Calendar

We will discuss an algebraic method by J. Huang to bound the singularity probability of the adjacency matrix of random d-regular digraphs. Although we can slightly improve the bounds, the current estimates are still far from best possible, especially toward the problem of bounding the least singular values.

The objects of arithmetic geometry are not manifolds. Some concepts from differential geometry admit analogues in arithmetic, but they are not straightforward. How then can we hope to make precise sense of quantum field theories on these objects? I will propose the beginnings of a mathematical framework via a general theory of factorization algebras. A new feature is a subtle piece of additional structure on our objects – what I call a world-structure – that is ordinarily left implicit.

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

Password: cmsa

The purpose of dimension reduction methods for data visualization is to project high dimensional data to 2 or 3 dimensions so that humans can understand some of its structure. In this talk, we will give an overview of some of the most popular and powerful methods in this active area. We will then focus on two algorithms: Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). Here, we will present new rigorous results that establish an equilibrium distribution for these methods when the number of data points diverge in the presence of pure noise or with a planted signal. Based on joint work with Daniel Fletcher.

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

Let C be a genus 3 curve whose Jacobian is geometrically simple and has geometric endomorphism algebra equal to an imaginary quadratic field. In particular, consider Picard curves y^3 = f_4(x) where the geometric endomorphism algebra is Q (zeta3). We study the associated mod-l Galois representations and their images. I will discuss an algorithm, developed in ongoing joint work with Pip Goodman, to compute the set of primes mod-l for which the images are not maximal. By running it on several datasets of Picard curves, the largest non-maximal prime we obtain is 13. This may be compared with genus 1, where Serre’s uniformity question asks if the mod-l Galois image of non-CM elliptic curves over Q is maximal for all primes l > 37

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html

Rescheduled from February 13th. Please note special time and location.

Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials).

In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (mASEP); (2) the q-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the q-Boson particle system; (3) the q-deformed Pushing Totally Asymmetric Simple Exclusion Process (q-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new “queue vertex models” on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. This is joint work with A. Aggarwal and L. Petrov.

See webpage for more details: https://people.math.harvard.edu/~ctm/sem/

We will discuss the following problem in enumerative geometry: if C is an algebraic curve, p_1,…,p_n are points on C, and X_1,…,X_n are linear subspaces of the projective space P^r, then how many maps f:C\to P^r are there with the property that f(p_i)\in X_i? It turns out that the problem can be reduced to the calculation of cohomology classes of certain subvarieties of Grassmannians. In the special case where the X_i are all points, these subvarieties are torus orbit closures, which are well-understood. The general case is more mysterious. We will describe degeneration techniques which give some new insight into the geometry of orbit closures, and also lead to a complete answer to the general problem when r=2 in terms of SSYT avoiding certain patterns. Similar techniques work in principle in more generality, but there are combinatorial obstacles.

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

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

How can we understand the set of curves on a Fano variety?  One perspective is provided by Geometric Manin’s Conjecture, a collection of conjectures with roots in arithmetic and topology.  While I will mention some recent progress, the main focus will be developing a conceptual framework for thinking about our question.

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

both as individual particles and collectively. I will discuss two examples of directed motion: one in cell migration, and one in collections of self-propelled colloids. First, I will show that cells lacking cell-substrate adhesions migrate along friction gradients. We call this phenomenon frictiotaxis, which is a new type of cell guidance. Second, I will present a new mechanism for flocking whereby self-propelled particles can align and move collectively despite turning away from each other.

This seminar will be held in person and on Zoom.

https://harvard.zoom.us/j/96657833341

Password: cmsa

Isabel Longbottom speaks on K-theory of the K(1)-local sphere via TC

Josh Tenenbaum will give the Fourth Annual Yip Lecture on February 29, 2024.

Title: How to grow a mind from a brain: From guessing and betting to thinking and talking

Time: 4:00-5:00 pm ET

Location: Harvard Science Center

Registration is required.

• Register here to attend in-person: In-person registration
• Register here to attend virtually: Zoom webinar registration

The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

Abstract TBA

What is the maximum proportion of d-dimensional space that can be covered by disjoint, identical spheres? In this talk I will discuss a new lower bound for this problem, which is the first asymptotically growing improvement to Rogers’ bound from 1947. Our proof is almost entirely combinatorial and reduces to a novel theorem about independent sets in graphs with bounded degrees and codegrees.

This is based on joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.

**Special Location**

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

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

The bordered Floer homology of Lipshitz, Ozsvath, and Thurston was interpreted by Auroux as defining an element in the partially wrapped Fukaya category of a symmetric product of the boundary. We naturally expect that this assignment should be functorial; e.g. a cobordism between two manifolds with torus boundary should induce a morphism between the corresponding Lagrangians. I’ll describe a framework for thinking about functoriality in terms of sutured manifolds and describe what it looks like for Heegaard Floer homology.