talk we study hyperuniformity on flat tori. Hyperuniform point sets on the unit sphere have been studied by J. Brauchart, P. Grabner, W. Kusner and J. Ziefle. It is shown that point sets which are hyperuniform for large balls, small balls or balls of threshold order on the flat tori are uniformly distributed.

Ilya Kapovich, University of Illinois at Urbana-Champaign
Location: Stevenson 5211

The problem of counting closed geodesics of bounded length, originally in the setting of negatively curved manifolds, goes back to the classic work of Margulis in 1960s about the dynamics of the geodesic flow. Since then Margulis’ results have been generalized to many other contexts where some whiff of hyperbolicity is present. Thus a 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface S of genus $g\ge 2$, the number $N(L)$ of closed Teichmuller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi$ in the mapping class group $MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)>1$ is the “dilatation” or “stretch factor” of $\phi$. We consider an analogous problem in the $Out(F_r)$ setting, for the action of $Out(F_r)$ on a “cousin” of Teichmuller space, called the Culler-Vogtmann outer space $X_r$. In this context being a “fully irreducible” element of $Out(F_r)$ serves as a natural counterpart of being pseudo-Anosov. Every fully irreducible $\phi\in Out(F_r)$ acts on $X_r$ as a loxodromic isometry with translation length $\log\lambda(\phi)$, where again $\lambda(\phi)$ is the stretch factor of $\phi$. We estimate the number $N_r(L)$ of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. We prove, for $r\ge 3$, that $N_r(L)$ grows \emph{doubly exponentially} in $L$ as $L\to\infty$, in terms of both lower and upper bounds. These bounds reveal new behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Denis Osin)

Rudy Rodsphon, Northeastern University
Location: SC 1310

Josh Edge, Indiana University
Location: Stevenson 1432

Bogdan Chornomaz, Vanderbilt University
Location: SC 1312

When estimating the size of a lattice, it is natural to consider not only the number of its join-irreducible, but also of its meet-irreducible elements. This generalization turns out to be much harder, so in this talk we will point out a bunch of unsolved problems in this direction, accompanied by the small number of positive results. Those are, to put it lightly, very modest comparing to the general scope. Still, to the speaker’s opinion they are neat and worth talking about.

Shamindra Ghosh, Indian Statistical Institute
Location: Stevenson 1432

Given a finite index subfactor $N \subset M$, one may consider the associated bimodule category which is a rigid semisimple C*-2-category generated by the object $L^2(M)$ as an $N$-$M$-bimodule. If we know that $N$ is isomorphic to $M$, then using the isomorphism we can make the bimodule category sit inside a rigid semisimple C*-tensor category; we will see that this is possible in general (that is, without any isomorphism) using some “free” type construction and the associated subfactor planar algebra. Moreover, if we start with a hyperfinite subfactor, then our constructed C*-tensor category sits as a full subcategory of the $R$-$R$-bimodules. This is a joint work with Corey Jones and Madhav Reddy.

Frank Wagner, Vanderbilt University
Location: SC 1206

Andrei Okounkov, Columbia University
Location: Stevenson 5211

Tea at 3:33 pm in Stevenson 1425. (Contact Person: Vaughan Jones)

Paramita Das, Indian Statistical Institute
Location: Stevenson 1432

Given a diagonal subfactor $N \subset M$, the corresponding $N$-$N$ bimodule category turns out to be equivalent to the tensor category of $G$-graded finite dimensional Hilbert spaces for some group $G$. However, the tensor structure might not be not be strict and the associativity constraint is given by a scalar 3-cocycle of $G$. I will describe the annular representations of this category (which was explicitly computed in my joint work with Dietmar Bisch, Shamindra Ghosh & Narayan Rakshit) and also discuss the effect of cocycle twist on the annular representations in the more general set up of group graded categories (in the work of Bhowmick-Ghosh-Rakshit-Yamashita).

Ryan Solava, Vanderbilt University
Location: SC 1206

Justin Lanier, Georgia Tech
Location: Stevenson 1308

Susan Friedlander, University of Southern California
Location: Stevenson 5211

We will briefly review Kolmogorov’s (41) theory of homogeneous turbulence and Onsager’s ( 49) conjecture that in 3-dimensional turbulent flows energy dissipation might exist even in the limit of vanishing viscosity. Although over the past 60 years there is a vast body of literature related to this subject, at present there is no rigorous mathematical proof that the solutions to the Navier-Stokes equations yield Kolmogorov’s laws. For this reason various models have been introduced that are more tractable but capture some of the essential features of the Navier-Stokes equations themselves. We will discuss one such stochastically driven dyadic model for turbulent energy cascades. We will describe how results for stochastic PDEs can be used to prove that this dyadic model is consistent with Kolmogorov’s theory and Onsager’s conjecture. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Giusy Mazzone)

Scott Wilson, CUNY Queens College
Location: Stevenson 1310

Yasu Kawahigashi, University of Tokyo
Location: Stevenson 1432

Andy Jarnevic, Vanderbilt University
Location: SC 1206

Alexander Olshanskiy, Vanderbilt University
Location: SC 1308

Wenqing Hu, Missouri University of Science and Technology.
Location: Stevenson 1307

Frank Thorne, University of South Carolina
Location: Stevenson 1310

Willie Wongu, Michigan State University
Location: Stevenson Center 1307

Gueo Grancharov, Florida International University
Location: Stevenson 1310

Dennis Sullivan, Suny at Stony Brook
Location: Stevenson 5211

Tea at 3:33 pm in Stevenson 1425. (Contact Person: Marcelo Disconzi)

Vladimir Sverak, University of Minnesota
Location: Stevenson 5211

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

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.