Past

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