Happy Hour Talk Scheduleon Zoom, Thursdays, 3:45–5:15 pm; Lecture 4:15–5
Titles and Abstracts
Michael Hutchings, 6/18: Weyl laws and dense periodic orbits
A hypersurface in R4 has a natural vector field on it, the Hamiltonian vector field. We explain a result of Kei Irie which asserts that for a generic star-shaped hypersurface, this vector field has dense periodic orbits. This uses a kind of "Weyl law" relating periods of orbits to volume, which was proved by myself and (at the time) graduate students Dan Cristofaro-Gardiner and Vinicius Ramos. This talk is supposed to be at least mostly accessible to all graduate students and faculty, so everyone should feel free to ask questions.
Bernd Sturmfels, 6/25: Sixty-four curves of degree six
This lecture is an invitation to real algebraic geometry and its computational aspects. Our journey takes us from Hilbert's 16th problem to eigenvectors of tensors and to K3 surfaces. We present an experimental study - with many pictures - of smooth curves of degree six in the real plane. Sixty-four is number of the rigid isotopy types in the Rokhlin-Nikulin classification.
Olga Holtz, 7/2, 1pm: Zeros of polynomials, from Descartes’
Rule to the Riemann Hypothesis
Classical methods for zero localization of polynomials go back to Newton and Descartes. Born from algebra and analysis, these methods evolved to spill over to functional analysis, probability, statistical mechanics, theoretical computer science, matrix theory, and combinatorics. I will review this evolution and discuss some problems where these methods have been proven or conjectured to shed a light, including a couple of notorious ones. No heavy technology will be deployed during the talk.
David Nadler, 7/9: Hyperkahler rotation in linear algebra
(or how not to teach Math 54)
To understand real matrices, it is helpful -- perhaps indispensable -- to study complex matrices. For some questions, it is also helpful to go one step further to the quaternions. This talk will explain an example of this, providing an excuse to introduce some key constructions in geometric representation theory: quiver varieties, hyperkahler quotients, Springer theory,... as time allows.
Martin Olsson, 7/16: Topological Reconstruction of varieties
A basic theme in algebraic geometry is to what extent various invariants capture the isomorphism classes of the varieties. In this talk I will discuss recent work with Kollár, Lieblich, and Sawin showing that in many cases algebraic varieties are determined by their Zariski topological spaces (the definition of which will be discussed in the talk). I will aim to make the talk accessible to all graduate students.
Maciej Zworski, 7/23: Magic angles, theta functions and
spectral instability of large matrices
Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting.
Please do not be scared by the physics though: I will present a very simple operator whose spectral properties are thought to determine which angles are magical. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hörmander's bracket condition in a very simple setting). All of this will be illustrated by colourful numerics which suggest some open problems. The talk is based on a ``summer relaxation project" with S Becker, M Embree and J Wittsten.
Thomas Scanlon, 7/30: Definability and (un)decidability in
rings and fields
Hilbert's Tenth Problem from the 1900 International Congress of Mathematicians asks for a procedure to decide given a polynomial equation in several variables whether there is a solution in integers. Fifty years ago, following up on work of Martin Davis, Hilary Putnam, and Julia Robinson, Yuri Matiyasevich showed that no such algorithm can exist. Rather than fully resolving the question, the MDRP theorem transformed the problem from "Is there a method to decide solvability of diophantine equations?" to "For which natural algebraic structures is the diophantine theory algorithmically tractable?" Some fields, for instance, the real numbers and the p-adic numbers, are known to have decidable theories, whereas others, for example, fields of rational functions over finite fields, are known to have undecidable diophantine theories. The central open problem in this area concerns the rational numbers: it is known from Julia Robinson's 1948 Berkeley PhD thesis that the full first-order theory of the rational numbers is undecidable, but Hilbert's Tenth Problem relativized to the rationals remains open.
With this talk, I will discuss some of this history, explain how number theory and ideas from logic interact, and highlight some of the open problems. This should be seen as invitation to the upcoming MSRI program on Decidability, Definability, and Computability in Number Theory (https://www.msri.org/programs/319).
Richard Bamler, 8/6, 4pm: Uniqueness of Weak Solutions to the
Ricci Flow and Topological Applications
(Plenary lecture at the Pacific Rim conference; link supplied in the email)
In this talk I will survey recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the diffeomorphism group of every 3-dimensional spherical space form up to homotopy. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, which was originally due to Hatcher. Second, we show that the space of metrics with positive scalar curvature on every 3-manifold is either contractible or empty. This completes work initiated by Marques.
Our proof is based on a new uniqueness theorem for singular Ricci flows, which I have previously obtained with Kleiner. Singular Ricci flows were inspired by Perelman’s proof of the Poincaré and Geometrization Conjectures, which relied on a flow in which singularities were removed by a certain surgery construction. Since this surgery construction depended on various auxiliary parameters, the resulting flow was not uniquely determined by its initial data. Perelman therefore conjectured that there must be a canonical, weak Ricci flow that automatically “flows through its singularities” at an infinitesimal scale. Our work on the uniqueness of singular Ricci flows gives an affirmative answer to Perelman’s conjecture and allows the study of continuous families of singular Ricci flows leading to the topological applications mentioned above. More details and historical background will be given in the talk.
Nicolai Reshetikhin, 8/13: On the limit shape phenomenon in
In many cases a large random system develops deterministic behaviour. In probability theory this phenomenon is known as the large deviation theory. This talk will focus on few examples from statistical mechanics when a random geometry becomes deterministic at large scale for large systems. Examples include the statistics of irreducible components in large tensor products, the Ising model and dimer models.
Lin Lin. 8/20: Quantum linear system problem
Google declared that "quantum supremacy" was reached in 2019, i.e. a programmable quantum device can solve a problem that no classical computer can solve in any feasible amount of time (irrespective of the usefulness of the problem). In this happy hour talk, I will discuss how to use a quantum computer to solve linear systems of equations, or the Math 54 problem: Ax=b. I will start with a toy problem, where A is merely a 2 x 2 matrix, and it is possible that the algorithm is unlike anything you have seen in the classical context. I will then talk about some recent progress of quantum linear system solvers. No prior knowledge on quantum computation is necessary.
Seminar and Reading Group Suggestions:
Please join us on Thursday 8/20 to discuss further and/or send me an email if interested. Your name will be added to the relevant seminar(s).