Mon, 06 Nov 2023
14:15
L4

The New $\mu$-Invariants: Infinite-Dimensional Morse Indices and New Invariants of $G_2$-Manifolds

Laurence Mayther
(Cambridge)
Abstract

There are two main methods of constructing compact manifolds with holonomy $G_2$, viz. resolution of singularities (first applied by Joyce) and twisted connect sum (first applied by Kovalev).  In the second case, there is a known invariant (the $\overline{\nu}$-invariant, introduced by Crowley–Goette–Nordström) which can, in many cases, be used to distinguish between different examples.  This invariant, however, has limitations; in particular, it cannot be computed on the $G_2$-manifolds constructed by resolution of singularities.

 

In this talk, I shall begin by discussing the notion of a $G_2$-manifold and the $\overline{\nu}$-invariant and its limitations.  In the context of this, I shall then introduce two new invariants of $G_2$-manifolds, termed $\mu$-invariants, and explain why these promise to overcome these limitations, in particular being well-suited to, and computable on, Joyce's examples of $G_2$-manifolds.  These invariants are related to $\eta$- and $\zeta$-invariants and should be regarded as the Morse indices of a $G_2$-manifold when it is viewed as a critical point of certain Hitchin functionals.  Time permitting, I shall explain how to prove a closed formula for the invariants on the orbifolds used in Joyce's construction, using Epstein $\zeta$-functions.

Mon, 01 May 2023
13:00
L1

Keeping matter in the loop in dS_3 quantum gravity

Alejandra Castro
(Cambridge)
Abstract

In this talk I will discuss a novel mechanism  that couples matter fields to three-dimensional de Sitter quantum gravity. This construction is based on the Chern-Simons formulation of three-dimensional Euclidean gravity, and it centers on a collection of Wilson loops winding around Euclidean de Sitter space. We coin this object a Wilson spool.  To construct the spool, we build novel representations of su(2). To evaluate the spool, we adapt and exploit several known exact results in Chern-Simons theory. Our proposal correctly reproduces the one-loop determinant of a free massive scalar field on S^3 as G_N->0. Moreover, allowing for quantum metric fluctuations, it can be systematically evaluated to any order in perturbation theory.   

Mon, 06 Mar 2023
13:00
L1

Bounds on quantum evolution complexity via lattice cryptography

Marine De Clerck
(Cambridge)
Abstract

I will present results from arXiv:2202.13924, where we studied the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. The notion of complexity of interest to us will be Nielsen’s complexity applied to the time-dependent evolution operator of the quantum systems. I will review Nielsen’s complexity, discuss the difficulties associated with this definition and introduce a simplified approach which appears to retain non-trivial information about the integrable properties of the dynamical systems.

Fri, 02 Dec 2022

16:00 - 17:00
L1

Strong cosmic censorship versus Λ

Mihalis Dafermos
(Cambridge)
Abstract

The strong cosmic censorship conjecture is a fundamental open problem in classical general relativity, first put forth by Roger Penrose in the early 70s. This is essentially the question of whether general relativity is a deterministic theory. Perhaps the most exciting arena where the validity of the conjecture is challenged is the interior of rotating black holes, and there has been a lot of work in the past 50 years in identifying mechanisms ensuring that at least some formulation of the conjecture be true. It turns out that when a nonzero cosmological constant Λ is added to the Einstein equations, these underlying mechanisms change in an unexpected way, and the validity of the conjecture depends on a detailed understanding of subtle aspects of black hole scattering theory, surprisingly involving, in the case of negative Λ, some number theory. Does strong cosmic censorship survive the challenge of non-zero Λ? This talk will try to address this Question!

Tue, 22 Nov 2022

14:00 - 14:30
L3

Regularization by inexact Krylov methods with applications to blind deblurring

Malena Sabate Landman
(Cambridge)
Abstract

In this talk I will present a new class of algorithms for separable nonlinear inverse problems based on inexact Krylov methods. In particular, I will focus in semi-blind deblurring applications. In this setting, inexactness stems from the uncertainty in the parameters defining the blur, which are computed throughout the iterations. After giving a brief overview of the theoretical properties of these methods, as well as strategies to monitor the amount of inexactness that can be tolerated, the performance of the algorithms will be shown through numerical examples. This is joint work with Silvia Gazzola (University of Bath).

Mon, 13 Jun 2022
14:15
L5

Open FJRW theory

Mark Gross
(Cambridge)
Abstract

I will describe joint work with Tyler Kelly and Ran Tessler. FJRW (Fan-Jarvis-Ruan-Witten) theory is an enumerative theory of quasi-homogeneous singularities, or alternatively, of Landau-Ginzburg models. It associates to a potential W:C^n -> C given by a quasi-homogeneous polynomial moduli spaces of (orbi-)curves of some genus and marked points along with some extra structure, and these moduli spaces carry virtual fundamental classes as constructed by Fan-Jarvis-Ruan. Here we specialize to the case W=x^r+y^s and construct an analogous enumerative theory for disks. We show that these open invariants provide perturbations of the potential W in such a way that mirror symmetry becomes manifest. Further, these invariants are dependent on certain choices of boundary conditions, but satisfy a beautiful wall-crossing formalism.

Mon, 16 May 2022
14:15
L5

Morava K-theory and Hamiltonian loops

Ivan Smith
(Cambridge)
Abstract

A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Tue, 08 Feb 2022
14:00
Virtual

Large hypergraphs without tight cycles

Barnabas Janzer
(Cambridge)
Abstract

An $r$-uniform tight cycle of length $k>r$ is a hypergraph with vertices $v_1,\ldots,v_k$ and edges $\{v_i,v_{i+1},…,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $k$. Sós, and independently Verstraëte, asked the following question: how many edges can there be in an $n$-vertex $r$-uniform hypergraph if it contains no tight cycles of any length? In this talk I will review some known results, and present recent progress on this problem.

Tue, 25 Jan 2022
14:00
Virtual

Induced Poset Saturation

Maria-Romina Ivan
(Cambridge)
Abstract

Given a fixed poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-free if it does not contain an (induced) copy of $\mathcal P$. And we say that $F$ is $\mathcal P$-saturated if it is maximal $\mathcal P$-free. How small can a $\mathcal P$-saturated family be? The smallest such size is the induced saturation number of $\mathcal P$, $\text{sat}^*(n, \mathcal P)$. Even for very small posets, the question of the growth speed of $\text{sat}^*(n,\mathcal P)$ seems to be hard. We present background on this problem and some recent results.

Tue, 03 May 2022

14:00 - 15:00
L6

Equivariant line bundles with connection on the Drinfeld upper half-space

Amy Zhu
(Cambridge)
Abstract

Ardakov and Wadsley developed a theory of D-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of SL_2 by taking global sections of Drinfeld line bundles. In this talk, we will extend their techniques to SL_3 by studying the Drinfeld upper half-space \Omega^{(3)} of dimension 2.

Subscribe to Cambridge