Past

We introduce constrained variational principles for fluid dynamics on rough paths. The advection of the fluid is constrained to be the sum of a vector field which represents coarse-scale motion and a rough (in time) vector field which parametrizes fine-scale motion. The rough vector field is regarded as fixed and the rough partial differential equation for the coarse-scale velocity is derived as a consequence of being a critical point of the action functional.

The action functional is perturbative in the sense that if the rough vector f ield is set to zero, then the corresponding variational principle agrees with the reduced (to the vector fields) Euler-Poincare variational principle introduced in Holm, Marsden and Ratiu (1998). More precisely, the Lagrangian in the action functional encodes the physics of the fluid and is a function of only the coarse-scale velocity. 

By parametrizing the fine-scales of fluid motion with a rough vector field, we preserve the pathwise nature of deterministic fluid dynamics and establish a flexible framework for stochastic parametrization schemes. The main benefit afforded by our approach is that the system of rough partial differential equations we derive satisfy essential conservation laws, including Kelvin’s circulation theorem. This talk is based on recent joint work with Dan Crisan, Darryl Holm, and Torstein Nilssen.

In corporate bond markets, which are mainly OTC markets, market makers play a central role by providing bid and ask prices for a large number of bonds to asset managers from all around the globe. Determining the optimal bid and ask quotes that a market maker should set for a given universe of bonds is a complex task. Useful models exist, most of them inspired by that of Avellaneda and Stoikov. These models describe the complex optimization problem faced by market makers: proposing bid and ask prices in an optimal way for making money out of the difference between bid and ask prices while mitigating the market risk associated with holding inventory. While most of the models only tackle one-asset market making, they can often be generalized to a multi-asset framework. However, the problem of solving numerically the equations characterizing the optimal bid and ask quotes is seldom tackled in the literature, especially in high dimension. In this paper, our goal is to propose a numerical method for approximating the optimal bid and ask quotes over a large universe of bonds in a model à la Avellaneda-Stoikov. Because we aim at considering a large universe of bonds, classical finite difference methods as those discussed in the literature cannot be used and we present therefore a discrete time method inspired by reinforcement learning techniques. More precisely, the approach we propose is a model-based actor-critic-like algorithm involving deep neural networks

Non-uniformly elliptic functionals are variational integrals like
\[
(1) \qquad \qquad W^{1,1}_{loc}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega} \left[F(x,Dw)-f\cdot w\right] \, \textrm{d}x,
\]
characterized by quite a wild behavior of the ellipticity ratio associated to their integrand $F(x,z)$, in the sense that the quantity
$$
\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}}\mathcal R(z, B):=\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}} \frac{\mbox{highest eigenvalue of}\ \partial_{z}^{2} F(x,z)}{\mbox{lowest eigenvalue of}\  \partial_{z}^{2} F(x,z)} $$
may blow up as $|z|\to \infty$. 
We analyze the interaction between the space-depending coefficient of the integrand and the forcing term $f$ and derive optimal Lipschitz criteria for minimizers of (1). We catch the main model cases appearing in the literature, such as functionals with unbalanced power growth or with fast exponential growth such as
$$
w \mapsto \int_{\Omega} \gamma_1(x)\left[\exp(\exp(\dots \exp(\gamma_2(x)|Dw|^{p(x)})\ldots))-f\cdot w \right]\, \textrm{d}x
$$
or
$$
w\mapsto \int_{\Omega}\left[|Dw|^{p(x)}+a(x)|Dw|^{q(x)}-f\cdot w\right] \, \textrm{d}x.
$$
Finally, we find new borderline regularity results also in the uniformly elliptic case, i.e. when
$$\mathcal{R}(z,B)\sim \mbox{const}\quad \mbox{for all balls} \ \ B\Subset \Omega.$$

The talk is based on:
C. De Filippis, G. Mingione, Lipschitz bounds and non-autonomous functionals. $\textit{Preprint}$ (2020).

A common theme in the definitions of larger large cardinals is the existence of elementary embeddings from the universe into an inner model. In contrast, smaller large cardinals, such as weakly compact and Ramsey cardinals, are usually characterized by their combinatorial properties such as existence of large homogeneous sets for colorings. It turns out that many familiar smaller large cardinals have elegant elementary embedding characterizations. The embeddings here are correspondingly ‘small’; they are between transitive set models of set theory, usually the size of the large cardinal in question. The study of these elementary embeddings has led us to isolate certain important properties via which we have defined robust hierarchies of large cardinals below a measurable cardinal. In this talk, I will introduce these types of elementary embeddings and discuss the large cardinal hierarchies that have come out of the analysis of their properties. The more recent results in this area are a joint work with Philipp Schlicht.

Let X_1 and X_2 be finite graphs with isomorphic universal covers.

Leighton's graph covering theorem states that X_1 and X_2 have a common finite cover.

I will discuss recent work generalizing this theorem and how myself and Sam Shepherd have been applying it to rigidity questions in geometric group theory.

I will talk about how to get large deviations estimates for randomly rotated matrix models using the asymptotics of spherical (aka orbital, aka HCIZ) integrals. Compared to the talk I gave last week in integrable probability conference I will concentrate on random  matrices rather than symmetric functions.

The classical Erdős-Szekeres theorem dating back almost a hundred years states that any sequence of $(n-1)^2+1$ distinct real numbers contains a monotone subsequence of length $n$. This theorem has been generalised to higher dimensions in a variety of ways but perhaps the most natural one was proposed by Fishburn and Graham more than 25 years ago. They raise the problem of how large should a $d$-dimesional array be in order to guarantee a "monotone" subarray of size $n \times n \times \ldots \times n$. In this talk we discuss this problem and show how to improve their original Ackerman-type bounds to at most a triple exponential. (Joint work with M. Bucic and T. Tran)

A Latin square of order n is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser, Brualdi and Stein from 60s which says that every Latin square of order $n \times n$ contains a transversal of order $n-1$. A closely related problem is 40 year old conjecture of Brouwer that every Steiner triple system of order $n$ contains a matching of size $\frac{n-4}{3}$. The third problem we'd like to mention asks how many distinct symbols in Latin arrays suffice to guarantee a full transversal? In this talk we discuss a novel approach to attack these problems. Joint work with Peter Keevash, Alexey Pokrovskiy and Benny Sudakov.

Generative Adversarial Networks (GANs) have celebrated great empirical success, especially in image generation and processing. Meanwhile, Mean-Field Games (MFGs),  as analytically feasible approximations for N-player games, have experienced rapid growth in theory of controls. In this talk, we will discuss a new theoretical connections between GANs and MFGs. Interpreting MFGs as GANs, on one hand, allows us to devise GANs-based algorithm to solve MFGs. Interpreting GANs as MFGs, on the other hand, provides a new and probabilistic foundation for GANs. Moreover, this interpretation helps establish an analytical connection between GANs and Optimal Transport (OT) problems, the connection previously understood mostly from the geometric perspective. We will illustrate by numerical examples of using GANs to solve high dimensional MFGs, demonstrating its superior performance over existing methodology.

Addressing a question of Baker and Reid,

we give a criterion to show that an arithmetic group 

has a congruence subgroup that is algebraically

fibered. Some examples to which the criterion applies

include a hyperbolic 4-manifold group containing infinitely

many Bianchi groups, and a complex hyperbolic surface group.

This is joint work with Matthew Stover.

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. 

We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/diffenretial geometry, and then to computational geometry, and now to data science and AI.
With the concrete playground of the Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of physical and mathematical interest.
 

In this session, Laura will explain the process of applying for an EPSRC fellowship. In particular, there will be a discussion on the Future Leaders Fellowships, New Investigator Awards and Standard Grant applications. There will also be a discussion on applying for EPSRC funding more generally. Laura will answer any questions that people have. 

Quiver varieties are one of the main objects of study in Geometric Representation Theory. Defined by Nakajima in 1994, there has been a lot of research on them, but there is still a lot to be yet discovered, especially about their geometry. In this seminar, I will talk about their first use in Geometric Representation Theory as providing geometric representations of symmetric Kac-Moody Lie algebras.

Email @email to get a link to the Jitsi meeting room (it is included in the weekly announcements).

Having its roots in classical operator theoretic questions, the theory of extensions of C*-algebras is now a powerful tool with applications in geometry and topology and of course within the theory of C*-algebras itself. In this talk I will give a gentle introduction to the topic highlighting some classical results and more recent applications and questions.

Abstract: The signature of a path is a sequence of iterated coordinate integrals along the path. We aim at reconstructing a path from its signature. In the special case of lattice paths, one can obtain exact recovery based on a simple algebraic observation. For general continuously differentiable curves, we develop an explicit procedure that allows to reconstruct the path via piecewise linear approximations. The errors in the approximation can be quantified in terms of the level of signature used and modulus of continuity of the derivative of the path. The main idea is philosophically close to that for the lattice paths, and this procedure could be viewed as a significant generalisation. A key ingredient is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales.We will also discuss possible simplifications and improvements that may be potentially significant. Based on joint works with Terry Lyons, and also with Jiawei Chang, Nick Duffield and Hao Ni.

The property of amenability is a cornerstone in the study and classification of II_1 factor von Neumann algebras. Likewise, ultraproduct analysis is an essential tool in the subject. We will discuss the history, recent results, and open questions on characterizations of amenability for separable II_1 factors in terms of embeddings into ultraproducts.

Abstract: Sequential and temporal data arise in many fields of research, such as quantitative finance, medicine, or computer vision. We will be concerned with a novel approach for sequential learning, called the signature method, and rooted in rough path theory. Its basic principle is to represent multidimensional paths by a graded feature set of their iterated integrals, called the signature. On the one hand, this approach relies critically on an embedding principle, which consists in representing discretely sampled data as paths, i.e., functions from [0,1] to R^d. We investigate the influence of embeddings on prediction accuracy with an in-depth study of three recent and challenging datasets. We show that a specific embedding, called lead-lag, is systematically better, whatever the dataset or algorithm used. On the other hand, in order to combine signatures with machine learning algorithms, it is necessary to truncate these infinite series. Therefore, we define an estimator of the truncation order and prove its convergence in the expected signature model.

The classification of NIP fields is a major open problem in model theory.  This talk will be an overview of an ongoing attempt to classify NIP fields of finite dp-rank.  Let $K$ be an NIP field that is neither finite nor separably closed.  Conjecturally, $K$ admits exactly one definable, valuation-type field topology (V-topology).  By work of Anscombe, Halevi, Hasson, Jahnke, and others, this conjecture implies a full classification of NIP fields.  We will sketch how this technique was used to classify fields of dp-rank 1, and what goes wrong in higher ranks.  At present, there are two main results generalizing the rank 1 case.  First, if $K$ is an NIP field of positive characteristic (and any rank), then $K$ admits at most one definable V-topology.  Second, if $K$ is an unstable NIP field of finite dp-rank (and any characteristic), then $K$ admits at least one definable V-topology.  These statements combine to yield the classification of positive characteristic fields of finite dp-rank. In characteristic 0, things go awry in a surprising way, and it becomes necessary to study a new class of "finite rank" field topologies, generalizing V-topologies.  The talk will include background information on V-topologies, NIP fields, and dp-rank.

I will discuss the percolation model on planar triangulations, and present a bijection that is key to relating this model to some fundamental probabilistic objects. I will attempt to achieve several goals:
1. Present the site-percolation model on random planar triangulations.
2. Provide an informal introduction to several probabilistic objects: the Gaussian free field, Schramm-Loewner evolutions, and the Brownian map.
3. Present a bijective encoding of percolated triangulations by certain lattice paths, and explain its role in establishing exact relations between the above-mentioned objects.
This is joint work with Nina Holden, and Xin Sun.

Random graphs with finite excess appear naturally in at least two different settings: random graphs in the critical window (aka critical percolation on regular and other classes of graphs), and unicellular maps of fixed genus. In the first situation, the scaling limit of such random graphs was obtained by Addario-Berry, Broutin and Goldschmidt based on a depth-first exploration of the graph and on the coding of the resulting forest by random walks. This idea originated in Aldous' work on the critical random graph, using instead a breadth-first search approach that seem less adapted to taking graph scaling limits. We show hat this can be done nevertheless, resulting in some new identities for quantities like the radius and the two-point function of the scaling limit. We also obtain a similar "breadth-first" construction of the scaling limit of unicellular maps of fixed genus. This is based on joint work with Sanchayan Sen.

We consider random graph models where graphs are generated by connecting not only pairs of nodes by edges but also larger subsets of
nodes by copies of small atomic subgraphs of arbitrary topology. More specifically we consider canonical and microcanonical ensembles
corresponding to constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such
models. We also introduce a procedure that enables the distributions of multiple atomic subgraphs to be combined resulting in more coarse
grained models. As a result we obtain a general class of models that can be parametrized in terms of basic building blocks and their
distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs (Karrer & Newman PRE 2010, Bollobas et al. RSA 2011), random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy expressions for all these models can be derived from a single expression that is characterized by the symmetry groups of their atomic subgraphs.

For a bundle E over a manifold M, the associated "configuration-section spaces" are spaces of configurations of points in M together with a section of E over the complement of the configuration. One often considers subspaces where the behaviour of the section near a configuration point -- a kind of "monodromy" -- is restricted or prescribed. These are examples of "non-local configuration spaces", and may be interpreted physically as moduli spaces of "fields with prescribed singularities" in an ambient manifold.

An important class of examples is given by Hurwitz spaces, which are moduli spaces of branched G-coverings of the 2-disc, and which are homotopy equivalent to certain configuration-section spaces on the 2-disc. Ellenberg, Venkatesh and Westerland proved that, under certain conditions, Hurwitz spaces are (rationally) homologically stable; from this they then deduced an asymptotic version of the Cohen-Lenstra conjecture for function fields, a purely number-theoretical result.

We will discuss another homological stability result for configuration-section spaces, which holds (with integral coefficients) whenever the base manifold M is connected and open. We will also show that the "stabilisation maps" are split-injective (in all degrees) whenever dim(M) is at least 3 and M is either simply-connected or its handle dimension is at most dim(M) - 2.

This represents joint work with Ulrike Tillmann.

In this talk, we will discuss two-dimensional theories with discrete

one-form symmetries, examples (which we have been studying

since 2005), their properties, and gauging of the one-form symmetry.

Their most important property is that such theories decompose into a

disjoint union of theories, recently deemed `universes.'

This decomposition has the effect of restricting allowed nonperturbative

sectors, in a fashion one might deem a `multiverse interference effect,'

which has had applications in topics including Gromov-Witten theory and

gauged linear sigma model phases.  After reviewing one-form symmetries

and decomposition in general, we will discuss a particular 

example in detail to explicitly illustrate these properties and 

to demonstrate how

gauging the one-form symmetry projects onto summands in the

decomposition.  If time permits, we will briefly review

analogous phenomena in four-dimensional theories with three-form symmetries,

as recently studied by Tanizaki and Unsal.

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.

Recent years have seen a great deal of progress in understanding the behavior of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite complete understanding of their evolution starting from a random initial condition, with a universality picture for their critical behavior. Here we will consider their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a $p$-coin iff it can be infected in the next step by the bootstrap model. In particular infection can also heal, hence the non-monotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own : when $p$ shrinks to 0 they display some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics.

We prove that for Bernoulli bond percolation on $\mathbb{Z}^d$, $d\geq2$, the percolation density $\theta(p)$ (defined as the probability of the origin lying in an infinite cluster) is an analytic function of the parameter in the supercritical interval $(p_c,1]$. This answers a question of Kesten from 1981.

The proof involves a little bit of elementary complex analysis (Weierstrass M-test), a few well-known results from percolation theory (Aizenman-Barsky/Menshikov theorem), but above all combinatorial ideas. We used a new notion of contours, bounds on the number of partitions of an integer, and the inclusion-exclusion principle, to obtain a refinement of a classical argument of Peierls that settled the 2-dimensional case in 2018. More recently, we coupled these techniques with a renormalisation argument to handle all dimensions.

Joint work with Christoforos Panagiotis.

This talk will give an easy introduction to non-commutative L_p spaces associated with a tracial von Neumann algebra. Then I will focus on non-commutative L_p spaces associated to locally compact groups and talk about some interesting completely bounded multipliers on them.

This talk will introduce some of the basic notions and results in the theory of C_0-semigroups, including generation theorems, growth and spectral bounds. If time permits, I will also try to discuss one or two classical results in the asymptotic theory of C_0-semigroups.

Color each vertex of an infinite graph blue with probability $p$ and red with probability $1-p$, independently among vertices. For which values of $p$ is there an infinite connected component of blue vertices? The talk will focus on this classical percolation problem for the class of planar graphs. Recently, Itai Benjamini made several conjectures in this context, relating the percolation problem to the behavior of simple random walk on the graph. We will explain how partial answers to Benjamini's conjectures may be obtained using the theory of circle packings. Among the results is the fact that the critical percolation probability admits a universal lower bound for the class of recurrent plane triangulations. No previous knowledge on percolation or circle packings will be assumed.

I'll discuss our recent proof of a conjecture of Talagrand, a fractional version of the "expectation-threshold" conjecture of Kahn and Kalai. As a consequence of this result, we resolve various (heretofore) difficult problems in probabilistic combinatorics and statistical physics.

I will describe how certain recursive distributional equations can be solved by using tools from numerical analysis on the convergence of approximation schemes for PDEs. This project is joint work with Luc Devroye, Hannah Cairns, Celine Kerriou, and Rivka Maclaine Mitchell.

A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers $\mathbb{Z}$. The study of these objects was initiated in 1950 by Erdős, and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called 'minimum modulus problem' was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$.

In this talk I will describe a simple and general method of attacking covering problems that was inspired by Hough's proof. We expect that this technique, which we call the 'distortion method', will have further applications in combinatorics.

This talk is based on joint work with Paul Balister, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.

Many real networks feature the property of nestedness, i.e. the neighbours of nodes with a few connections are hierarchically nested within the neighbours of nodes with more connections. Despite the abstract simplicity of this notion, different mathematical definitions of nestedness have been proposed, sometimes giving contrasting results. Moreover, there is an ongoing debate on the statistical significance of nestedness, since even random networks where the number of connections (degree) of each node is fixed to its empirical value are typically as nested as real-world ones. In this talk we show unexpected effects due to the recent finding that random networks where the degrees are enforced as hard constraints (microcanonical ensembles) are thermodynamically different from random networks where the degrees are enforced as soft constraints (canonical ensembles). We show that if the real network is perfectly nested, then the two ensembles are trivially equivalent and the observed nestedness, independently of its definition, is indeed an unavoidable consequence of the empirical degrees. On the other hand, if the real network is not perfectly nested, then the two ensembles are not equivalent and alternative definitions of nestedness can be even positively correlated in the canonical ensemble and negatively correlated in the microcanonical one. This result disentangles distinct notions of nestedness captured by different metrics and highlights the importance of making a principled choice between hard and soft constraints in null models of ecological networks.

[1] Bruno, M., Saracco, F., Garlaschelli, D., Tessone, C. J., & Caldarelli, G. (2020). Nested mess: thermodynamics disentangles conflicting notions of nestedness in ecological networks. arXiv preprint arXiv:2001.11805.
 

For the problem of prediction with expert advice in the adversarial setting with geometric stopping, we compute the exact leading order expansion for the long time behavior of the value function using techniques from stochastic analysis and PDEs. Then, we use this expansion to prove that as conjectured in Gravin, Peres and Sivan the comb strategies are indeed asymptotically optimal for the adversary in the case of 4 experts.
 

We provide a detailed, probabilistic interpretation, based on stochastic calculus, for the variational characterization of conservative diffusion as entropic gradient flux. Jordan, Kinderlehrer, and Otto showed in 1998 that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features. These are valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by taking expectations. As a bonus, we obtain the HWI inequality of Otto and Villani relating relative entropy, Fisher information and Wasserstein distance; and from it the celebrated log-Sobolev, Talagrand and Poincare inequalities of functional analysis. (Joint work with W. Schachermayer and B. Tschiderer, from the University of Vienna.)

 

Speaker: Thomas Oliver

Title: Hyperbolic circles and non-trivial zeros

Abstract: L-functions can often be considered as generating series of arithmetic information. Their non-trivial zeros are the subject of many famous conjectures, which offer countless applications to number theory. Using simple geometric observations in the hyperbolic plane, we will study the relationship between the zeros of L-functions and their characterisation amongst more general Dirichlet series.
 

Speaker: Ebrahim Patel

Title: From trains to brains: Adventures in Tropical Mathematics.

Abstract: Tropical mathematics uses the max and plus operator to linearise discrete nonlinear systems; I will present its popular application to solve scheduling problems such as railway timetabling. Adding the min operator generalises the system to allow the modelling of processes on networks. Thus, I propose applications such as disease and rumour spreading as well as neuron firing behaviour.