Colloquium: Elijah Liflyand (Bar-Ilan University) @ Keller 401
Nov 27 @ 3:30 pm – 4:30 pm

Speaker: Elijah Liflyand (Bar-Ilan University)
Title: A tale of two Hardy spaces

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.

Colloquium: Nate Brown (Penn State)
Dec 6 @ 3:30 pm – 4:30 pm

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.

Logic seminar: David Ross
Dec 11 @ 2:30 pm – 3:30 pm
Colloquium: Prasit Bhattacharya (U. Virginia) @ Keller 301
Dec 20 @ 2:30 pm – 3:30 pm

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.

Colloquium: Anna Puskas (Kavli Institute) @ Keller 401
Jan 22 @ 3:30 pm – 4:30 pm

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

Title: Demazure-Lusztig operators and Metaplectic Whittaker functions


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.

Colloquium: Vasu Tewari (U. Penn) @ Keller 401
Jan 24 @ 3:30 pm – 4:30 pm

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.

Colloquium: François Greer (SUNY Stony Brook)
Jan 27 @ 3:30 pm – 4:30 pm
Logic seminar: Jack Yoon
Jan 31 @ 2:30 pm – 3:30 pm

The first meeting of the logic seminar will be today at 2:30–3:20 in Keller 314. Our speaker will be Jack Yoon, who will give an introductory lecture on reverse mathematics. An abstract for his talk is below.

I will introduce the basics of reverse mathematics and begin Hunter’s paper on higher order reverse topology, which can be found here:

Reverse mathematics is a study of foundations of mathematics by assessing the “strength” of the theorems from ordinary mathematics. Rather than starting from given axioms to prove a theorem, it asks a reverse question “which axioms are necessary to prove the theorem?”. Traditionally, reverse mathematics has played out within the second order arithmetic, but further progress has been made on higher order systems as well. For example, Hunter’s paper above branches out to higher order systems to study the theorems of topology.