Abstract: I will describe a machine that lets us go between certain problems in topology and problems in non-commutative algebra. One main question of interest to topologists is how to distinguish between different knotted objects – a powerful tool that has been developed to address this are the so-called universal finite-type invariants. On the algebraic side, this structure translates to finding solutions of sets of equations in quantum algebra, in the graded setting. In this talk, we will in particular consider a certain class of knotted tubes in 4-space, pictured as movies of flying circles, and show how they are related to a class of equations originally introduced by Masaki Kashiwara and Michele Vergne.
Abstract: Deep learning has had transformative impacts in many fields including computer vision, computational biology, and dynamics by allowing us to learn functions directly from data. However, there remain many domains in which learning is difficult due to poor model generalization or limited training data. We’ll explore two applications of representation theory to neural networks which help address these issues. Firstly, consider the case in which the data represent a group equivariant function. In this case, we can consider spaces of equivariant neural networks which may more easily be fit to the data using gradient descent. Secondly, we can consider symmetries of the parameter space as well. Exploiting these symmetries can lead to models with fewer free parameters, faster convergence, and more stable optimization.
Abstract: A common theme in the field of low-dimensional topology is to study embedded surfaces in 4-manifolds. Knotted surfaces can be used to distinguish different 4-dimensional objects or to create new ones. I’ll talk about why they might be interesting, some elementary tricks for distinguishing surfaces that can be used to prove surprising results, and briefly describe a few recent developments in the area.
Abstract: Suppose you take a 1 x L rectangular strip of paper, twist it around in space, and tape the (length 1) ends together to make a paper Moebius band. How small can you make L? In this talk I’ll show that you can do it if and only if L > sqrt(3). Ben Halpern and Charles Weaver conjectured this answer in 1977, though I guess that most people would eventually come to this conclusion after doing some experiments on their own. I’ll also talk about some still unsolved problems that have a similar flavor.
Abstract:
Chat GPT, AlphaGo, and AlphaGeometry have attracted a lot of
attention recently. Computers are getting better and better at problem
solving. But what of problem posing? Can computers help us arrive at
interesting conjectures and state more nuanced theorems?
In this talk, I’ll introduce SAT solvers and talk about their increasing
utility in solving combinatorial problems. In particular, we can ask for
the minimum number of convex pentagons when n points are placed in the
plane in general position. SAT can solve the problem for large enough
sets of points that allow us to conjecture the minimum for all n.
We’ll discuss other open problems and see how SAT may be applied.
Title: The flexibility of caustics and its applications
Abstract: I will begin by reviewing the story of the flexibility of singularities of smooth mappings, from the immersion theory of Smale-Hirsch to the Igusa-Lurie contractibility of the space of framed functions. I will then explain similar flexibility results for caustics: the singularities of wavefronts, and discuss its applications to symplectic and contact topology, notably to the arborealization program for skeleta of Weinstein manifolds.
Abstract: We make predictions of the future by using initial conditions and the probabilities of transitioning from one state to another via some process. To make inference about the past, we can use final conditions and knowledge of the process together with Bayes’ rule, given by the formula P(y|x)P(x)=P(x|y)P(y). But can we define an intuitive, yet completely rigorous, mathematical framework for what it means to make consistent inferences about the past using only the idea of processes, without any reference to probabilities? In this talk, I will introduce ideas of categorical thinking to indicate how Bayes’ rule might be characterized in such a way.
