Tue, 19 May 2020
15:30
Virtual

Maximum height of 3D Ising interfaces

Eyal Lubetzky
(Courant Institute)
Further Information

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

Abstract

Dobrushin (1972) showed that, at low enough temperatures, the interface of the 3D Ising model - the random surface separating the plus and minus phases above and below the $xy$-plane - is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height $M_n$ on a box of side length $n$ is $O_P(\log n)$. We study this interface and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to analyze the maximum height $M_n$ of the interface, and prove that at low temperature $M_n/\log n$ converges to $c\beta$ in probability. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tails bounds.
Joint work with Reza Gheissari.

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.

Tue, 19 May 2020
12:00

Feynman propagators from the worldsheet

Yvonne Geyer
(Chulalongkorn University)
Abstract

Ambitwistor strings are a class of holomorphic worldsheet models that directly describe massless quantum field theories, such as supergravity and super Yang-Mills. Their correlators give remarkably compact amplitude representations, known as the CHY formulas: characteristic worldsheet integrals that are fully localized on a set of polynomial constraints known as the scattering equations. Moreover, the ambitwistor string models provide a natural way of extending these formulas to loop level, where the constraints can be used to simplify the formulas (originally on higher genus curves) to 'forward limit-like' constructions on nodal spheres. After reviewing these developments, I will discuss one of the peculiar features of this approach: the worldsheet formulas on nodal spheres result in a non-standard integrand representation that makes it difficult to e.g. apply established integration techniques. While several approaches for addressing this look feasible or have been put forward in the literature, they only work for the simplest toy models. Taking inspiration from these attempts, I want to discuss a novel strategy to overcome this difficulty, and formulate compact worldsheet formulas with standard Feynman propagators.

Mon, 18 May 2020

16:00 - 17:00

The functional Breuer-Major theorem

Ivan Nourdin
(University of Luxembourg)
Abstract


Let ?={??}?∈ℤ be zero-mean stationary Gaussian sequence of random variables with covariance function ρ satisfying ρ(0)=1. Let φ:R→R be a function such that ?[?(?_0)2]<∞ and assume that φ has Hermite rank d≥1. The celebrated Breuer–Major theorem asserts that, if ∑|?(?)|^?<∞ then
the finite dimensional distributions of the normalized sum of ?(??) converge to those of ?? where W is
a standard Brownian motion and σ is some (explicit) constant. Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the
space ?([0,1]) of càdlàg functions endowed with the Skorohod topology is still missing. Our main goal in this paper is to fill this gap. More precisely, by using suitable boundedness properties satisfied by the generator of the Ornstein–Uhlenbeck semigroup,
we show that tightness holds under the sufficient (and almost necessary) natural condition that E[|φ(X0)|p]<∞ for some p>2.

Joint work with D Nualart
 

Mon, 18 May 2020
15:45
Virtual

Boundaries and 3-dimensional topological field theories

Dan Freed
(University of Texas at Austin)
Abstract

Just as differential equations often boundary conditions of various types, so too do quantum field theories often admit boundary theories. I will explain these notions and then discuss a theorem proved with Constantin Teleman, essentially characterizing certain 3-dimensional topological field theories which admit nonzero boundary theories. One application is to gapped systems in condensed matter physics.

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, 18 May 2020
12:45
Virtual

Compensating strong coupling with large charge -- ZOOM SEMINAR

Susanne Reffert
(Bern)
Abstract

Over the last few years, it has become clear that working in sectors of large global charge leads to significant simplifications when studying strongly coupled CFTs. It allows us in particular to calculate the CFT data as an expansion in inverse powers of the large charge.
In this talk, I will introduce the large-charge expansion via the simple example of the O(2) model and will then apply it to a number of other systems which display a richer structure, such as non-Abelian global symmetry groups.
 

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.

Fri, 15 May 2020

14:00 - 15:00

To be announced

To be announced
(To be announced)
Fri, 15 May 2020

11:45 - 13:15
Virtual

InFoMM CDT Group Meeting

Giancarlo Antonucci, Helen Fletcher, Alexandru Puiu, Yu Tian
(Mathematical Institute)
Thu, 14 May 2020

16:00 - 16:45
Virtual

An introduction to Cuntz--Pimsner algebras

Francesca Arici
(Universiteit Leiden)
Further Information

Part of UK virtual operator algebras seminar: https://sites.google.com/view/uk-operator-algebras-seminar/home

Abstract

In 1997 Pimsner described how to construct two universal C*-algebras associated with an injective C*-correspondence, now known as the Toeplitz--Pimsner and Cuntz--Pimsner algebras. In this talk I will recall their construction, focusing for simplicity on the case of a finitely generated projective correspondence. I will describe the associated six-term exact sequence in K(K)-theory and explain how these can be used in practice for computational purposes. Finally, I will describe how, in the case of a self-Morita equivalence, these exact sequences can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles.

Thu, 14 May 2020
16:00
Virtual

Replica-exchange for non-convex optimization

Jing Dong
(Columbia Business School)
Abstract

Abstract: Gradient descent is known to converge quickly for convex objective functions, but it can be trapped at local minimums. On the other hand, Langevin dynamic can explore the state space and find global minimums, but in order to give accurate estimates, it needs to run with small discretization step size and weak stochastic force, which in general slows down its convergence. This work shows that these two algorithms can “collaborate” through a simple exchange mechanism, in which they swap their current positions if Langevin dynamic yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighbourhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. This is joint work with Xin Tong at National University of Singapore.

Thu, 14 May 2020

16:00 - 17:00

Dynamic default contagion: From Eisenberg--Noe to the Mean field

Andreas Sojmark
((Imperial College, London))
Abstract

 

Abstract: In this talk we start by introducing a simple model for interbank default contagion in the vein of the  seminal clearing frameworks of Eisenberg & Noe (2001) and Rogers & Veraart (2013). The key feature, and main novelty, consists in combining stochastic dynamics of the external assets with a simple but realistic balance sheet methodology for determining early defaults. After first developing the model for a finite number of banks, we present a natural way of passing to the mean field limit such that the original network structure (of the interbank obligations) is maintained in a meaningful way. Thus, we provide a clear connection between the more classical network-based literature on systemic risk and the recent approaches rooted in stochastic particle systems and mean field theory.

Thu, 14 May 2020

12:00 - 13:00
Virtual

Augmented systems and surface tension

Prof. Didier Bresch
(Savoie University)
Abstract

In this talk, I will present different PDE models involving surface tension where it may be efficient to consider augmented versions.

Wed, 13 May 2020

17:00 - 18:00

Renaud Lambiotte - Smartphones vs COVID-19

Renaud Lambiotte
(University of Oxford)
Further Information

For several weeks news media has been full of how contact tracing Smartphone apps could help fight COVID-19. However, mobile phones can do more than just trace - they are vital tools in the measurement, prediction and control of the virus.

Looking at recent epidemics as well as COVID-19, Renaud will discuss the different types of data that researchers have been collecting, demonstrating their pros and cons as well as taking a wider view of where mobile data can help us understand the impact of lockdowns on social behaviour and improve our ways of calibrating and updating our epidemiological models. And he will discuss the issue that underpins all this and which is vital for widespread take-up from the Public: privacy and data protection.

Renaud Lambiotte is Associate Professor of Networks and Nonlinear Systems in Oxford.

Watch live:
https://twitter.com/OxUniMaths
https://www.facebook.com/OxfordMathematics/
https://livestream.com/oxuni/lambiotte

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Wed, 13 May 2020
10:00
Virtual

A Mapping Class Group Presentation from Fatgraphs

Adele Jackson
(University of Oxford)
Abstract

The mapping class group of a surface with boundary acts freely and properly discontinuously on the fatgraph complex, which is a contractible cell complex arising from a cell decomposition of Teichmuller space. We will use this action to get a presentation of the mapping class group in terms of fat graphs, and convert this into one in terms of chord diagrams. This chord slide presentation has potential applications to computing bordered Heegaard Floer invariants for open books with disconnected binding.

Tue, 12 May 2020

15:30 - 16:30

Interacting particle systems and random walks on Hecke algebras

Alexey Bufetov
(Hausdorff Center for Mathematics)
Abstract

In the last thirty years there was a lot of progress in understanding the asymmetric simple exclusion process (ASEP). Much less is currently known about the multi-species extension of ASEP. In the talk I will discuss the connection of such an extension to random walks on Hecke algebras and its probabilistic applications. 

Tue, 12 May 2020
15:30
Virtual

Approximate subgroups with bounded VC dimension

Anand Pillay
(Notre Dame)
Further Information

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

Abstract

This is joint with Gabe Conant. We give a structure theorem for finite subsets A of arbitrary groups G such that A has "small tripling" and "bounded VC dimension". Roughly, A will be a union of a bounded number of translates of a coset nilprogession of bounded rank and step (up to a small error).

Tue, 12 May 2020
15:30

Approximate subgroups with bounded VC dimension

Anand Pillay
(Notre Dame)
Further Information

Part of joint combinatorics - logic seminar.  See 

http://people.maths.ox.ac.uk/scott/dmp.htm

Abstract

This is joint with Gabe Conant. We give a structure theorem for finite subsets A of arbitrary groups G such that A has "small tripling" and "bounded VC dimension". Roughly, A will be a union of a bounded number of translates of a coset nilprogession of bounded rank and step (up to a small error).

Tue, 12 May 2020
14:00
Virtual

Sections of high rank varieties and applications

Tamar Ziegler
(Hebrew University of Jerusalem)
Further Information

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

Abstract

I will describe some recent work with D. Kazhdan where we obtain results in algebraic geometry, inspired by questions in additive combinatorics, via analysis over finite fields. Specifically we are interested in quantitative properties of polynomial rings that are independent of the number of variables. A sample application is the following theorem : Let $V$ be a complex vector space, $P$ a high rank polynomial of degree $d$, and $X$ the null set of $P$, $X=\{v \mid P(v)=0\}$. Any function $f:X\to C$ which is polynomial of degree $d$ on lines in $X$ is the restriction of a degree $d$ polynomial on $V$.

Tue, 12 May 2020
12:00

Summing scalar Feynman diagrams

Hadleigh Frost
(Oxford)
Abstract

A motivation in the development of string theory was the 'duality' flip, exchanging the s- and t-channels, which relates all the cubic Feynman graphs at each order in perturbation theory, with fixed planar structure. In string theory, we can understand this as coming from the moduli spaces of marked surfaces, with the cubic diagrams corresponding to complete triangulations. I will describe how geometric-type cluster algebras give a surprising 'linear' way to talk about the same combinatorial problem, using results from work with N Arkani-Hamed and H Thomas and G Salvatori. This gives new ways to compute cubic scalar amplitudes, and new families of integrals generalizing the Veneziano amplitude.