Behrang Noohi: What is derived geometry?
You may have heard of "derived/higher geometry" and wondered what it is. A brainchild of Alexander Grothendieck, it is more of a "new viewpoint in mathematics" rather than being a specific subject. A recent flurry of activities, aiming to set up the right foundations, has seen the subject transform enormously, resulting in spectacular applications to algebraic geometry,
Abhishek Saha: Bounding the heights of peaks of vibration modes on hyperbolic membranes
I will give a gentle introduction to the "sup-norm problem" in a setting where number theory plays a key role. The sup-norm problem asks for non-trivial bounds on the sup-norms of eigenfunctions of the Laplacian on Riemannian manifolds. In the special case when the manifold is a surface of constant negative structure, and isconstructed from "quaternion algebras", a famous result of Iwaniec and Sarnak improves upon the trivial bound using number-theoretic techniques. I will explain this result, and then talk about recent progress on an analogous question where the underlying surface is itself allowed to vary.
Mark Jerrum (1:00-1:30)
Title: Developments in rejection sampling
Abstract: I’ll attempt to ingratiate myself with the maximum number of people by touching on rejection sampling, the Lovász Local Lemma, the Tutte polynomial, etc. The talk will include joint work with Heng Guo, lately a postdoc in the School.
Justin Ward (1:30-2:00)
Title: Approximation, Hardness, and Proof Theory
Abstract: In this talk, I will provide a brief overview of the main ideas underpinning the modern field of “approximation algorithms.” From a practical perspective, approximation algorithms can be viewed as a way of coping with computationally “hard” problems. More generally, however, the study of so-called “approximability” is about developing a finely-grained notion of what it means for a problem to be “hard.” This notion relies on a collection of fundamental results linking proof theory, computation, and optimisation, which I will briefly describe and discuss.
Martin Benning: What do you meme? Nonlinear scale-space methods in practice. (1:05-1:30)
Abstract: We discuss PDE-based approaches for decomposing signals into components with features at different scales. Starting with linear scale-space and inverse scale-space methods, we see shortcomings when using those to decompose structures with discontinuities. We use this as a motivation for considering nonlinear inverse scale-space methods and their extensions to spectral decomposition methods. Those methods allow to obtain more suitable decompositions of signals with discontinuities, and we show how this can be exploited in a variety of applications in image-processing. We particularly focus on the creation of memes via facial image fusion.
Primoz Skraba: Stability in Applied Topology. (1:30-1:55)
Abstract: The goal of this talk is to give a brief overview of applied topology, introducing persistence and various notions of stability. I will try to explain the type of problems people in the field are working on and how it intersects with other areas of mathematics - including probability, algebra, and algorithms.