Thu, 15 Feb 2024
16:00
Lecture Room 4, Mathematical Institute

Strong Bounds for 3-Progressions

Zander Kelley
(UIUC)
Abstract
Suppose you have a set $A$ of integers from $\{1, 2,\ldots, N\}$ that contains at least $N / C$ elements.
Then for large enough $N$, must $A$ contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?
In 1953, Roth showed that this is indeed the case when $C \approx \log \log N$, while Behrend in 1946 showed that $C$ can be at most $2^{\sqrt{\log N}}$ by giving an explicit construction of a large set with no 3-term progressions.
Since then, the problem has been a cornerstone of the area of additive combinatorics.
Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on $C$ to $C = (\log N)^{1 + c}$, for some constant $c > 0$.
This talk will describe our work which shows that the same holds when $C \approx 2^{(\log N)^{1/12}}$, thus getting closer to Behrend's construction.
Based on a joint work with Raghu Meka.
Mon, 26 May 2008

10:00 - 11:00
L3

Computation in quotients of polynomial rings and enumerative geometry

Daniel Grayson
(UIUC)
Abstract
Abstract: I will describe how computations are done using "Groebner bases" in quotient rings of polynomial rings, and I will describe explicitly the form of a particular Groebner basis for the ideal defining the ring parametrizing all factorizations of a monic polynomial of degree a+b+...+e into monic factors of degree a,b,...,e. That can be and is used in practice to compute intersection numbers involving of algebraic cycles arising as Chern classes on flag bundles of vector bundles. Simplest example: how many lines in 3-space meet four fixed lines?
Tue, 20 May 2008
15:45
L3

Mirabolic Langlands duality and the Quantum Calogero-Moser system II

Thomas Nevins
(UIUC)
Abstract

The geometric Langlands program aims at a "spectral decomposition" of certain derived categories, in analogy with the spectral decomposition of function spaces provided by the Fourier transform. I'll explain such a geometrically-defined spectral decomposition of categories for a particular geometry that arises naturally in connection with integrable systems (more precisely, the quantum Calogero-Moser system) and representation theory (of Cherednik algebras). The category in this case comes from the moduli space of vector bundles on a curve equipped with a choice of ``mirabolic'' structure at a point. The spectral decomposition in this setting may be understood as a case of ``tamely ramified geometric Langlands''. In the talk, I won't assume any prior familiarity with the geometric Langlands program, integrable systems or Cherednik algebras.

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