Fri, 08 May 2020

15:00 - 16:00
Virtual

Graph Filtrations with Spectral Wavelet Signatures

Ambrose Yim
(Oxford)
Abstract

We present a recipe for constructing filter functions on graphs with parameters that can optimised by gradient descent. This recipe, based on graph Laplacians and spectral wavelet signatures, do not require additional data to be defined on vertices. This allows any graph to be assigned a customised filter function for persistent homology computations and data science applications, such as graph classification. We show experimental evidence that this recipe has desirable properties for optimisation and machine learning pipelines that factors through persistent homology. 

Fri, 15 May 2020

15:00 - 16:00
Virtual

From dynamics to combinatorics and back again

Kelly Spendlove
(Oxford)
Abstract

The last fifty years of dynamical systems theory have established that dynamical systems can exhibit extremely complex behavior with respect to both the system variables (chaos theory) and parameters (bifurcation theory). Such complex behavior found in theoretical work must be reconciled with the capabilities of the current technologies available for applications. For example, in the case of modelling biological phenomena, measurements may be of limited precision, parameters are rarely known exactly and nonlinearities often cannot be derived from first principles. 

The contrast between the richness of dynamical systems and the imprecise nature of available modeling tools suggests that we should not take models too seriously. Stating this a bit more formally, it suggests that extracting features which are robust over a range of parameter values is more important than an understanding of the fine structure at some particular parameter.

The goal of this talk is to present a high-level introduction/overview of computational Conley-Morse theory, a rigorous computational approach for understanding the global dynamics of complex systems.  This introduction will wander through dynamical systems theory, algebraic topology, combinatorics and end in game theory.

Mon, 11 May 2020
15:45
Virtual

Torus knots in contact topology

Irena Matkovic
(Oxford)
Abstract

Tight contact structures on knot complements arise both from Legendrian realizations of the knot in the standard tight contact structure and from the non-loose Legendrian realizations in the overtwisted structures on the sphere. In this talk, we will deal with negative torus knots. We wish to concentrate on the relations between these various Legendrian realizations of a knot and the contact structures on the surgeries along the knot. In particular, we will build every contact structure by a single Legendrian surgery, and relate the knot properties to the properties of surgeries; namely, tightness, fillability and non-vanishing Heegaard Floer invariant.

Tue, 19 May 2020
14:00
Virtual

The maximum length of K_r-Bootstrap Percolation

Gal Kronenberg
(Oxford)
Further Information

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

How long does it take for a pandemic to stop spreading? When modelling an infection process, especially these days, this is one of the main questions that comes to mind. In this talk, we consider this question in the bootstrap percolation setting.

Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobás in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E_t \subseteq E(Kn)$, we infect a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n] , E_t \cup \{e\})$. A question raised by Bollobás asks for the maximum time the process can run before it stabilizes. Bollobás, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H=K_r$. They answered the question for $r \leq 4$ and gave a non-trivial lower bound for every $r \geq 5$. They also conjectured that the maximal running time is $o(n^2)$ for every integer $r$. We disprove their conjecture for every $r \geq 6$ and we give a better lower bound for the case $r=5$; in the proof we use the Behrend construction. This is a joint work with József Balogh, Alexey Pokrovskiy, and Tibor Szabó.

Mon, 18 May 2020
14:15
Virtual

Some constructions of Calabi--Yau threefolds and real Lagrangian submanifolds

Thomas Prince
(Oxford)
Abstract

I will describe the results of two projects on the construction of Calabi-Yau threefolds and certain real Lagrangian submanifolds. The first concerns the construction of a novel dataset of Calabi-Yau threefolds via an application of the Gross-Siebert algorithm to a reducible union of toric varieties obtained by degenerating anti-canonical hypersurfaces in a class of (around 1.5 million) Gorenstein toric Fano fourfolds. Many of these constructions correspond to smoothing such a hypersurface; in contrast to the famous construction of Batyrev-Borisov which exploits crepant resolutions of such hypersurfaces. A central ingredient here is the construction of a certain 'integral affine structure with singularities' on the boundary of a class of polytopes from which one can form a topological model, due to Gross, of the corresponding Calabi-Yau threefold X. In general, such topological models carry a canonical (anti-symplectic) involution i and in the second project, which is joint work with H. Argüz, we describe the fixed point locus of this involution. In particular, we prove that the map i*-1 on graded pieces of a Leray filtration of H^3(X,Z2) can be identified with the map D -> D^2, where D is an element of H^2(X',Z2) and X' is mirror-dual to X. We use this to compute the Z2 cohomology group of the fixed locus, answering a question of Castaño-Bernard--Matessi.

Mon, 25 May 2020
14:15
Virtual

Quantum K-theory and 3d A-model

Cyril Closset
(Oxford)
Abstract

I will discuss some ongoing work on three-dimensional supersymmetric gauge theories and their relationship to (equivariant) quantum K-theory. I will emphasise the interplay between the physical and mathematical motivations and approaches, and attempt to build a dictionary between the two.  As an interesting example, I will discuss the quantum K-theory of flag manifolds. The QK ring will be related to the vacuum structure of a gauge theory with Chern-Simons interactions, and the (genus-0) K-theoretic invariants will be computed in terms of explicit residue formulas that can be derived from the relevant supersymmetric path integrals.

Mon, 11 May 2020
14:15
Virtual

Universal structures in enumerative invariant theories

Dominic Joyce
(Oxford)
Abstract

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some geometric problem, by means of a virtual class $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ of the moduli spaces ${\mathcal M}_\alpha^{\rm st}(\tau)\subseteq{\mathcal M}_\alpha^{\rm ss}(\tau)$ of $\tau$-(semi)stable objects in some homology theory. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds.

We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces ${\mathcal M},{\mathcal M}^{\rm pl}$, where my big vertex algebras project http://people.maths.ox.ac.uk/~joyce/hall.pdf gives $H_*({\mathcal M})$ the structure of a graded vertex algebra, and $H_*({\mathcal M}^{\rm pl})$ a graded Lie algebra, closely related to $H_*({\mathcal M})$. The virtual classes $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ take values in $H_*({\mathcal M}^{\rm pl})$. In most such theories, defining $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ when ${\mathcal M}_\alpha^{\rm st}(\tau)\ne{\mathcal M}_\alpha^{\rm ss}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in homology over $\mathbb Q$, and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*({\mathcal M}^{\rm pl})$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles.

This is joint work with Jacob Gross and Yuuji Tanaka.

 

Mon, 04 May 2020
14:15
Virtual

Homology of moduli stacks of complexes

Jacob Gross
(Oxford)
Abstract

There are many known ways to compute the homology of the moduli space of algebraic vector bundles on a curve. For higher-dimensional varieties however, this problem is very difficult. It turns out that the moduli stack of objects in the derived category of a variety X, however, is topologically simpler than the moduli stack of vector bundles on X. We compute the rational homology of the moduli stack of complexes in the derived category of a smooth complex projective variety. For a certain class of varieties X including curves, surfaces, flag varieties, and certain 3- and 4-folds we get that the rational cohomology is freely generated by Künneth components of Chern characters of the universal complex––this allows us to identify Joyce's vertex algebra construction with a super-lattice vertex algebra on the rational cohomology of X in these cases. 

Mon, 27 Apr 2020
14:15
Virtual

Evanescent ergosurfaces and waves

Joe Keir
(Oxford)
Abstract

Certain exotic Lorentzian manifolds, including some of importance to string theory, possess an unusual geometric feature called an "evanescent ergosurface". In this talk I will introduce this feature and motivate the study of the wave equation on the associated geometries. It turns out that the presence of an evanescent ergosurface prevents the energy of waves from being uniformly bounded in terms of their initial energy; I will outline the proof of this statement. An immediate corollary is that there do not exist manifolds with both an evanescent ergosurface and a globally timelike Killing vector field.

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