Thu, 04 Aug 2022
15:00
S2.27

### K-theoretic classification of inductive limit actions of fusion categories on AF-algebras

Roberto Hernandez Palomares
(Texas A&M University)
Abstract

I will introduce a K-theoretic complete invariant of inductive limits of finite dimensional actions of fusion categories on unital AF-algebras. This framework encompasses all such actions by finite groups on AF-algebras. Our classification result essentially follows from applying Elliott's Intertwining Argument adapted to this equivariant context, combined with tensor categorical techniques.

Our invariant roughly consists of a finite list of pre-ordered abelian groups and positive homomorphisms, which can be computed in principle. Under certain conditions this can be done in full detail. For example, using our classification theorem, we can show torsion-free fusion categories admit a unique AF-action on certain AF-algebras.

Connecting with subfactors, inspired by Popa’s classification of finite-depth hyperfinite subfactors by their standard invariant, we study unital inclusions of AF-algebras with trivial centers, as natural analogues of hyperfinite II_1 subfactors. We introduce the notion of strongly AF-inclusions and an Extended Standard Invariant, which characterizes them up to equivalence.

Thu, 07 Jul 2022
12:00
C2

### Resonances and unitarity from celestial amplitude

Dr Jinxiang Wu
((Oxford University))

Note: we would recommend to join the meeting using the Zoom client for best user experience.

Abstract

We study the celestial description of the O(N) sigma model in the large N limit. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand unitarity, we propose a form of the optical theorem that controls the imaginary part of the correlator based solely on the presence of these resonances. We also explicitly analyze the conformal block expansions and factorization of four-point functions into three-point functions. We find that summing over resonances is key for these factorization properties to hold. This is a joint work with D. García-Sepúlveda, A. Guevara, J. Kulp.

Wed, 06 Jul 2022
12:00
C2

### Pushing Forward Rational Differential Forms

Robert Moermann
(University of Hertfordshire)

Note: we would recommend to join the meeting using the Zoom client for best user experience.

Abstract

The scattering equations connect two modern descriptions of scattering amplitudes: the CHY formalism and the framework of positive geometries. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry in the CHY moduli space. In this talk, I consider the general problem of pushing forward rational differential forms via the scattering equations. I will present some recent results (2206.14196) for achieving this without ever needing to explicitly solve any scattering equations. These results use techniques from computational algebraic geometry, and they extend the application of similar results for rational functions to rational differential forms.

Wed, 29 Jun 2022

16:00 - 17:00

### Information theory with kernel methods

Francis Bach
(INRIA - Ecole Normale Supérieure)
Further Information
Abstract

I will consider the analysis of probability distributions through their associated covariance operators from reproducing kernel Hilbert spaces. In this talk, I will show that the von Neumann entropy and relative entropy of these operators are intimately related to the usual notions of Shannon entropy and relative entropy, and share many of their properties. They come together with efficient estimation algorithms from various oracles on the probability distributions. I will also present how these new notions of relative entropy lead to new upper-bounds on log partition functions, that can be used together with convex optimization within variational inference methods, providing a new family of probabilistic inference methods (based on https://arxiv.org/pdf/2202.08545.pdf, see also https://francisbach.com/information-theory-with-kernel-methods/).

Tue, 28 Jun 2022

14:00 - 15:00
C3

### The temporal rich club phenomenon

Nicola Pedreschi
(Mathematical Institute (University of Oxford))
Abstract

Identifying the hidden organizational principles and relevant structures of complex networks is fundamental to understand their properties. To this end, uncovering the structures involving the prominent nodes in a network is an effective approach. In temporal networks, the simultaneity of connections is crucial for temporally stable structures to arise. In this work, we propose a measure to quantitatively investigate the tendency of well-connected nodes to form simultaneous and stable structures in a temporal network. We refer to this tendency as the temporal rich club phenomenon, characterized by a coefficient defined as the maximal value of the density of links between nodes with a minimal required degree, which remain stable for a certain duration. We illustrate the use of this concept by analysing diverse data sets and their temporal properties, from the role of cohesive structures in relation to processes unfolding on top of the network to the study of specific moments of interest in the evolution of the network.

Mon, 27 Jun 2022
16:15
St Catherine's

### The Reddick Lecture 2022: The Benefits of Applied Mathematics in Product Development

Dr Uwe Beuscher, W.L. Gore & Associates, Inc.
Further Information

Abstract

Throughout a product development project, many decisions must be made. These include whether to start, stop, continue, or re-direct a project based on the learnings of the project team. Some of these decisions are related to the risk of achieving certain product performance attributes and they are often based on experimental observations in the laboratory or in field applications of early prototypes. Sometimes, these observations provide sufficient insight but often a significant uncertainty remains. Mathematical simulation can provide deeper insight into the mechanisms, may indicate limiting parameters and transport steps, and allows exploration of novel prototypes without actually making them. This talk will illustrate how Mathematics have been used to inform project development projects and their guiding decisions at WL Gore by describing examples from three very different applications.

Mon, 27 Jun 2022

12:45 - 13:45
L3

### Marginal quenches and drives in Tomonaga-Luttinger liquids/free boson CFTs

Apoorv Tivari
(Stockholm)
Abstract

I will discuss the free compact boson CFT thrown out of equilibrium by marginal deformations, modeled by quenching or periodically driving the compactification radius of the free boson between two different values. All the dynamics will be shown to be crucially dependent on the ratio of the compactification radii via the Zamolodchikov distance in the space of marginal deformations. I will present various exact analytic results for the Loschmidt echo and the time evolution of energy density for both the quench and the periodic drive. Finally, I will present a non-perturbative computation of the  Rényi divergence, an information-theoretic distance measure, between two marginally deformed thermal density matrices.

The talk will be based on the recent preprint: arXiv:2206.11287

Mon, 27 Jun 2022 09:00 -
Fri, 22 Jul 2022 17:00
Mathematical Institute, Ground and Mezzanine levels

### All we ever wanted was everything / 24.02.22 (for Ukraine)

Andy Bullock
Further Information

On June 27th, in the Reception area of the Mathematical Institute, Oxford artist Andy Bullock unveiled his most ambitious knot sculpture to date, a large floor-based work titled ‘All we ever wanted was everything / 24.02.22 (for Ukraine)’ constructed using 70 metres of metal trunking. As with all his knot sculptures they often reference issues of complexity with situations and people, the personal and interpersonal; focusing on what it means to be human.

In a first for the artist, Bullock will be inviting members of the recently arrived Ukrainian refugee community to contribute to the artwork by incorporating items of personal relevance. Bullock is reaching out to Oxfordshire’s Ukrainian community in a collaboration with Yulia Astasheva, a recent arrival herself from the Dnipropetrovsk region, where she still has close family living only miles from the Russian-occupied region.

The idea for the work came initially from a commission from Oxford Mathematics for Bullock to create an exhibition of his maths-related painting, photography and sculpture to be open to the public this summer. The core of his fine art master’s degree show last year was a creative examination and exploration of the topological subject of knot theory, and in particular the work of Clifford Hugh Dowker (1912-82) an eminent mathematician whose work is still studied today. “I find a poetic beauty in the mathematics I researched even though my understanding of the subject is virtually nil” said Bullock. “My final dissertation for my master’s degree examined the similarities in thought of mathematicians working in these areas and that of artists working in a more conceptual arena”.

In the lower ground floor space of the building there is an exhibition of some of Andy Bullock’s ‘knot variation’ paintings and photographs and a display of original handwritten manuscripts from Dowker’s personal archive alongside Andy's own sketchbooks, allowing an insight into the respective processes of mathematician and artist.

For further information:

Andy Bullock - @email - 07582 526957 - www.bullockstudio.com

Yulia Astasheva - @email

Tue, 21 Jun 2022

16:30 - 17:30
C1

### Amenable actions and purely infinite crossed products

Julian Kranz
(University of Münster)
Abstract

Since the completion of the Elliott classification programme it is an important question to ask which C*-algebras satisfy the assumptions of the classification theorem. We will ask this question for the case of crossed-product C*-algebras associated to actions of nonamenable groups and focus on two extreme cases: Actions on commutative C*-algebras and actions on simple C*-algebras. It turns out that for a large class of nonamenable groups, classifiability of the crossed product is automatic under the minimal assumptions on the action. This is joint work with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro.

Tue, 21 Jun 2022

15:30 - 16:30
L4

Oliver Gäfvert
(Oxford)
Tue, 21 Jun 2022

14:00 - 15:00
L6

### The orbit method and normality of closures of nilpotent orbits

Dan Barbasch
(Cornell University, USA)
Abstract

The work of Kraft-Procesi classifies closures of nilpotent orbits that are normal in the cases of classical complex Lie algebras. Subsequent work of Ranee Brylinsky combines this work with the Theta correspondence as defined by Howe to attach a representation of the corresponding complex group. It provides a quantization of the closure of a nilpotent orbit. In joint work with Daniel Wong, we carry out a detailed analysis of these representations viewed as (\g,K)-modules of the complex group viewed as a real group. One consequence is a "representation theoretic" proof of the classification of Kraft-Procesi.

Tue, 21 Jun 2022

14:00 - 15:00
C6

### Sequential Motifs in Observed Walks

Timothy LaRock
(Mathematical Institute (University of Oxford))
Abstract

The structure of complex networks can be characterized by counting and analyzing network motifs, which are small graph structures that occur repeatedly in a network, such as triangles or chains. Recent work has generalized motifs to temporal and dynamic network data. However, existing techniques do not generalize to sequential or trajectory data, which represents entities walking through the nodes of a network, such as passengers moving through transportation networks. The unit of observation in these data is fundamentally different, since we analyze observations of walks (e.g., a trip from airport A to airport C through airport B), rather than independent observations of edges or snapshots of graphs over time. In this work, we define sequential motifs in trajectory data, which are small, directed, and sequenced-ordered graphs corresponding to patterns in observed sequences. We draw a connection between counting and analysis of sequential motifs and Higher-Order Network (HON) models. We show that by mapping edges of a HON, specifically a kth-order DeBruijn graph, to sequential motifs, we can count and evaluate their importance in observed data, and we test our proposed methodology with two datasets: (1) passengers navigating an airport network and (2) people navigating the Wikipedia article network. We find that the most prevalent and important sequential motifs correspond to intuitive patterns of traversal in the real systems, and show empirically that the heterogeneity of edge weights in an observed higher-order DeBruijn graph has implications for the distributions of sequential motifs we expect to see across our null models.

Mon, 20 Jun 2022
15:30
L5

### Coxeter groups acting on CAT(0) cube complexes

Michah Sageev
Abstract

We will give a general overview of how one gets groups to act on CAT(0) cube complexes, how compatible such actions are and how this plays out in the setting of Coxeter groups.

Mon, 20 Jun 2022

12:45 - 13:45
L4

### Large N Partition Functions, Holography, and Black Holes

Nikolay Bobev
Abstract

I will discuss the large N behavior of partition functions of the ABJM theory on compact Euclidean manifolds. I will pay particular attention to the S^3 free energy and the topologically twisted index for which I will present closed form expressions valid to all order in the large N expansion. These results have important implications for holography and the microscopic entropy counting of AdS_4 black holes which I will discuss. I will also briefly discuss generalizations to other SCFTs arising from M2-branes.

Fri, 17 Jun 2022

16:00 - 17:00
L5

### Defect Central Charges

(Southampton University)
Abstract

Conformal defects can be characterised by their contributions to the Weyl anomaly. The coefficients of these terms, often called defect central charges, depend on the particular defect insertion in a given conformal field theory. I will review what is currently known about defect central charges across dimensions, and present novel results. I will discuss many examples where they can be computed exactly without requiring any approximations or limits. Particular emphasis will be placed on recently developed tools for superconformal defects as well as defects in free theories.

Fri, 17 Jun 2022

14:00 - 15:00
L6

### Data-driven early detection of potential high risk SARS-CoV-2 variants

Dr Marcin J. Skwark
Abstract

Recent advances in Deep Learning have enabled us to explore new application domains in molecular biology and drug discovery - including those driven by complex processes that defy analytical modelling.  However, despite the combined forces of increased data, improving compute resources and continuous algorithmic innovation all bringing previously intractable problems into the realm of possibility, many advances are yet to make a tangible impact for life science discovery.  In this talk, Dr Marcin J. Skwark will discuss the challenge of bringing machine learning innovation to tangible real-world impact.  Following a general introduction of the topic, as well as newly available methods and data, he will focus on the modelling of COVID-19 variants and, in particular, the DeepChain Early Warning System (EWS) developed by InstaDeep in collaboration with BioNTech.  With thousands of new, possibly dangerous, SARS-CoV-2 variants emerging each month worldwide, it is beyond humanities combined capacity to experimentally determine the immune evasion and transmissibility characteristics of every one.  EWS builds on an experimentally tested AI-first computational biology platform to evaluate new variants in minutes, and is capable of risk monitoring variant lineages in near real-time.  This is done by combining an AI-driven protein structure prediction framework with large, spike protein sequence-oriented Transformer models to allow for rapid simulation-free assessment of the immune escape risk and expected fitness of new variants, conditioned on the current state of the world.  The system has been extensively validated in cooperation with BioNTech, both in terms of host cell infection propensity (including experimental assays of receptor binding affinity), and immune escape (pVNT assays with monoclonal antibodies and real-life donor sera). In these assessments, purely unsupervised, data-first methods of EWS have shown remarkable accuracy. EWS flags and ranks all but one of the SARS-CoV-2 Variants of Concern (Alpha, Beta, Gamma, Delta… Omicron), discriminates between subvariants (e.g. BA.1/BA.2/BA.4 etc. distinction) and for most of the adverse events allows for proactive response on the day of the observation. This allows for appropriate response on average six weeks before it is possible for domain experts using domain knowledge and epidemiological data. The performance of the system, according to internal benchmarks, improves with time, allowing for example for supporting the decisions on the emerging Omicron subvariants on the first days of their occurrence. EWS impact has been notable in general media [2, 3] for the system's applicability to a novel problem, ability to derive generalizable conclusions from unevenly distributed, sparse and noisy data, to deliver insights which otherwise necessitate long and costly experimental assays.

Fri, 17 Jun 2022

14:00 - 15:00
L4

### Cancelled

Fri, 17 Jun 2022

14:00 - 17:00
Large Lecture Theatre, Department of Statistics, University of Oxford

### CDT in Mathematics of Random Systems June Workshop 2022

Ziheng Wang, Professor Ian Melbourne, Dr Sara Franceschelli
Further Information

Abstract

2:00 Ziheng Wang, EPSRC CDT in Mathematics of Random Systems Student

Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

Abstract: We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using a stochastic estimate for the gradient of the stationary distribution. The gradient estimate satisfies an SDE and is simultaneously updated, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of our online algorithm for dissipative SDE models and present numerical results for other nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm.

2:45 Ian Melbourne,  Professor of Mathematics, University of Warwick

Interpretation of stochastic integrals, and the Levy area

Abstract: An important question in stochastic analysis is the appropriate interpretation of stochastic integrals. The classical Wong-Zakai theorem gives sufficient conditions under which smooth integrals converge to Stratonovich stochastic integrals. The conditions are automatic in one-dimension, but in higher dimensions it is necessary to take account of corrections stemming from the Levy area. The first part of the talk covers work with Kelly 2016, where we justified the Levy area correction for large classes of smooth systems, bypassing any stochastic modelling assumptions. The second part of the talk addresses a much less studied question: is the Levy area zero or nonzero for systems of physical interest, eg Hamiltonian time-reversible systems? In recent work with Gottwald, we classify (and clarify) the situations where such structure forces the Levy area to vanish. The conclusion of our work is that typically the Levy area correction is nonzero.

3:45 Break

4:15 Sara Franceschelli, Associate Professor,  École Normale Supérieure de Lyon

When is a model is a good model? Epistemological perspectives on mathematical modelling

When a model is a good model? Must it represent a specific target system? Allow to make predictions? Provide an explanation for observed behaviors?  After a brief survey of general epistemological questions on modelling, I will consider examples of mathematical modelling in physics and biology from the perspective of dynamical systems theory. I will first show that even if it has been little noticed by philosophers, dynamical systems theory itself as a mathematical theory has been a source of questions and criteria in order to assess the goodness of a model (notions of stability, genericity, structural stability). I will then discuss the theoretical fruitfulness of arguments of (in)stability in the mathematical modelling of morphogenesis.

Fri, 17 Jun 2022

10:00 - 11:00
L4

### Silt build up at Peel Ports locks

David Porter (Carbon Limiting Technologies), Chris Breward, Daniel Alty (Peel Ports; joining remotely)
(Peel Ports)
Abstract

Peel Ports operate a number of locks that allow ships to enter and leave the port. The lock gates comprise a single caisson structure which blocks the waterway when closed and retracts into the dockside as the gate opens. Build up of silt ahead of the opening lock gate can prevent it from fully opening or requiring excessive power to move. If the lock is not able to fully open, ships are unable to enter the port, leading to significant operational impacts for the whole port. Peel ports are interested in understanding, and mitigating, this silt build up.

Thu, 16 Jun 2022

16:00 - 17:00
L4

### Ax-Schanuel and exceptional integrability

Jonathan Pila
(University of Oxford)
Abstract

In joint work with Jacob Tsimerman we study when the primitive
of a given algebraic function can be constructed using primitives
from some given finite set of algebraic functions, their inverses,
algebraic functions, and composition. When the given finite set is just {1/x}
this is the classical problem of "elementary integrability".
We establish some results, including a decision procedure for this problem.

Thu, 16 Jun 2022

14:00 - 15:00
L2

Carmen Jorge Diaz
((Oxford University))
Abstract
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome
Thu, 16 Jun 2022

14:00 - 15:00
L5

### Recent results on finite element methods for incompressible flow at high Reynolds number

Erik Burman
(University College London)
Abstract

The design and analysis of finite element methods for high Reynolds flow remains a challenging task, not least because of the difficulties associated with turbulence. In this talk we will first revisit some theoretical results on interior penalty methods using equal order interpolation for smooth solutions of the Navier-Stokes’ equations at high Reynolds number and show some recent computational results for turbulent flows.

Then we will focus on so called pressure robust methods, i.e. methods where the smoothness of the pressure does not affect the upper bound of error estimates for the velocity of the Stokes’ system. We will discuss how convection can be stabilized for such methods in the high Reynolds regime and, for the lowest order case, show an interesting connection to turbulence modelling.

Thu, 16 Jun 2022

12:00 - 13:00
L2

### Repulsive Geometry

Keenan Crane
(Carnegie Mellon Univeristy, School of Computer Science)
Further Information

Keenan Crane is the Michael B. Donohue Associate Professor in the School of Computer Science at Carnegie Mellon University, and a member of the Center for Nonlinear Analysis in the Department of Mathematical Sciences.  He is a Packard Fellow and recipient of the NSF CAREER Award, was a Google PhD Fellow in the Department of Computing and Mathematical Sciences at Caltech, and was an NSF Mathematical Postdoctoral Research Fellow at Columbia University.  His work applies insights from differential geometry and computer science to develop fundamental algorithms for working with real-world geometric data.  This work has been used in production at Fortune 500 companies, and featured in venues such as Communications of the ACM and Notices of the AMS, as well as in the popular press through outlets such as WIRED, Popular Mechanics, National Public Radio, and Scientific American.

Abstract

Numerous applications in geometric, visual, and scientific computing rely on the ability to nicely distribute points in space according to a repulsive potential.  In contrast, there has been relatively little work on equidistribution of higher-dimensional geometry like curves and surfaces—which in many contexts must not pass through themselves or each other.  This talk explores methods for optimization of curve and surface geometry while avoiding (self-)collision. The starting point is the tangent-point energy of Buck & Orloff, which penalizes pairs of points that are close in space but distant with respect to geodesic distance. We develop a discretization of this energy, and introduce a novel preconditioning scheme based on a fractional Sobolev inner product.  We further accelerate this scheme via hierarchical approximation, and describe how to incorporate into a constrained optimization framework. Finally, we explore how this machinery can be applied to problems in mathematical visualization, geometric modeling, and geometry processing.

Wed, 15 Jun 2022
14:00
L5

### The heterotic $G_2$ system and coclosed $G_2$-structures on cohomogeneity one manifolds

Izar Alonso Lorenzo
(Oxford)
Abstract

When considering compatifications of heterotic string theory down to 3D, the heterotic $G_2$ system arises naturally. It is a system for both geometric fields and gauge fields over a manifold with a $G_2$-structure. In particular, it asks for the $G_2$-structure to be coclosed. We will begin this talk defining this system and giving a description of the geometry of cohomogeneity one manifolds. Then, we will look for coclosed $G_2$-structures in the cohomogeneity one setting. We will end up by proving the existence of a family of coclosed $G_2$-structures which are invariant under a cohomogeneity one action of $\text{SU}(2)^2$ on certain seven-dimensional simply connected manifolds.

Tue, 14 Jun 2022

16:00 - 17:00
C1

### Semifinite tracial ultraproducts

James Gabe
(University of Southern Denmark)
Abstract

One of the most important constructions in operator algebras is the tracial ultrapower for a tracial state on a C*-algebra. This tracial ultrapower is a finite von Neumann algebra, and it appears in seminal work of McDuff, Connes, and more recently by Matui-Sato and many others for studying the structure and classification of nuclear C*-algebras. I will talk about how to generalise this to unbounded traces (such as the standard trace on B(H)). Here the induced tracial ultrapower is not a finite von Neumann algebra, but its multiplier algebra is a semifinite von Neumann algebra.