Much of the recent rigorous progress on the classical Ising model was driven by a new detailed understanding of its stochastic geometric representations. The extent of couplings between the Ising model and its FK, random current and high-temperature representations is ever-increasing. A framework unifying the relations will be presented, along with some surprising implications including:

• percolation of the uniform even graph is not monotone in the domain;
• percolation of the high-temperature expansion and single current is not monotone in the parameter;
• absence of exponential decay for the random current corresponding to a supercritical Ising model.

Chathumini Kondasinghe, Temple University

This talk aims to explore how long exact sequences of group cohomology provide insights into field theory and elliptic curves. We begin with a proof of Hilbert's theorem 90, and then use the long exact sequence of Galois cohomology to establish Kummer theory for fields. In the second half, we will talk about the idea that goes into the proof of the weak Mordell-Weil theorem, a finiteness result for elliptic curves.

Abstract: We'll begin by looking at the lacunary zeta function $\sum 1/F(n)^s$, where $F(n)$ is the $n$th Fibonacci number. Surprisingly (at least to me), this is deeply connected to modular forms, and a small generalization is connected to counting 3-term arithmetic progressions of squares. This includes work with Eran Assaf, Chan Ieong Kuan, Thomas Hulse, Alexander Walker, and Raphael Steiner.
 

Branching Brownian motion describes a growing swarm of particles that move and multiply stochastically. In multiple dimensions, the frontier of this population is governed by an associated "derivative martingale," which converges to a random field on the sphere. In this talk, we will explore the irregularity of this limiting field: it is almost surely discontinuous almost everywhere in d≥3.

Join us for a talk on billiards played on non-rectangles. And, of course, there will be free pizza!

Abstract: For far too long has Big Rectangle dominated the industry of billiard tables. In this talk we shall dare to play pool on other shapes and along the way, maybe unearth some geometric properties of the space and the space of these spaces we're playing on. And we will do this all based on the fundamental principles of reflection, a little group theory, hyperbolic geometry and complex analysis. 

Sean O'Donnell, Temple University

This talk will introduce the long exact cohomology sequence and some basic theory surrounding it. We will begin by defining the cochain and cohomology functors, and then introduce the long exact sequence and discuss its naturality. The structure of this exact sequence will motivate us to investigate acyclic, cohomologically trivial, and induced modules, which we will then use to introduce the technique of dimension shifting. During this, we will also discuss the application of these structures to the cohomology of Galois groups acting on the additive groups of their relevant fields.

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Background talk (11am): Waist inequalities, Property T, and Higher Expansion

Abstract: Suppose we are given two simplicial complexes $X$ and $Y$, where $X$ is "complicated” and $Y$ is lower dimensional than $X$. Then must a map $f: X \to Y$ have at least one "complicated” fiber? This talk will give an overview of a program Gromov initiated to prove quantitative statements of this kind, called waist inequalities. Both this talk and the next will feature examples where what "complicated" means can be either geometric (e.g. volume) or topological (e.g., a measure of the largeness of the fundamental group.) To give a sense for the theory I will first talk about what is known for concrete examples like the sphere and the torus, before moving towards the wilder setting of negative curvature. Along the way, I will as time permits describe connections to scalar curvature, (higher versions of) property $T$, systolic geometry, and various notions of higher expander simplicial complexes and manifolds originating in computer science. This talk will move slowly and not assume prior knowledge in this area.

Research talk (3:45pm): Minimal Submanifolds and Waist Inequalities for Locally Symmetric Spaces

Abstract: This talk will focus on the case of nonpositively curved locally symmetric spaces. In addition to being the most natural non-positively curved spaces to study from the perspective of differential geometry, they also have strong connections to geometric group theory, number theory, and algebraic geometry. I will describe recent joint work with Mikolaj Fraczyk that establishes a number of different kinds of higher expansion properties for families of manifolds in this setting by bringing new tools into the picture from representation theory and minimal surface theory. One goal will be to explain how knowledge of the unitary representations of a semisimple Lie group can be used to study the geometry of the associated locally symmetric spaces. On the minimal surface side, we establish new monotonicity formulas, or volume growth estimates, for minimal submanifolds of low-codimension in nonpositively curved symmetric spaces. I will explain how this can be played against the information coming from representation theory to prove waist inequalities. This talk may not move as slowly as the first talk but it will not assume prior knowledge in this area.

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Background talk (10am): Thinking about dimension 4, stuck in dimension 3: Knots, Concordances, and Homology Cobordisms

Abstract: It is a common theme in topology to study n-manifolds based on the n-1-manifolds they bound, or the n+1-manifolds bounded by them. We’ll explore together why this is an interesting and useful thing to do in dimensions 3 and 4! More specifically, we will talk about knots in the 3-sphere which are secretly related in 4-dimensional ways and how this can help us think about 3-manifolds that are similarly mysteriously connected. 
 

Research talk (2pm): 0-Surgeries on Links

Abstract: In work in progress with Ryan Stees, we show that every closed, oriented 3-manifold can be obtained by 0-surgery on a link. Since the 0-surgery of a link can capture the data of many of the typical isotopy and concordance invariants of a link, particularly in the pairwise linking number 0 case, this result gives us a nice lens through which to study both 3-manifolds and links. However, 0-surgery on a link is certainly not a complete link invariant, and we also give multiple constructions for non-isotopic (and even non-concordant) links with homeomorphic 0-surgeries.

Abstract: In this talk, we will discuss some recent applications of machine learning to number theory. In particular, we will discuss murmurations of Arithmetic L-functions and experiments on the Tate–Shafarevich groups. We will also introduce the recent results of applying transformer models to predict information about Frobenius traces a_p from elliptic curves given information about other traces a_q. Our experiments reveal that these models achieve high accuracy, even in the absence of explicit number-theoretic tools like functional equations of L-functions. We also present partial interpretability findings on the patterns learned by the machine learning models. No prior knowledge of machine learning is assumed.

This week's meeting of the Temple Math Club will be a game night! Come by to play a variety of games, and, of course, there will be free pizza!