Title: Symmetries of Surfaces

Abstract: There are many ways to study surfaces: topologically, geometrically, dynamically, algebraically, and combinatorially, just to name a few. We will touch on some of the motivation for studying surfaces and their associated mapping class groups, which is the collection of symmetries of a surface. We will also describe a few of the ways that these different perspectives for studying surfaces come together in beautiful and sometimes unexpected ways.

Speaker: Elijah Liflyand (Bar-Ilan University)

Title: A tale of two Hardy spaces

Abstract:

New relations between the Fourier transform of a function of bounded

variation and the Hilbert transform of its derivative are revealed.

If we do not distinguish between the cosine and sine transforms and consider

the general Fourier transform of $f$, direct calculations give the belonging

of the derivative $f’$ to the real Hardy space $H^1$ as a sufficient condition

for the integrability of the Fourier transform. Our analysis is more delicate.

The main result is an asymptotic formula for the {bf cosine} Fourier

transform, while much earlier known results gives an asymptotic formula

for the sine Fourier transform. The difference is achieved by assuming that

the derivative belongs to different subspaces of $H^1$. However, this tale of

each of the two subspaces were impossible if we would not have a new proof

even for the old result. The known proofs used to give strong priority just to

the sine transform. Interrelations of various function spaces are studied

in this context, first of all of these two types of Hardy spaces. The obtained

results are used for proving completely new results on the integrability

of trigonometric series.

Speaker: Nate Brown (Penn State)

Title: Tomorrow’s STEM leaders are diverse

Abstract: Thirty years ago a radical experiment began at the University of Maryland Baltimore County (UMBC). The aim was to prepare undergraduates from underrepresented groups to be successful graduate students in STEM fields. The pillars of the program were unorthodox and the results have been stunning. In this talk I will discuss the Driving Change Initiative, funded by the Howard Hughes Medical Institute, which aims to replicate UMBC’s experiment at research institutions across the country.

Speaker: Prasit Bhattacharya (U. Virginia)

Title: Stable homotopy groups of spheres, finite CW-complexes and periodic self-maps

Abstract: Patterns in the stable homotopy groups of spheres are hard to detect. However chromatic homotopy theory gives a theoretical framework which justifies existence of a robust pattern. In theory, elements of stable homotopy groups are arranged in layers called the chromatic layers (one for each natural number). However, not much is known beyond chromatic layer 1. One way to detect elements in the stable homotopy groups is via finite CW-complexes which admit special self-maps, called v_n-self-maps. This talk will introduce a new class of CW-complexes which has the potential to detect elements in chromatic layer 2 of the stable homotopy group localized at the prime 2.

Speaker: Anna Puskas (Kavli Institute for the Physics and Mathematics of the Universe)

Title: Demazure-Lusztig operators and Metaplectic Whittaker functions

Abstract:

The study of objects from Number Theory such as metaplectic Whittaker

functions has led to surprising applications of Combinatorial

Representation Theory. Classical Whittaker functions can be expressed in

terms of symmetric polynomials, such as Schur polynomials via the

Casselman-Shalika formula. Tokuyama’s theorem is an identity that links

Schur polynomials to highest-weight crystals, a symmetric structure that

has interesting combinatorial parameterizations.

Approaches to generalizing the Casselman-Shalika formula resemble the

two sides of Tokuyama’s identity. Connecting these approaches with

purely combinatorial tools motivates the search for a generalization of

Tokuyama’s theorem. This talk will discuss how the introduction of

certain algebraic tools (Demazure and Demazure-Lusztig operators) yields

such a result. We shall see how these tools can further be used to

investigate questions in the infinite-dimensional setting.

Speaker: Vasu Tewari (U. Penn)

Title: Divided symmetrization and Schubert polynomials

Abstract: Divided symmetrization is an algebraic operation that takes a multivariate polynomial as input and outputs a scalar, which in many cases is a combinatorially interesting quantity. In this talk, I will describe how divided symmetrization arises in different areas of mathematics, ranging from discrete geometry, where it is intimately tied to computing volumes of permutahedra, to algebraic geometry, where it makes an appearance in the cohomology class of a certain variety.

I will then focus on the divided symmetrization of Schubert polynomials. The emphasis throughout is on the combinatorics involved.