University of Oxford

Empirical networks often exhibit different meso-scale structures, such as community and core-periphery structure. Core-periphery typically consists of a well-connected core, and a periphery that is well-connected to the core but sparsely connected internally. Most core-periphery studies focus on undirected networks. In this talk we discuss  a generalisation of core-periphery to directed networks which  yields a family of core-periphery blockmodel formulations in which, contrary to many existing approaches, core and periphery sets are edge-direction dependent. Then we shall  focus on a particular structure consisting of two core sets and two periphery sets, and we introduce  two measures to assess the statistical significance and quality of this  structure in empirical data, where one often has no ground truth. The idea will be illustrated on three empirical networks --  faculty hiring, a world trade data-set, and political blogs.

This is based on joint work with Andrew Elliott, Angus Chiu, Marya Bazzi and Mihai Cucuringu, available at https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0783

The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting.

We introduce the notion of δ-excursion, defined as a path which deviates by δ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into δ-excursions, which is useful for scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss and drawdown. As δ is decreased to zero, properties of this decomposition relate to the local time of the path. When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent δ-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursion properties match those observed in empirical data.

Joint work with Anna Ananova and Rama Cont: https://ssrn.com/abstract=3723980

We analyze the regularity of solutions and discrete-time approximations of extended mean-field control (extended MFC) problems, which seek optimal control of McKean-Vlasov dynamics with coefficients involving mean-field interactions both on the  state and actions, and where objectives are optimized over
open-loop strategies.

We show for a large class of extended MFC problems that the unique optimal open-loop control is 1/2-Hölder continuous in time. Based on the regularity of the solution, we prove that the value functions of such extended MFC problems can be approximated by those with piecewise constant controls and discrete-time state processes arising from Euler-Maruyama time stepping up to an order 1/2 error, which is optimal in our setting. Further, we show that any epsilon-optimal control of these discrete-time problems
converge to the optimal control of the original problems.

To establish the time regularity of optimal controls and the convergence of time discretizations, we extend the canonical path regularity results to general coupled 
McKean-Vlasov forward-backward stochastic differential equations, which are of independent interest.

This is based on join work joint work with C. Reisinger and Y. Zhang.

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.

'Quasi-isometric rigidity' in group theory is the slogan for questions of the following nature: let A be some class of groups (e.g. finitely presented groups). Suppose an abstract group H is quasi-isometric to a group in A: does it imply that H is in A? Such statements link the coarse geometry of a group with its algebraic structure. 

Much is known in the case A is some class of lattices in a given Lie group. I will present classical results and outline ideas in their proofs, emphasizing the geometric nature of the proofs. I will focus on one key ingredient, the quasi-flat rigidity, and discuss some geometric objects that come into play, such as neutered spaces, asymptotic cones and buildings. I will end the talk with recent developments and possible generalizations of these results and ideas.

Primitive elements are elements that are part of a basis for a free group. We present the classical Whitehead algorithm for the recognition of such elements, and discuss the ideas behind the proof. We also present a second algorithm, more recent and completely different in the approach.

Operads are tools to encode operations satisfying algebro-homotopic relations. They have proved to be extremely useful tools, for instance for detecting spaces that are iterated loop spaces. However, in many natural examples, composition of operations is only associative up to homotopy and operads are too strict to captured these phenomena. This leads to the notion of infinity operads. While they are a well-established tool, there are few examples of infinity operads in the literature that are not the nerve of an actual operad. I will introduce new topological operad of bracketed trees that can be used to identify and construct natural examples of infinity operads. The key example for this talk will be the normalised cacti model for genus 0 surfaces.

Glueing surfaces along their boundaries defines composition laws that have been used to construct topological field theories and to compute the homology of the moduli space of Riemann surfaces. Normalised cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By using the operad of bracketed trees, I will show that this operation is associative up to all higher homotopies and hence that normalised cacti form an infinity operad.

We study a group theoretic analog of Dehn fillings of 3-manifolds and derive a spectral sequence to compute the cohomology of Dehn fillings of hyperbolically embedded subgroups. As applications, we generalize the results of Dahmani-Guirardel-Osin and Hull on SQ-universality and common quotients of acylindrically hyperbolic groups by adding cohomological finiteness conditions. This is a joint work with Nansen Petrosyan.

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.