Past

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

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

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

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

https://arxiv.org/abs/2003.02860

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

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

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

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

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

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

Many classical fractal functions, such as the Weierstrass and Takagi-van der Waerden functions, admit a finite p-th variation along a natural sequence of partitions. They can thus serve as integrators in pathwise Itô calculus. Motivated by this observation, we
introduce a new class of stochastic processes, which we call Weierstrass bridges. They have continuous sample paths and arbitrarily low regularity and so provide a new example class of “rough” stochastic processes. We study some of their sample path properties
including p-th variation and moduli of continuity. This talk includes joint work with Xiyue Han and Zhenyuan Zhang.

For a given vector field $h$ on a manifold $M$ and an initial point $x \in M$, let $t \mapsto \exp th(x)$ denote the solution to the Cauchy problem $y' = h(y)$, $y(0) = x$. Given two vector fields $f$, $g$, the flows $\exp(tf)$, $\exp(tg)$ in general are not commutative. That is, it may happen that, for some initial point $x$,

$$\exp(-tg) \circ \exp(-tf) \circ \exp(tg) \circ \exp(tf) (x) ≠ x,$$

for small times $t ≠ 0$.

         As is well-known, the Lie bracket $[f,g] := Dg \cdot f - Df \cdot g$ measures the local non-commutativity of the flows. Indeed, one has (on any coordinate chart)

$$\exp(-tg) \circ \exp(-tf) \circ \exp(tg) \circ \exp(tf) (x) - x = t^2 [f,g](x) + o(t^2)$$

         The non-commutativity of vector fields lies at the basis of many nonlinear issues, like propagation of maxima for solutions of degenerate elliptic PDEs, controllability sufficient conditions in Nonlinear Control Theory, and higher order necessary conditions for optimal controls. The fundamental results concerning commutativity (e.g. Rashevski-Chow's Theorem, also known as Hörmander's full rank condition, or Frobenius Theorem) assume that the vector fields are smooth enough for the involved iterated Lie brackets to be well defined and continuous: for instance, if the bracket $[f,[g,h]]$ is to be used, one posits $g,h \in C^2$ and $f \in C^{1..}$.

         We propose a notion of (set-valued) Lie bracket (see [1]-[3]), through which we are able to extend some of the mentioned fundamental results to families of vector fields whose iterated brackets are just measurable and defined almost everywhere.

References.

[1]  Rampazzo, F. and Sussmann, H., Set-valued differentials and a nonsmooth version of Chow’s Theorem (2001), Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, 2001 (IEEE Publications, New York), pp. 2613-2618.

[2] Rampazzo F.  and Sussmann, H.J., Commutators of flow maps of nonsmooth vector fields (2007), Journal of Differential Equations, 232, pp. 134-175.

[3] Feleqi, E. and Rampazzo, F., Iterated Lie brackets for nonsmooth vector fields (2017), Nonlinear Differential Equations and Applications NoDEA, 24-6.

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

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

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

This is joint work with Jacob Gross and Yuuji Tanaka.

Vafa-Witten theory is a topologically twisted version of 4d N=4 super Yang-Mills theory. In my talk I will tell how to derive a holomorphic anomaly equation for its partition function on a Kaehler 4-manifold with b_2^+=1 and b_1=0 from the path integral of the effective theory on the Coulomb branch. I will also briefly mention an alternative and somewhat similar computation of the same holomorphic anomaly in the effective 2d theory obtained by compactification of the corresponding 6d (2,0) theory on the 4-manifold.
 

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

In this talk I will briefly sketch the philosophy and methods in which derived enhancements of classical moduli problems are produced. I will then discuss the character variety and distinguish two of its enhancements; one of these will represent a derived moduli stack for local systems. Lastly, I will mention how variations of this moduli space have been represented in number theoretic and rigid analytic contexts.

I will present a conjectural picture of what a classification theory for NIP could look like, in the spirit of Shelah's classification theory for stable structures. Though most of it is speculative, there are some encouraging initial results about the lower levels of the classification, in particular concerning structures which, in some strong sense, do not contain trees.

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.