Past

I will report on recent joint work with Karen Vogtmann on the Euler characteristic of $Out(F_n)$ and the moduli space of graphs. A similar study has been performed in the seminal 1986 work of Harer and Zagier on the Euler characteristic of the mapping class group and the moduli space of curves. I will review a topological field theory proof, due to Kontsevich, of Harer and Zagier´s result and illustrate how an analogous `renormalized` topological field theory argument can be applied to $Out(F_n)$.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.
 

I will introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions and, at the same time, accommodate “real-time” incoming market information. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, I will first propose two distinct definitions, one based on the preservation of performance value and the other on the time-consistency of policies and, in turn, establish their equivalence. I will then fully characterize the viable class of probability distortion processes, providing a bifurcation-type result. This will also characterize the candidate optimal wealth process, whose structure motivates the introduction of a new, distorted measure and a related dynamic market. I will, then, build a striking correspondence between the forward rank-dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection provides a direct construction method for forward rank-dependent criteria with dynamic incoming information. Furthermore, a direct by-product of our work are new results on the so-called dynamic utilities and time-inconsistent problems in the classical (backward) setting. Indeed, it turns out that open questions in the latter setting can be directly addressed by framing the classical problem as a forward one under suitable information rescaling.

In the talk I will discuss the  following result and related analytic and geometric questions:   On the boundary of any convex body in the Euclidean space there exists at least one infinite geodesic.

We show the existence of an infinite family of complex saddle-points at large N, for the matrix model of the superconformal index of SU(N) N = 4 super Yang-Mills theory on S3 × S1 with one chemical potential τ. The saddle-point configurations are labelled by points (m,n) on the lattice Λτ = Z τ + Z with gcd(m, n) = 1. The eigenvalues at a given saddle are uniformly distributed along a string winding (m, n) times along the (A, B) cycles of the torus C/Λτ . The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and its values at (m,n) saddles are determined by Fourier averages of the latter along directions of the torus. The actions of (0,1) and (1,0) agree with that of pure AdS5 and the Gutowski-Reall AdS5 black hole, respectively. The actions of the other saddles take a surprisingly simple form. Generically, they carry non vanishing entropy. The Gutowski-Reall black hole saddle dominates the canonical ensemble when τ is close to the origin, and other saddles dominate when τ approaches rational points. 

In this talk, linear algebra for persistence modules will be introduced, together with a generalization of persistent homology. This theory permits us to handle the Mayer-Vietoris spectral sequence for persistence modules, and solve any extension problems that might arise. The result of this approach is a distributive algorithm for computing persistent homology. That is, one can break down the underlying data into different covering subsets, compute the persistent homology for each cover, and join everything together. This approach has the added advantage that one can recover extra geometrical information related to the barcodes. This addresses the common complaint that persistent homology barcodes are 'too blind' to the geometry of the data.

Despite the stereotype of the lone genius working by themselves, most professional mathematicians collaborate with others.  But when you're learning maths as a student, is it OK to work with other people, or is that cheating?  And if you're not used to collaborating with others, then you might feel shy about discussing your ideas when you're not confident about them.  In this session, we'll explore ways in which you can get the most out of collaborations with your fellow students, whilst avoiding inadvertently passing off other people's work as your own.  This session will be suitable for undergraduate and MSc students at any stage of their degree who would like to increase their confidence in collaboration.  Please bring a pen or pencil!

The space of complete collineations is an important and beautiful chapter of algebraic geometry, which has its origins in the classical works of Chasles, Schubert and many others, dating back to the 19th century. It provides a 'wonderful compactification' (i.e. smooth with normal crossings boundary) of the space of full-rank maps between two (fixed) vector spaces. More recently, the space of complete collineations has been studied intensively and has been used to derive groundbreaking results in diverse areas of mathematics. One such striking example is L. Lafforgue's compactification of the stack of Drinfeld's shtukas, which he subsequently used to prove the Langlands correspondence for the general linear group. 

In joint work with M. Kapranov, we look at these classical spaces from a modern perspective: a complete collineation is simply a spectral sequence of two-term complexes of vector spaces. We develop a theory involving more full-fledged (simply graded) spectral sequences with arbitrarily many terms. We prove that the set of such spectral sequences has the structure of a smooth projective variety, the 'variety of complete complexes', which provides a desingularization, with normal crossings boundary, of the 'Buchsbaum-Eisenbud variety of complexes', i.e. a 'wonderful compactification' of the union of its maximal strata.
 

In this talk we consider a mean field optimal control problem with an aggregation-diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton-Jacobi and a Fokker-Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. We will discuss the existence and regularity of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker-Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.

Let $E/k$ be an elliptic curve over a number field and $K/k$ a Galois extension with group $G$. What can we say about $E(K)$ as a Galois module? Not just what complex representations appear, but its structure as a $\mathbb{Z}[G]$-module. We will look at some examples with small $G$.

One of the key unsolved challenges at the interface of physical and life sciences is to formulate comprehensive computational modeling of cells of higher organisms that is based on microscopic molecular principles of chemistry and physics. Towards addressing this problem, we have developed a unique reactive mechanochemical force-field and software, called MEDYAN (Mechanochemical Dynamics of Active Networks: http://medyan.org).  MEDYAN integrates dynamics of multiple mutually interacting phases: 1) a spatially resolved solution phase is treated using a reaction-diffusion master equation; 2) a polymeric gel phase is both chemically reactive and also undergoes complex mechanical deformations; 3) flexible membrane boundaries interact mechanically and chemically with both solution and gel phases.  In this talk, I will first outline our recent progress in simulating multi-micron scale cytosolic/cytoskeletal dynamics at 1000 seconds timescale, and also highlight the outstanding challenges in bringing about the capability for routine molecular modeling of eukaryotic cells. I will also report on MEDYAN’s applications, in particular, on developing a theory of contractility of actomyosin networks and also characterizing dissipation in cytoskeletal dynamics. With regard to the latter, we devised a new algorithm for quantifying dissipation in cytoskeletal dynamics, finding that simulation trajectories of entropy production provide deep insights into structural evolution and self-organization of actin networks, uncovering earthquake-like processes of gradual stress accumulation followed by sudden rupture and subsequent network remodeling.
 

A fine compactified Jacobian is a proper open substack of the moduli space of simple sheaves. We will see that fine compactified Jacobians correspond to a certain combinatorial datum, essentially obtained by taking multidegrees of all elements of the compactified Jacobian. This picture generalizes to flat families of curves. We will discuss a classification result in the case when the family is the universal family over the moduli space of curves. This is a joint work with Jesse Kass.

We'll discuss the large charge expansion in CFTs with supersymmetry, focussing on 1908.10306 by Grassi, Komargodski and Tizzano.

The development of effective solvers for high frequency wave propagation problems, such as those described by the Helmholtz equation, presents significant challenges. One promising class of solvers for such problems are parallel domain decomposition methods, however, an appropriate coarse space is typically required in order to obtain robust behaviour (scalable with respect to the number of domains, weakly dependant on the wave number but also on the heterogeneity of the physical parameters). In this talk we introduce a coarse space based on generalised eigenproblems in the overlap (GenEO) for the Helmholtz equation. Numerical results within FreeFEM demonstrate convergence that is effectively independent of the wave number and contrast in the heterogeneous coefficient as well as good performance for minimal overlap.

Let A be a Tannakian category. Any exact tensor functor defined on A is either zero, or faithful. In this talk, I want to draw attention to a derived analogue of this statement (in characteristic zero) due to Jack Hall and David Rydh, and discuss some remarkable consequences for certain classification problems in algebraic geometry.

The Anderson Hamiltonian is used to model particles moving in
disordered media, it can be thought of as a Schrödiger operator with an
extremely irregular random potential. Using the recently developed theory of
"Paracontrolled Distributions" we are able to define the Anderson
Hamiltonian as a self-adjoint non-positive operator on the 2- and
3-dimensional torus and give an explicit description of its domain.
Then we use these results to solve some semi-linear PDEs whose linear part
is given by the Anderson Hamiltonian, more precisely the multiplicative
stochastic NLS and nonlinear Wave equation.
This is joint work with M. Gubinelli and B. Ugurcan.

In this talk I will explain a category theoretic perspective on geometry.  Starting with a category of local objects (of and algebraic nature), and a (Grothendieck) 
topology on it, one can define global objects such as schemes and stacks. Examples of this  approach are algebraic, analytic, differential geometries and also more exotic geometries  such as analytic and differential geometry over the integers and analytic geometry over  the field with one element. In this approach the notion of a point is not primary but is  derived from the local to global structure. The Zariski and Huber spectra are recovered  in this way, and we also get new spectra which might be of interest in model theory.

We will answer the following question: given a finite simplicial complex X acted on by a finite group G, which object stores the minimal amount of information about the symmetries of X in such a way that we can reconstruct both X and the group action? The natural first guess would be the quotient X/G, which remembers one representative from each orbit. However, it does not tell us the size of each orbit or how to glue together simplices to recover X. Our desired object is, in fact, a complex of groups. We will understand two processes: compression and reconstruction and see primarily through an example how to answer our initial question.

In 2010 Coecke, Sadrzadeh, and Clark formulated a new model of natural language which operates by combining the syntactics of grammar and the semantics of individual words to produce a unified ''meaning'' of sentences. This they did by using category theory to understand the component parts of language and to amalgamate the components together to form what they called a ''distributional compositional categorical model of meaning''. In this talk I shall introduce the model of Coecke et. al., and use it to compare the meaning of sentences in Irish and in English (and thus ascertain when a sentence is the translation of another sentence) using a cosine similarity score.

The Irish language is a member of the Gaelic family of languages, originating in Ireland and is the official language of the Republic of Ireland.

I will give an overview of the localization technique: a powerful dimension-reduction tool for proving geometric and functional inequalities.  Having its roots in a  pioneering work of Payne-Weinberger in the 60ies about sharp Poincare’-Wirtinger inequality on Convex Bodies in Rn, recently such a technique found new applications for a range of sharp geometric and functional inequalities in spaces with Ricci curvature bounded below.

This talk will be a survey on the applications of random matrix theory in neuroscience. We will explain why and how we use random matrices to model networks of neurons in the brain. We are mainly interested in the study of neuronal dynamics, and we will present results that cover two parallel directions taken by the field of theoretical neuroscience. First, we will talk about the critical point of transitioning to chaos in cases of random matrices that aim to be more "biologically plausible". And secondly, we will see how a deterministic and a random matrix (corresponding to learned structure and noise in a neuronal network) can interact in a dynamical system.