Past events

This seminar is part of our Frontiers in Quantitative Finance. Attendance is free of charge but requires prior online registration.

Abstract
Empirical studies consistently find that the price impact of large trades approximately follows a nonlinear power law. Yet, tractable formulas for the portfolios that trade off predictive trading signals, risk, and trading costs in an optimal manner are only available for quadratic costs corresponding to linear price impact. In this paper, we show that the resulting linear strategies allow to achieve virtually optimal performance also for realistic nonlinear price impact, if the “effective” quadratic cost parameter is chosen appropriately. To wit, for a wide range of risk levels, this leads to performance losses below 2% compared to the numerical Viterbi algorithm of Kolm and Ritter (2014) run at very high accuracy. The effective quadratic cost depends on the portfolio risk, but can be computed without any sophisticated numerics by simply maximizing an explicit scalar function.
Read more on this work here.

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves contains a Zariski-dense set of special points if and only if it is special. I'll explain this result, and discuss an effective version that Gal Binyamini, Harry Schmidt, Margaret Thomas and I proved.

We use the term shape optimization when we want to find a minimizer of an objective function that assigns real values to shapes of domains. Solving shape optimization problems can be quite challenging, especially when the objective function is constrained to a PDE, in the sense that evaluating the objective function for a given domain shape requires first solving a boundary value problem stated on that domain. The main challenge here is that shape optimization methods must employ numerical methods capable of solving a boundary value problem on a domain that changes after each iteration of the optimization algorithm.

The first part of this talk will provide a gentle introduction to shape optimization. The second part of this talk will highlight how the finite element framework leads to automated numerical shape optimization methods, as realized in the open-source library fireshape. The talk will conclude with a brief overview of some academic and industrial applications of shape optimization.

In order for artificial neural networks to learn a task, one must solve an inverse design problem. What network will produce the desired output? We have harnessed AI approaches to design physical systems to perform functions inspired by biology, such as protein allostery. But artificial neural networks require a computer in order to learn in top-down fashion by the global process of gradient descent on a cost function. By contrast, the brain learns by local rules on its own, with each neuron adjusting itself and its synapses without knowing what all the other neurons are doing, and without the aid of an external computer. But the brain is not the only biological system that learns by local rules; I will argue that the actin cortex and the amnioserosa during the dorsal closure stage of Drosophila development can also be viewed this way.

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

It is a truth universally acknowledged, that an infinite group in possession of a good algebraic structure, must be in want of a hyperbolic space to act on. For the mapping class group of a surface, one of the most popular choices is the curve graph. This is a combinatorial object, built from curves on the surface and intersection patterns between them.
Hyperbolicity of the curve graph was proved by Masur and Minsky in a celebrated paper in 1999. In the same article, they showed how the geometry of the action on this graph reflects dynamical/topological properties of the mapping class group; in particular, loxodromic elements are precisely the pseudo-Anosov mapping classes.
In light of this, one would like to better understand distances in the curve graph. The graph is locally infinite, and finding a shortest path between two vertices is highly non-trivial. In this talk, we will see how to use the machinery of train tracks to overcome this issue and compute (approximate) distances in the curve graph. If time permits -- which, somehow, it never does -- we will also analyse this construction from an algorithmic perspective.

I will talk about a family of measures on partitions (specifically, a case of Okounkov's Schur measures) which are in one-to-one correspondence with models of random unitary matrices and lattice fermions. Under these measures, as the expected size of a partition goes to infinity, the first part of a random partition generically exhibits the same universal asymptotic fluctuations as the largest eigenvalue of a GUE random Hermitian matrix. First, I'll describe how we can tune these measures to exhibit new edge fluctuations at a smaller scale, which naturally generalise the GUE edge behaviour. These new fluctuations are universal, having previously been found for trapped fermions, and when a measure is tuned to have them, the corresponding unitary matrix model is "multicritical". Then, I'll describe how our measures can escape these more general universality classes, when tuned to have several cuts in a certain "Fermi sea". In this case, the breakdown in universality arises from an oscillation phenomenon previously observed in multi-cut Hermitian matrix models. Moreover, we have a one-to-one correspondence with multi-cut unitary matrix models. This is partly based on joint work with Dan Betea and Jérémie Bouttier. 

For almost 10 years, it has been known that if a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct a hyperbolic space where said element acts WPD. For a long time, the following question remained unanswered: if Morse is equivalent to strongly contracting, does there exist a space where all generalized loxodromics act WPD? In this talk, I will present a construction of a hyperbolic space, that answers this question positively.

Diffusion is one of the most common phenomenon in natural sciences and large part of applied mathematics have been interested in the tools to model it. Trying to study different types of diffusions, the mathematical ways to describe them and the numerical methods to simulate them is an appealing challenge, giving a wide range of applications. The aim of our work is the design of a finite-volume numerical scheme to model non-local diffusion given by the fractional Laplacian and to build numerical solutions for the Lévy-Fokker-Planck equation that involves it. Numerical methods for fractional diffusion have been indeed developed during the last few years and large part of the literature has been focused on finite element methods. Few results have been rather proposed for different techniques such as finite volumes.

 
We propose a new fractional Laplacian for bounded domains, which is expressed as a conservation law. This new approach is therefore particularly suitable for a finite volumes scheme and allows us also to prescribe no-flux boundary conditions explicitly. We enforce our new definition with a well-posedness theory for some cases to then capture with a good level of approximation the action of fractional Laplacian and its anomalous diffusion effect with our numerical scheme. The numerical solutions we get for the Lévy-Fokker-Planck equation resemble in fact the known analytical predictions and allow us to numerically explore properties of this equation and compute stationary states and long-time asymptotics.

We determine the minimal forbidden minors characterising the class of countable graphs that embed into some compact surface. We will also discuss Thomas’s conjecture that the Robertson—Seymour Graph Minor Theorem extends to countably infinite graphs. [https://arxiv.org/abs/2301.11042]

The theory of D-modules has found remarkable applications in various mathematical areas, for example, the representation theory of complex semi-simple Lie algebras. Two pivotal theorems in this field are the Beilinson-Bernstein Localisation Theorem and the Riemann-Hilbert Correspondence. This talk will explore a p-adic analogue. Ardakov-Wadsley introduced the sheaf D-cap of infinite order differential operators on a given smooth rigid-analytic variety to develop a p-adic counterpart for the Beilinson-Bernstein localisation. However, the classical approach to the Riemann-Hilbert Correspondence does not apply in the p-adic context. I will present an alternative approach, introducing a solution functor for D-cap-modules using new methods from p-adic Hodge theory.

Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in AdS3 pure gravity, which we formulate as a topological quantum field theory. We explain how gravitational entropy can be viewed as bulk entanglement entropy of gravitational edge modes.  These edge modes transform under a quantum group symmetry. This suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.  Time permitting we will discuss a proposal for how our bulk TQFT arises from an ensemble of approximate CFT’s, generalizing the relation between JT gravity and random matrix ensemble.

Following Rosati and Perkowski’s work on constructing the first version of a rough super Brownian motion, we generalize the rough super Brownian motion to the case when the branching mechanism has infinite variance. In both case, we can prove the compact support properties and the exponential persistence.

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain, coupled to a Fokker-Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent.

The micro-macro interaction is reflected by the presence of a drag term in the Fokker-Planck equation and the divergence of a polymeric extra-stress tensor in the Navier-Stokes balance of momentum equation. In a simplified case where the drag term is corotational, we prove global existence of weak solutions and discuss some of their properties: we use the relative energy method to deduce a weak-strong uniqueness type result, and derive the macroscopic closure of the kinetic model: a corotational Oldroyd-B model with stress-diffusion.

In the general noncorotational case, we consider “generalised dissipative solutions” — a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation, which accounts for the lack of compactness in the polymeric extra-stress tensor. Joint work with Endre Suli (Oxford).

I will discuss a basic problem in analytic number theory which has appeared recently in my work. This will be a gentle introduction to the Gauss circle problem, hopefully with a discussion of some extensions and applications to understanding L-functions.

One way of studying infinite groups is by analysing
 their actions on classes of interesting spaces. This is the case
 for Kazhdan's property (T) and for Haagerup's property (also called a-T-menability),
 formulated in terms of actions on Hilbert spaces and relevant in many areas
(e.g. for the Baum-Connes conjectures, in combinatorics, for the study of expander graphs, in ergodic theory, etc.)
 
Recently, these properties have been reformulated for actions on Banach spaces,
with very interesting results. This talk will overview some of these reformulations
 and their applications. Part of the talk is on joint work with Ashot Minasyan and Mikael de la Salle, and with John Mackay.
 

We consider a model of network of interacting  neurons based on jump processes. Briefly, the membrane potential $V^i_t$ of each individual neuron evolves according to a one-dimensional ODE. Neuron $i$ spikes at rate which only depends on its membrane potential, $f(V^i_t)$. After a spike, $V^i_t$ is reset to a fixed value $V^{\mathrm{rest}}$. Simultaneously, the membrane potentials of any (post-synaptic) neuron $j$ connected to the neuron $i$ receives a kick of value $J^{i,j}$.

We study the limit (mean-field) equation obtained where the number of neurons goes to infinity. In this talk, we describe the long time behaviour of the solution. Depending on the intensity of the interactions, we observe convergence of the distribution to a unique invariant measure (small interactions) or we characterize the occurrence of spontaneous oscillations for  interactions in the neighbourhood of critical values.

There are two main methods of constructing compact manifolds with holonomy $G_2$, viz. resolution of singularities (first applied by Joyce) and twisted connect sum (first applied by Kovalev).  In the second case, there is a known invariant (the $\overline{\nu}$-invariant, introduced by Crowley–Goette–Nordström) which can, in many cases, be used to distinguish between different examples.  This invariant, however, has limitations; in particular, it cannot be computed on the $G_2$-manifolds constructed by resolution of singularities.

In this talk, I shall begin by discussing the notion of a $G_2$-manifold and the $\overline{\nu}$-invariant and its limitations.  In the context of this, I shall then introduce two new invariants of $G_2$-manifolds, termed $\mu$-invariants, and explain why these promise to overcome these limitations, in particular being well-suited to, and computable on, Joyce's examples of $G_2$-manifolds.  These invariants are related to $\eta$- and $\zeta$-invariants and should be regarded as the Morse indices of a $G_2$-manifold when it is viewed as a critical point of certain Hitchin functionals.  Time permitting, I shall explain how to prove a closed formula for the invariants on the orbifolds used in Joyce's construction, using Epstein $\zeta$-functions.

We will be joined by people with mentoring experience to discuss the importance of both having and being a good mentor.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.