A Pythagorean triple is a set of positive integers a, b, and c that are the sides of a right triangle. Two well-known examples are (3,4,5) and (5,12,13). There are infinitely many more examples, and in fact there is a formula for all of them.

In this talk, a formula for all Pythagorean triples will be presented and a few derivations of the formula will be given, using both algebra and geometry. One of the derivations has an unexpected connection to integral calculus.

Note: Free refreshments. The talk starts at 5:40.

A “combinatorial triangle” is a system of positive integers (or 0) that count something and are convenient to place into a triangular array. The most famous combinatorial triangle is the system of binomial coefficients, but there are many other important examples.

In this week’s meeting, we’ll look together at some specific combinatorial triangles besides the binomial coefficients: Stirling numbers and Eulerian numbers.

Note: Free refreshments.

Join us for a talk in Logic Colloquium!

Andrew Tedder (Institut Wiener Kreis, Vienna)

Negated Implications in Connexive Relevant Logics

This talk investigates the odd fact that one may add connexive theses to relevant logics, giving rise to contraclassical systems, and obtain logics which are not trivial, still obey many of the desired relevance properties, and yet allow one to prove every negated implication. I’ll show why this is the case, and investigate alternative connexive relevant logics in the area that don’t have this undesirable property.

All welcome!

Abstract: In this talk I will present quantitative convergence results for a class of mean field games with common noise and controlled volatility. The basic strategy we employ is the one introduced recently by Laurière and Tangpi – roughly speaking, we use a synchronous coupling argument to prove a “forward-backward propagation of chaos” result for the FBSDEs which characterize the (open-loop) equilibria of the N-player and mean field games. Unlike in earlier works which have adopted this strategy, we do not require smallness conditions, and instead rely on monotonicity. In particular, (displacement) monotonicity of the Hamiltonian and the terminal cost allow us to establish a (uniform in N) stability estimate for the N-player FBSDEs, which implies the convergence result. The arguments are relatively simple, and flexible enough to yield similar results in the setting of mean field control and infinite horizon (discounted) mean field games.

This talk is based on a join work with Joe Jackson

Abstract: In this talk, we will describe an elliptic PDE that models electric conduction, and the electric field concentration phenomenon between closely spaced inclusions of high contrast. In joint work with Hongjie Dong (Brown University) and Yanyan Li (Rutgers University), we considered the case when inclusions are insulators, and obtained optimal gradient estimates in terms of the distance between inclusions. This solved one of the major open problems in this area. 

A prime number is a number bigger than 1 that is only divisible by 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

This list never ends: there are infinitely many primes.In this talk, multiple proofs of the infinitude of primes will be presented.

Note: Free refreshments. The talk starts at 5:40.

Speaker: Antonio Auffinger (Northwestern University)

Title: Dimension Reduction Methods for Data Visualization

Abstract:
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 overview some of the most popular and powerful methods in this active area. We will then the focus on two algorithms: Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). Here, we will present 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 (Northwestern).

Abstract: The Laplacian flow is a geometric flow of G2-Structures that aims to flow closed G2-Structures to torsion-free G2-Structures, the latter of which have metrics with special holonomy. A special class of closed G2-Structures are the Extremally Ricci-Pinched (ERP) G2-Structures, which are as Ricci-pinched as possible without being torsion-free. We introduce an ansatz valid on any manifold with an ERP G2-Structure and consider how the Laplacian flow behaves with regards to this ansatz. As a result, we are able to better understand the significance of the ERP condition and motivate more general results to prove concerning long-time existence and convergence of the Laplacian flow to torsion-free G2-Structures.

https://www.physics.ucsb.edu/people/aaron-kennon

This week in the math club, we will look at an important number called Euler’s constant, which is around .57721. We’ll see where it comes from, different ways to estimate it, and some famous unsolved problems about this number.

Note: Free refreshments.

We will present some recent progress on two problems in minimal surface theory posed by S. T. Yau in 1982. In particular, we will discuss the existence of infinitely many closed minimal hypersurfaces in a closed Riemannian manifold and the existence of four closed minimal two-spheres in a Riemannian three-sphere.

Philippe Schlenker