Past

We consider continuous time financial models with continuous paths, in a pathwise setting using functional Ito calculus. We look at applications of optimal transport duality in context of robust pricing and hedging and that of calibration. First, we explore exntesions of the discrete-time results in Aksamit et al. [Math. Fin. 29(3), 2019] to a continuous time setting. Second, we addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton--Jacobi--Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options and VIX futures simultaneously.

Based on joint works with Ivan Guo, Gregoire Loeper, Shiyi Wang.
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Tissue folding during animal development involves an intricate interplay
of cell shape changes, cell division, cell migration, cell
intercalation, and cell differentiation that obfuscates the underlying
mechanical principles. However, a simpler instance of tissue folding
arises in the green alga Volvox: its spherical embryos turn themselves
inside out at the close of their development. This inversion arises from
cell shape changes only.

In this talk, I will present a model of tissue folding in which these
cell shape changes appear as variations of the intrinsic stretches and
curvatures of an elastic shell. I will show how this model reproduces
Volvox inversion quantitatively, explains mechanically the arrest of
inversion observed in mutants, and reveals the spatio-temporal
regulation of different biological driving processes. I will close with
two examples illustrating the challenges of nonlinearity in tissue
folding: (i) constitutive nonlinearity leading to nonlocal elasticity in
the continuum limit of discrete cell sheet models; (ii) geometric
nonlinearity in large bending deformations of morphoelastic shells.
 

One of the standard methods for the solution of elliptic boundary value problems calls for reformulating them as systems of integral equations.  The integral operators that arise in this fashion typically have singular kernels, and, in many cases of interest, the solutions of these equations are themselves singular.  This makes the accurate discretization of the systems of integral equations arising from elliptic boundary value problems challenging.

Over the last decade, Generalized Gaussian quadrature rules, which are n-point quadrature rules that are exact for a collection of 2n functions, have emerged as one of the most effective tools for discretizing singular integral equations. Among other things, they have been used to accelerate the discretization of singular integral operators on curves, to enable the accurate discretization of singular integral operators on complex surfaces and to greatly reduce the cost of representing the (singular) solutions of integral equations given on planar domains with corners.

We will first briefly outline a standard method for the discretization of integral operators given on curves which is highly amenable to acceleration through generalized Gaussian quadratures. We will then describe a numerical procedure for the construction of Generalized Gaussian quadrature rules.

Much of this is joint work with Zydrunas Gimbutas (NIST Boulder) and Vladimir Rokhlin (Yale University).

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

In the study of geometric flows it is often important to understand when a flow which converges along a sequence of times going to infinity will, in fact, converge along every such sequence of times to the same limit. While examples of finite dimensional gradient flows that asymptote to a circle of critical points show that this cannot hold in general, a positive result can be obtained in the presence of a so-called Lojasiewicz-Simon inequality. In this talk we will introduce this problem of uniqueness of asymptotic limits and discuss joint work with Melanie Rupflin and Peter M. Topping in which we examined the situation for a geometric flow that is designed to evolve a map describing a closed surface in a given target manifold into a parametrization of a minimal surface.

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The Landau equation is an important PDE in kinetic theory modelling plasma particles in a gas. It can be derived as a limiting process from the famous Boltzmann equation. From the mathematical point of view, the Landau equation can be very challenging to study; many partial results require, for example, stochastic analysis as well as a delicate combination of kinetic and parabolic theory. The major open question is uniqueness in the physically relevant Coulomb case. I will present joint work with Jose Carrillo, Matias Delgadino, and Laurent Desvillettes where we cast the Landau equation as a generalized gradient flow from the optimal transportation perspective motivated by analogous results on the Boltzmann equation. A direct outcome of this is a numerical scheme for the Landau equation in the spirit of de Giorgi and Jordan, Kinderlehrer, and Otto. An extended area of investigation is to use the powerful gradient flow techniques to resolve some of the open problems and recover known results.

I will explain what it means for a manifold to have an affine structure and give an introduction to Benzecri's theorem stating that a closed surface admits an affine structure if and only if its Euler characteristic vanishes. I will also talk about an algebraic-topological generalization, due to Milnor and Wood, that bounds the Euler class of a flat circle bundle. No prior familiarity with the concepts is necessary.

Classical random matrix theory begins with a random matrix model and analyzes the distribution of the resulting eigenvalues.  In this work, we treat the reverse question: if the eigenvalues are specified but the matrix is "otherwise random", what do the entries typically look like?  I will describe a natural model of random matrices with prescribed eigenvalues and discuss a central limit theorem for projections, which in particular shows that relatively large subcollections of entries are jointly Gaussian, no matter what the eigenvalue distribution looks like.  I will discuss various applications and interpretations of this result, in particular to a probabilistic version of the Schur--Horn theorem and to models of quantum systems in random states.  This work is joint with Mark Meckes.

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property $H$, the speed of $H$ is the function which sends an integer n to the number of distinct elements in $H$ with underlying set $\{1,...,n\}$. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

In the classical setting of real semisimple Lie groups, the Dirac inequality (due to Parthasarathy) gives a necessary condition that the infinitesimal character of an irreducible unitary representation needs to satisfy in terms of the restriction of the representation to the maximal compact subgroup. A similar tool was introduced in the setting of representations of p-adic groups in joint work with Barbasch and Trapa, where the necessary unitarity condition is phrased in terms of the semisimple parameter in the Kazhdan-Lusztig parameterization and the hyperspecial parahoric restriction. I will present several consequences of this inequality to the problem of understanding the unitary dual of the p-adic group, in particular, how it can be used in order to exhibit several isolated "extremal" unitary representations and to compute precise "spectral gaps" for them.

Let $G_n$ be a sequence of finite, simple, connected, regular graphs with degrees tending to infinity and let $T_n$ be a uniformly drawn spanning tree of $G_n$. In joint work with Yuval Peres we show that the local limit of $T_n$ is the $\text{Poisson}(1)$ branching process conditioned to survive forever (that is, the asymptotic frequency of the appearance of any small subtree is given by the branching process). The proof is based on electric network theory and I hope to show most of it.

We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. As a particular application, we define the maximum-variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum-variance distribution can be interpreted as a core-periphery measure, illustrated by the fact that these distributions are supported on the leaf nodes of tree graphs, low-degree nodes in a configuration-like graph and boundary nodes in random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts. Finally, I will draw connections to related notion of assortativity on networks, a network analogue of correlation used to describe how the presence and absence of edges covaries with the properties of nodes.

https://arxiv.org/abs/2008.09155

We revisit the variational characterization of conservative diffusion as entropic gradient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin–Smoluchowski type, the Fokker–Planck probability density flow maximizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in the backward direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker–Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. As a bonus of our trajectorial approach we derive the HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).

In the talk, we will discuss the connection between quantitative hypoelliptic PDE methods and the long-time dynamics of stochastic differential equations (SDEs). In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a degenerate Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a  quantitative version of Hörmander's hypoelliptic regularity theory in an L1 framework which estimates this (degenerate) Fisher information from below by a W^{s,1} Sobolev norm using the associated Kolmogorov equation for the stationary density. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE and we prove that this class includes the classical Lorenz 96 model in any dimension greater than 6, provided the additive stochastic driving is applied to any consecutive pair of modes. This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for Lorenz 96 and, more recently, for finite dimensional truncations of the shell models GOY and SABRA (stochastically driven or otherwise), despite the overwhelming numerical evidence. If time permits, I will also discuss joint work with Kyle Liss, in which we obtain sharp, quantitative estimates on the spectral gap of the Markov semigroups. In both of these works, obtaining various kinds of quantitative hypoelliptic regularity estimates that are uniform in certain parameters plays a pivotal role.  

The Dirichlet class number formula gives an expression for the residue at s=1 of the Dedekind zeta function of a number field K in terms of certain quantities associated to K. Among those is the regulator of K, a certain determinant involving logarithms of units in K. In the 1980s, Don Zagier gave a conjectural expression for the values at integers s $\geq$ 2 in terms of "higher regulators", with polylogarithms in place of logarithms. The goal of this talk is to give an algebraic-geometric interpretation of these polylogarithms. Time permitting, we will also discuss a similar picture for Hasse--Weil L-functions of elliptic curves.
 

I will discuss connections between ambient geometry of Moduli spaces and Teichmuller dynamics. This includes the recent resolution of the Siu's conjecture about convexity of Teichmuller spaces, and the (conjectural) topological description of the Caratheodory metric on Moduli spaces of Riemann surfaces.

This is a report on joint work with Martijn Kool. 

Recently, Marian-Oprea-Pandharipande established a generalization of Lehn’s conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of higher rank. 

Using Mochizuki’s formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg- Witten invariants and intersection numbers on products of Hilbert schemes of points. We use this to  verify our conjectures in examples. 

I will discuss an analogue of the CHY formalism in AdS. Considering the biadjoint scalar theory on AdS, I will explain how to rewrite all the tree-level amplitudes as an integral over the moduli space of punctured Riemann spheres. Contrary to the flat space, the scattering equations are operator-valued. The resulting formula is motivated via a bosonic ambitwistor string on AdS and can be proven to be equivalent to the corresponding Witten diagram computation by applying a series of contour deformations.

Persistence theory provides useful tools to extract information from real-world data sets, and profits of techniques from different mathematical disciplines, such as Morse theory and quiver representation. In this seminar, I am going to present a new approach for studying persistence theory using model categories. I will briefly introduce model categories and then describe how to define a model structure on the category of the tame parametrised chain complexes, which are chain complexes that evolve in time. Using this model structure, we can define new invariants for tame parametrised chain complexes, which are in perfect accordance with the standard barcode when restricting to persistence modules. I will illustrate with some examples why such an approach can be useful in topological data analysis and what new insight on standard persistence can give us. 

Cuntz introduced pure infiniteness for simple C*-algebras as a C*-algebraic analogue of type III von Neumann factors. Notable examples include the Calkin algebra B(H)/K(H), the Cuntz algebras O_n, simple Cuntz-Krieger algebras, and other C*-algebras you would encounter in the wild. The separable, nuclear ones were classified in celebrated work by Kirchberg and Phillips in the mid 90s. I will talk about these topics including the non-simple case if time permits.

Unitary solutions of the Yang-Baxter equation ("R-matrices") play a prominent role in several fields, such as quantum field theory and topological quantum computing, but are difficult to find directly and remain somewhat mysterious. In this talk I want to explain how one can use subfactor techniques to learn something about unitary R-matrices, and a research programme aiming at the classification of unitary R-matrices up to a natural equivalence relation. This talk is based on joint work with Roberto Conti, Ulrich Pennig, and Simon Wood.

Liouville quantum gravity (LQG) is a theory of random fractal surfaces with origin in the physics literature in the 1980s. Most literature is about LQG with matter central charge $c\in (-\infty,1]$. We study a discretization of LQG which makes sense for all $c\in (-\infty,25)$. Based on a joint work with Gwynne, Pfeffer, and Remy.

Systems with lattice geometry can be renormalized exploiting their embedding in metric space, which naturally defines the coarse-grained nodes. By contrast, complex networks defy the usual techniques because of their small-world character and lack of explicit metric embedding. Current network renormalization approaches require strong assumptions (e.g. community structure, hyperbolicity, scale-free topology), thus remaining incompatible with generic graphs and ordinary lattices. Here we introduce a graph renormalization scheme valid for any hierarchy of coarse-grainings, thereby allowing for the definition of block-nodes across multiple scales. This approach reveals a necessary and specific dependence of network topology on an additive hidden variable attached to nodes, plus optional dyadic factors. Renormalizable networks turn out to be consistent with a unique specification of the fitness model, while they are incompatible with preferential attachment, the configuration model or the stochastic blockmodel. These results highlight a deep conceptual distinction between scale-free and scale-invariant networks, and provide a geometry-free route to renormalization. If the hidden variables are annealed, the model spontaneously leads to realistic scale-free networks with cut-off. If they are quenched, the model can be used to renormalize real-world networks with node attributes and distance-dependence or communities. As an example we derive an accurate multiscale model of the International Trade Network applicable across arbitrary geographic resolutions.

https://arxiv.org/abs/2009.11024 (23 sept.)

There is a growing body of results in extremal combinatorics and Ramsey theory which give better bounds or stronger conclusions under the additional assumption of bounded VC-dimension. Schur and Erdős conjectured that there exists a suitable constant $c$ with the property that every graph with at least $2^{cm}$ vertices, whose edges are colored by $m$ colors, contains a monochromatic triangle. We prove this conjecture for edge-colored graphs such that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension. This result is best possible up to the value of $c$.
    Joint work with Jacob Fox and Andrew Suk.

This talk is a personal how-to (and how-not-to) manual for doing Maths with industry, or indeed with government. The Maths element is essential but lots of other skills and activities are equally necessary. Examples: problem elicitation; understanding the environmental constraints; power analysis; understanding world-views and aligning personal motivations; and finally, understanding the wider systems in which the Maths element will sit. These issues have been discussed for some time in the management science community, where their generic umbrella name is Problem Structuring Methods (PSMs).