Past

According to the correspondence principle, classical physics emerges in the limit of large quantum numbers. We examine three examples of the semiclassical description of conformal field theory data: large charge boundary operators in the O(2) model, large spin impurities in the free triplet scalar field theory and large charge Wilson lines in QED. By simultaneously taking the coupling to zero and quantum numbers to infinity, we can connect the microscopic to the emergent classical description smoothly.

Abstract: 

Welcome (back) to Fridays@4! To start the new academic year in this session we’ll introduce what Fridays@4 is for our new students and colleagues. This session will be a chance to meet current students and ECRs from across Maths and Stats who will share their hints and tips on conducting successful research in Oxford. There will be lots of time for questions, discussions and generally meeting more people across the two departments – everyone is welcome!

Discrete structures have a long history of use in applied mathematics. Graphs and hypergraphs provide models of social networks, biological systems, academic collaborations, and much more. Network science, and more recently hypernetwork science, have been used to great effect in analyzing these types of discrete structures. Separately, the field of applied topology has gathered many successes through the development of persistent homology, mapper, sheaves, and other concepts. Recent work by our group has focused on the convergence of these two areas, developing and applying topological concepts to study discrete structures that model real data.

This talk will survey our body of work in this area showing our work in both the theoretical and applied spaces. Theory topics will include an introduction to hypernetwork science and its relation to traditional network science, topological interpretations of graphs and hypergraphs, and dynamics of topology and network structures. I will show examples of how we are applying each of these concepts to real data sets.

Cell fate decision-making is responsible for development and homeostasis, and is dysregulated in disease. Despite great promise, we are yet to harness the high-resolution cell state information that is offered by single-cell genomics data to understand cell fate decision-making as it is controlled by gene regulatory networks. We describe how we leveraged joint dynamics + genomics measurements in single cells to develop a new framework for single-cell-informed Bayesian parameter inference of Ca2+ pathway dynamics in single cells. This work reveals a mapping from transcriptional state to dynamic cell fate. But no cell is an island: cell-internal gene regulatory dynamics act in concert with external signals to control cell fate. We developed a multiscale model to study the effects of cell-cell communication on gene regulatory network dynamics controlling cell fates in hematopoiesis. Specifically, we couple cell-internal ODE models with a cell signaling model defined by a Poisson process. We discovered a profound role for cell-cell communication in controlling the fates of single cells, and show how our results resolve a controversy in the literature regarding hematopoietic stem cell differentiation. Overall, we argue for the need to consider single-cell-resolved models to understand and predict the fates of cells.

For positive integers $k$ and $n$ let $\sigma_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erd\H{o}s and Kac conjectured that, for every $k$, the number $\alpha_k = \sum_{n\geq 1} \frac{\sigma_k(n)}{n!}$ is irrational. This is known conditionally for all $k$ assuming difficult conjectures like the Hardy-Littlewood prime $k$-tuples conjecture. Before our work it was known unconditionally that $\alpha_k$ is irrational if $k\leq 3$. We discuss some of the ideas in our recent proof that $\alpha_4$ is irrational. The proof involves sieve methods and exponential sum estimates.

Theory of mean-field games (MFGs) has recently experienced an exponential growth. Existing analytical approaches to find Nash equilibrium (NE) solutions for MFGs are, however, by and large restricted to contractive or monotone settings, or rely on the uniqueness of the NE. We propose a new mathematical paradigm to analyze discrete-time MFGs without any of these restrictions. The key idea is to reformulate the problem of finding NE solutions in MFGs as solving an equivalent optimization problem, called MF-OMO (Mean-Field Occupation Measure Optimization), with bounded variables and trivial convex constraints. It is built on the classical work of reformulating a Markov decision process as a linear program, and by adding the consistency constraint for MFGs in terms of occupation measures, and by exploiting the complementarity structure of the linear program. This equivalence framework enables finding multiple (and possibly all) NE solutions of MFGs by standard algorithms such as projected gradient descent, and with convergence guarantees under appropriate conditions. In particular, analyzing MFGs with linear rewards and with mean-field independent dynamics is reduced to solving a finite number of linear programs, hence solvable in finite time. This optimization reformulation of MFGs can be extended to variants of MFGs such as personalized MFGs.

I will discuss the following statement, a definable version of the (p,q) theorem of Jiří Matoušek from combinatorics, conjectured by Chernikov and Simon:

Suppose that T is NIP and that phi(x,b) does not fork over a model M. Then there is some formula psi(y) in tp(b/M) such that the partial type {phi(x,b’) : psi(b’)} is consistent.

It has long been appreciated that in QCD-like theories without fundamental matter, confinement can be given a sharp characterization in terms of symmetry. More recently, such symmetries have been identified as 1-form symmetries, which fit into the broader category of generalized global symmetries.  In this talk I will discuss obstructions to the existence of a 1-form symmetry in large N QCD, where confinement is a sharp notion. I give general arguments for this disconnect between 1-form symmetries and confinement, and use 2d scalar QCD on the lattice as an explicit example.  

Hilbert complexes are chains of spaces linked by operators, with properties that are crucial to establishing the well-posedness of certain systems of partial differential equations. Designing stable numerical schemes for such systems, without resorting to nonphysical stabilisation processes, requires reproducing the complex properties at the discrete level. Finite-element complexes have been extensively developed since the late 2000's, in particular by Arnold, Falk, Winther and collaborators. These are however limited to certain types of meshes (mostly, tetrahedral and hexahedral meshes), which limits options for, e.g., local mesh refinement.

In this talk we will introduce the Discrete De Rham complex, a discrete version of one of the most popular complexes of differential operators (involving the gradient, curl and divergence), that can be applied on meshes consisting of generic polytopes. We will use a simple magnetostatic model to motivate the need for (continuous and discrete) complexes, then give a presentation of the lowest-order version of the complex and sketch its links with the CW cochain complex on the mesh. We will then briefly explain how this lowest-order version is naturally extended to an arbitrary-order version, and briefly present the associated properties (Poincaré inequalities, primal and adjoint consistency, commutation properties, etc.) that enable the analysis of schemes based on this complex.

Come along for free pizza and to hear about the Mathematrix events this term.

We present a framework to generate and evaluate thematic recommendations based on multilayer network representations of knowledge graphs (KGs).  We represent the relative importance of different types of connections (e.g., Directing/acting) with an intuitive salience matrix that can be learnt from data, tuned to incorporate domain knowledge, address different use cases, or respect business logic. We apply an adaptation of the personalised PageRank algorithm to multilayer network models of KGs to generate item-item recommendations. These recommendations reflect the knowledge we hold about the content, and are suitable for thematic or cold-start settings.

Evaluating thematic recommendations from user data presents unique challenges. Our method only recommends items that are 'thematically' related; that is, easily reachable following connections in the KG. We develop a variant of the widely-used Normalised Discounted Cumulative Gain (NDCG) to evaluate recommendations based on user-item ratings, respecting their thematic nature.

We apply our methods to a KG of the movie industry and MovieLens ratings and in an internal AB test. We learn the salience matrix and demonstrate that our approach outperforms existing thematic recommendation methods and is competitive with collaborative filtering approaches.

Profinite rigidity is essentially the study of which groups can be distinguished from each other by their finite quotients. This talk is meant to give a gentle introduction to the area - I will explain which questions are the right ones to ask and give an overview of some of the main results in the field. I will assume knowledge of what a group presentation is.

In this talk, I will discuss the notion of quantum limits from different viewpoints: Cordes' work on the Gelfand theory for pseudo-differential operators dating from the 70’s as well as the micro-local defect measures and semi-classical measures of the 90’s. I will also explain my motivation and strategy to obtain similar notions in subRiemannian or subelliptic settings. 

I will describe the interaction between a single soliton and a gas of solitons, providing for the first time a mathematical justification for the kinetic theory as posited by Zakharov in the 1970s.  Then I will explain how to use random matrix theory to introduce randomness into a large collection of solitons.

We formulate and sketch the proof of the K-theoretic Farrell-Jones Conjecture for
for the Hecke algebras of reductive p-adic groups. This is the first time that
a version of the farrell-Jones Conjecture for topological groups is formulated. It implies that
the reductive projective class group of the Hecke algebra of a reductive p-adic group
is the colimit of these for all compact open subgroups. This has been proved rationally by
Bernstein and Dat using representation theory. The main applications of our result
will concern the theory of smooth representations
In particular we will prove a conjecture of Dat.

The proof is much more involved than the one for instance for discrete CAT(0)-groups.
We will only give a very brief sketch of it and the new problems occurring in the setting of
totally disconnected groups. Most of the talk will be devoted
an introduction to the Farrell-Jones Conjecture and the theory of
smooth representations of reductive p-adic groups, and
discussion of  applications.

This is a joint project with Arthur Bartels.

An occupational hazard of mathematicians is the investigation of objects that are "optimal" in a mathematically precise sense, yet may be far from optimal in practice. This talk will discuss an extreme example of this effect: Gauss-Hermite quadrature on the real line. For large numbers of quadrature points, Gauss-Hermite quadrature is a very poor method of integration, much less efficient than simply truncating the interval and applying Gauss-Legendre quadrature or the periodic trapezoidal rule. We will present a theorem quantifying this difference and explain where the standard notion of optimality has failed.

In this talk we explore structural results about sets with small doubling in k dimensions. We start in the continuous world with a sharp stability result for the Brunn-Minkowski inequality conjectured by Figalli and Jerison and work our way to the discrete world, where we discuss the natural extension: we show that non-degenerate sets in Z^k with doubling close to 2^k are close to convex progressions i.e. convex sets intersected with a sub-lattice. This talk is based on joint work with Peter van Hintum and Hunter Spink.

Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general. We then give a block decomposition of the supercuspidal subcategory, by introducing a partition on each GL_n(F)-equivalent supercuspidal class through type theory, and we interpret this partition by the sense of l-blocks of finite groups. We give an example where a block of Rep_k(SL_2(F)) is defined with several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A.

Starting from the principle of locality in quantum field theory, which
states that an object is influenced directly only by its immediate

surroundings, I will first briefly review some features of the notion of
locality arising in physics and mathematics. These are then encoded
in  locality relations, given by symmetric binary relations whose graph
consists of pairs of "mutually independent elements".

I will mention challenging questions that arise from  enhancing algebraic
structures to their locality counterparts, such as i) when  is the quotient
of a locality vector space by a linear subspace, a locality vector space, if
equipped with the quotient locality relation,  ii) when does  the locality
tensor product of two locality vector spaces  define a locality vector
space. These are discussed in recent joint work  with Pierre Clavier, Loïc
Foissy and Diego López.

Locality morphisms, namely maps that factorise on   products of  pairs of
"mutually independent" elements, play a key role in the context of
renormalisation in
multiple variables. They include "locality evaluators", which we use to

consistently evaluate meromorphic germs in several variables at
their poles. I will  also report on recent joint work with Li Guo and Bin
Zhang. which gives a classification of locality evaluators on certain
classes of algebras of meromorphic germs.

*** Cancelled *** We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean field limits of recent Ensemble Kalman methods for inverse problems. We show how the transport problem splits into two separate minimisation problems: one for the evolution of mean and covariance of the interpolating curve, and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities, as well as the dynamics of the associated gradient flows. This is joint work of Martin Burger, Matthias Erbar, Daniel Matthes and André Schlichting.

Modular forms are holomorphic functions on the upper half plane satisfying a transformation property under the action of Mobius transformations. While they are a priori complex-analytic objects, they have applications to number theory thanks to their connection with Galois representations. Weight one modular forms are special because their Galois representations factor through a finite quotient. In this talk, we will explain a different degeneracy: they contribute to the cohomology of a line bundle over the modular curve in degrees 0 and 1. We propose an arithmetic explanation for this: an action of a unit group associated to the Galois representation of the modular form. This extends the conjectures of Venkatesh, Prasanna, and Harris. Time permitting, we will discuss a generalization to Hilbert modular forms.

Bestvina--Feighn proved that Aut(F_n) is a rational duality group, i.e. there is a Q[Aut(F_n)]-module, called the rational dualizing module, and a form of Poincare duality relating the rational cohomology of Aut(F_n) to its homology with coefficients in this module. Bestvina--Feighn's proof does not give an explicit combinatorial description of the rational dualizing module of Aut(F_n). But, inspired by Borel--Serre's description of the rational dualizing module of arithmetic groups, Hatcher--Vogtmann constructed an analogous module for Aut(F_n) and asked if it is the rational dualizing module. In work with Miller, Nariman, and Putman, we show that Hatcher--Vogtmann's module is not the rational dualizing module.

We study elastic manifolds with self-repelling terms and estimate their effective radius. This class of manifolds is modelled by a self-repelling vector-valued Gaussian free field with Neumann boundary conditions over the domain [−N,N]^d∩Z^d, that takes values in R^D. Our main results state that for two dimensional domain and range (D=2 and d=2), the effective radius R_N​ of the manifold is approximately N. When the dimension of the domain is d=2 and the dimension of the range is D=1, the effective radius of the manifold is approximately N^{4/3}. This verifies the conjecture of Kantor, Kardar and Nelson (Phys. Rev. Lett. ’86). We also provide results for the case where d≥3 and D≤d. These results imply that self-repelling elastic manifolds with a low dimensional range undergo a significantly stronger stretching than in the case where d=D. 

This is a joint work with Carl Mueller.

For geometric variational problems one often only has weak, rather than strong, compactness results and hence has to deal with the problem that sequences of (almost) critical points $u_j$ can converge to a limiting object with different topology.

A major challenge posed by such singular behaviour is that the seminal results of Simon on Lojasiewicz inequalities, which are one of the most powerful tools in the analysis of the energy spectrum of analytic energies and the corresponding gradient flows, are not applicable.

In this talk we present a method that allows us to prove Lojasiewicz inequalities in the singular setting of almost harmonic maps that converge to a simple bubble tree and explain how these results allow us to draw new conclusions about the energy spectrum of harmonic maps and the convergence of harmonic map flow for low energy maps from surfaces of positive genus into general analytic manifolds.

I will discuss the use of partitioned schemes for neural networks. This work is in the tradition of multrate numerical ODE methods in which different components of system are evolved using different numerical methods or with different timesteps. The setting is the training tasks in deep learning in which parameters of a hierarchical model must be found to describe a given data set. By choosing appropriate partitionings of the parameters some redundant computation can be avoided and we can obtain substantial computational speed-up. I will demonstrate the use of the procedure in transfer learning applications from image analysis and natural language processing, showing a reduction of around 50% in training time, without impairing the generalization performance of the resulting models. This talk describes joint work with Tiffany Vlaar.

Liouville conformal field theory models two-dimensional gravity with a cosmological constant and conformal matter. In its timelike regime, it reproduces the characteristic negative kinetic term of the conformal factor of the metric in the Einstein-Hilbert action, the sign which infamously makes the gravity path integral ill-defined. In this talk, I will first discuss the perturbative computation of the timelike Liouville partition function around the sphere saddle and propose an all-orders result. I will then turn to the disk and present the bulk 1-point functions of this CFT, and discuss possible interpretations in terms of boundary conditions.

Waldspurger proved maximality and minimality results for certain generalised Springer representations of $\text{Sp}(2n,\mathbb{C})$. We will discuss analogous results for $G = \text{SO}(N,\mathbb{C})$ and sketch their proofs.

Let $C$ be a unipotent class of $G$ and $E$ an irreducible $G$-equivariant local system on $C$. Let $\rho$ be the generalised Springer representation corresponding to $(C,E)$. We call $C$ the support of $\rho$. It is well-known that $\rho$ appears in the top cohomology of a certain variety. Let $\bar\rho$ be the representation obtained by summing the cohomology groups of this variety.

We show that if $C$ is parametrised by an orthogonal partition consisting of only odd parts, then $\bar\rho$ has a unique irreducible subrepresentation $\rho^{\text{max}}$ whose support is maximal among the supports of the irreducible subrepresentations of $\rho^{\text{max}}$. We also show that $\text{sgn}\otimes\rho^{\text{max}}$ is the unique subrepresentation of $\text{sgn}\otimes\bar\rho$ with minimal support. We will also present an algorithm to compute $\rho^{\text{max}}$.

I will review the basic assumptions and spell out the arguments that lead to the bound on the Regge growth of gravitational scattering amplitudes. I will discuss the Regge bounds both at fixed transfer momentum and smeared over it. Our basic conclusion is that gravitational scattering amplitudes admit dispersion relations with two subtractions. For a sub-class of smeared amplitudes, black hole formation reduces the number of subtractions to one. Finally, I will discuss bounds on local growth derived using dispersion relations. This talk is based on https://arxiv.org/abs/2202.08280.

Irreversible processes are accompanied by an increase in the internal entropy of a continuum, and as such the entropy production function is fundamental in determining the overall state of the system. In this talk, it will be shown that the entropy production function can be utilized for a variational analysis of certain dissipative continua in two different ways. Firstly, a novel unified Lagrangian-Hamiltonian formalism is constructed giving phase space extra structure, and applied to the study of fluid flow and brittle fracture.  Secondly, a maximum entropy production principle is presented for simple bodies and its implications to the study of fluid flow discussed. 

Second-order regularity results are established for solutions to elliptic equations and systems with the principal part having a Uhlenbeck structure and square-integrable right-hand sides. Both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems under minimal regularity assumptions on the boundary of the domain. In particular, if the domain is convex, no regularity of its boundary is needed. A critical step in the approach is a sharp pointwise inequality for the involved elliptic operator. This talk is based on joint investigations with A.Kh.Balci, L.Diening, and V.Maz'ya.

A common observation that wider (in the number of hidden units/channels/attention heads) neural networks perform better motivates studying them in the infinite-width limit.

Remarkably, infinitely wide networks can be easily described in closed form as Gaussian processes (GPs), at initialization, during, and after training—be it gradient-based, or fully Bayesian training. This provides closed-form test set predictions and uncertainties from an infinitely wide network without ever instantiating it (!).

These infinitely wide networks have become powerful models in their own right, establishing several SOTA results, and are used in applications including hyper-parameter selection, neural architecture search, meta learning, active learning, and dataset distillation.

The talk will provide a high-level overview of our work at Google Brain on infinite-width networks. In the first part I will derive core results, providing intuition for why infinite-width networks are GPs. In the second part I will discuss challenges and solutions to implementing and scaling up these GPs. In the third part, I will conclude with example applications made possible with infinite width networks.

The talk does not assume familiarity with the topic beyond general ML background.

4th IMA Conference on The Mathematical Challenges of Big Data

The 4th Ima Conference on The Mathematical Challenges of Big Data is issuing a Call For Papers for both contributed talks and posters. Mathematical foundations of data science and its ongoing challenges are rapidly growing fields, encompassing areas such as: network science, machine learning, modelling, information theory, deep and reinforcement learning, applied probability and random matrix theory. Applying deeper mathematics to data is changing the way we understand the environment, health, technology, quantitative humanities, the natural sciences, and beyond ‐ with increasing roles in society and industry. This conference brings together researchers and practitioners to highlight key developments in the state‐of‐the art and find common ground where theory and practice meet, to shape future directions and maximize impact. We particularly welcome talks aimed to inform on recent developments in theory or methodology that may have applied consequences, as well as reports of diverse applications that have led to interesting successes or uncovered new challenges.

Contributed talks and posters are welcomed from the mathematically oriented data science community. Contributions will be selected based on brief abstracts and can be based on previously unpresented results, or recent material originally presented elsewhere. We encourage contributions from both established and early career researchers. Contributions will be assigned to talks or posters based on the authors request as well as the views of the organizing committee on the suitability of the results. The conference will be held in person with the option to attend remotely where needed.

Inducement of sparsity, Heather Battey
Sparsity, the existence of many zeros or near-zeros in some domain, is widely assumed throughout the high-dimensional literature and plays at least two roles depending on context. Parameter orthogonalisation (Cox and Reid, 1987) is presented as inducement of population-level sparsity. The latter is taken as a unifying theme for the talk, in which sparsity-inducing parameterisations or data transformations are sought. Three recent examples are framed in this light: sparse parameterisations of covariance models; systematic construction of factorisable transformations for the elimination of nuisance parameters; and inference in high-dimensional regression. The solution strategy for the problem of exact or approximate sparsity inducement appears to be context specific and may entail, for instance, solving one or more partial differential equation, or specifying a parameterised path through transformation or parameterisation space.

Confirmed Invited Speakers
Dr Heather Battey, Imperial College London
Prof. Lenka Zdebrova, EPFL (Swiss Federal Institute Technology)
Prof. Nando de Freitas, Google Deep Mind
Prof. Tiago de Paula Peixoto, Central European University

Please follow the link below for Programme and Registration:

https://ima.org.uk/17625/4th-ima-conference-on-the-mathematical-challen…

Join us in Oxford in September 2022  for displays, demonstrations and a lively workshop discussions celebrating the minds, manuscripts and machines that made the dreams and realities that  we now call artificial intelligence. 

PDE Workshop in Stability Analysis for Nonlinear PDEs will be running Monday 15th - Friday 19th August.

Location: L3, AWB

Our goal was to bring together leading experts in the stability analysis of nonlinear partial differential equations across multi-scale applications. Some of the topics to be addressed include: 

• Stability analysis of shock wave patterns of reflections/diffraction.
• Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities for multidimensional hyperbolic systems of conservation laws.
• Stability analysis of particle to continuum limits including the quantifying asymptotic/mean-field/large-time limits for pairwise interactions and particle limits for general interactions among multi-agent systems
• Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from multidimensional compressible viscous to inviscid flows with large initial data.

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.

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.

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.

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/).

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.

Article link: https://www.nature.com/articles/s41567-022-01634-8

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.

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

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. 

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.

ArXiv link: https://arxiv.org/abs/2112.05642

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.