Calendar

Suppose you have a set S of integers from {1,2,…,N} that contains at least N / C elements. Then for large enough N, must S contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?

In 1953, Roth showed this is the case when C is roughly (log log N). Behrend in 1946 showed that C can be at most exp(sqrt(log N)). 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 a new work showing that C can be much closer to Behrend’s construction. Based on joint work with Zander Kelley.

Zoom: https://harvard.zoom.us/j/7855806609

Password: cmsa

Zoom: https://harvard.zoom.us/j/7855806609

Password: cmsa

Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the “one force-one solution principle” for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09
Password: cmsa

There is a beautiful story connecting the permutohedron to several different objects in algebra, geometry, and combinatorics: namely, to the symmetric group, to the geometry of a particular moduli space of curves, and to the theory of matroids.  Somewhat more recently, the analogue of this story in type B was developed, where the role of the permutohedron is played by the signed permutohedron and the corresponding moduli space parameterizes curves with an involution.  I will discuss joint work with C. Damiolini, C. Eur, D. Huang, S. Li, and R. Ramadas that develops a further generalization, defining a “permutohedral complex” that relates to a certain family of complex reflection groups, to the geometry of a moduli space of curves with finite-order automorphism, and to the combinatorics of multimatroid

===============================

For more info, see https://math.mit.edu/combin/

The Rouquier dimension of a toric variety is recently shown to be achieved by the Frobenius pushforward of O via coherent-constructible correspondence. From the perspective of noncommutative geometry, this result leads to a geometric construction of toric NCR of the invariant ring of the Cox ring with respect to a multi-grading which also gives the information about its global dimension. From the perspective of mirror symmetry, the same construction provides a universal “wall skeleton” capturing VGIT wall-crossings, which contains a window for each chamber as a full subcategory. From the perspective of commutative algebra, the same construction indicates the existence of virtual resolutions of the multigraded diagonal bimodule, which agrees with a recent result of Hanlon-Hicks-Larzarev constructing one such resolution explicitly. In this talk, I will survey these perspectives. The talk is based on joint works with P. Zhou, joint works with D. Favero, and work in progress with D. Favero.

Zoom: https://harvard.zoom.us/j/93338480366?pwd=NEROWElhWStQVjVLRVZFSm1tV1ZCdz09
Passcode: 564263

In joint work with Greg Moore and Constantin Teleman we show how ideas and techniques in topological field theory apply to the study of symmetry in quantum field theory.  I will discuss how this came about, beginning with some discussion of symmetry in mathematics more generally, and give some examples.

Friday, March 8 at 12pm, with lunch, in the lounge at CMSA (20 Garden Street).

Also by Zoom: https://harvard.zoom.us/j/92410768363

In joint work with Greg Moore and Constantin Teleman we show how ideas and techniques in topological field theory apply to the study of symmetry in quantum field theory.  I will discuss how this came about, beginning with some discussion of symmetry in mathematics more generally, and give some examples.

Friday, March 8 at 12pm, with lunch, in the lounge at CMSA (20 Garden Street).

Also by Zoom: https://harvard.zoom.us/j/92410768363

We will begin by briefly introducing the use of higher-order Fourier analysis in additive combinatorics for a general audience. In particular, we will discuss the arithmetic regularity lemma and how it identifies a certain class of arithmetically-structured functions — nilsequences — as extremal objects for problems in additive combinatorics. We will then discuss how it has recently come to light that the analysis of nilsequences on certain additive patterns — those which satisfy a certain algebraic criterion known as the flag condition — is easier than the general case, and discuss some recent ideas and developments in overcoming the difficulties that arise when the additive pattern of interest is not flag.

===============================

For more info, see https://math.mit.edu/combin/

This talk investigates two types of proposed invariants of Calabi-Yau 3-folds:

• from gauge theory: Calabi-Yau monopoles,
• from calibrated geometry: count of special Lagrangians weighted with their Fueter sections.

Here, we focus on three conjectures central to the definition of these invariants and their relations:

1. The Donaldson-Segal conjecture on gauge theory/calibrated geometry duality: Calabi-Yau monopoles = weighted count of special Lagrangians,
2. The Donaldson-Scaduto conjecture: the existence of the pair of pants special Lagrangians, related to the formation of special Lagrangian singularities,
3. A hyperkähler variation of the Atiyah-Floer conjecture for Fueter sections: monopole Fueter Floer homology = Lagrangian Fueter Floer homology.

The discussion explores recent progress on these conjectures. This is joint work with Yang Li.