Past

In this interactive session, Jenny Roberts from the Institute of Mathematics and its Applications will offer guidance on what employers are looking for in mathematical graduates, and how best to sell yourself for those jobs!

There is a well-known relationship between finite W-algebras and Yangians. The work of Rogoucy and Sorba on the "rectangular case" in type A eventually led Brundan and Kleshchev to introduce shifted Yangians, which surject onto the finite W-algebras for general linear Lie algebras. Thus, these W-algebras can be realised as truncated shifted Yangians. In parallel, the work of Ragoucy and then Brown showed that truncated twisted Yangians are isomorphic to the finite W-algebra associated to a rectangular nilpotent element in a Lie algebra of type B, C or D. For many years there has been a hope that this relationship can be extended to other nilpotent elements.

I will report on a joint work with Lewis Topley in which we introduced the shifted twisted Yangians, following the work of Lu-Wang-Zhang, and described Poisson isomorphisms between their truncated semiclassical degenerations and the functions Slodowy slices associated with even nilpotent elements in classical simple Lie algebras( which can be viewed as semiclassical W-algebras). I will also mention a work in progress with Lu-Peng-Topley-Wang which deals with the quantum analogue of our theorem.

I will also recall what Poisson algebras and (filtered) quantizations are and give a brief intro to Slodowy slices, finite W-algebras and Yangians so that the talk should be quite accessible.

Tumour microenvironment is characterised by heterogeneity at various scales: from various cell populations (immune cells, cancerous cells, ...) and various molecules that populate the microenvironment (cytokines, chemokines, extracellular vesicles, …); to phenotype heterogeneity inside the same cell population (e.g., immune cells with different phenotypes and different functions); as well as temporal heterogeneity in cells’ phenotypes (as cancer evolves through time) and spatial heterogeneity.
In this talk we overview some mathematical models and computational approaches developed to investigate different single-scale and multi-scale aspects related to heterogeneous immune responses during cancer evolution. Throughout the talk we emphasise the qualitative vs. quantitative results, and data availability across different scales

The open core of a structure is the reduct generated by the open definable sets. Tame topological structures (e.g. o-minimal) are often inter-definable with their open core. Structures such as M = (ℝ,<, +, ℚ) are wild in the sense that they define a dense co-dense set. Still, M is NIP and its open core is o-minimal. In this talk, we push forward the thesis that the open core of an NTP2 (a generalization of NIP) topological structure is tame. Our main result is that, under suitable conditions, the open core has quantifier elimination (every definable set is constructible), and its definable functions are generically continuous.

We introduce a novel approach to global optimization  via continuous-time dynamic programming and Hamilton-Jacobi-Bellman (HJB) PDEs. For non-convex, non-smooth objective functions,  we reformulate global optimization as an infinite horizon, optimal asymptotic stabilization control problem. The solution to the associated HJB PDE provides a value function which corresponds to a (quasi)convexification of the original objective.  Using the gradient of the value function, we obtain a  feedback law driving any initial guess towards the global optimizer without requiring derivatives of the original objective. We then demonstrate that this HJB control law can be integrated into other global optimization frameworks to improve its performance and robustness. 

In this talk, I will discuss a conjecture of Penrose, which asserts a lower bound on the mass of a spacetime in terms of the area of a suitable horizon. Whilst Penrose presented a physical motivation for this inequality in the 1970s, the only proofs heavily rely upon PDE arguments, and in particular the use of geometric flows. I hope to show in this talk, through this concrete example (and without unpleasant technical details!), how ideas from geometric PDE theory can be helpful in obtaining results in physics.
 

I will talk about the connections between the Siu inequality and existence of the model companion for GVFs. The talk will be partially based on a joint work with Antoine Sedillot.

The hyperbolic plane and its higher-dimensional analogues are well-known
objects. They belong to a larger class of spaces, called rank-one
symmetric spaces, which include not only the hyperbolic spaces but also
their complex and quaternionic counterparts, and the octonionic
hyperbolic plane. By a result of Pansu, two of these families exhibit
strong rigidity properties with respect to their self-quasiisometries:
any self-quasiisometry of a quaternionic hyperbolic space or the
octonionic hyperbolic plane is at uniformly bounded distance from an
isometry. The goal of this talk is to give an overview of the rank-one
symmetric spaces and the tools used to prove Pansu's rigidity theorem,
such as the subRiemannian structure of their visual boundaries and the
analysis of quasiconformal maps.

This is  a notion we defined with Johan de Jong. If a finitely presented group  is the topological fundamental group of a smooth quasi-projective complex variety, then we prove that it is weakly integral. To this aim we use the Langlands program (both arithmetic to produce companions and geometric to use de Jong’s conjecture). On the other hand there are finitely presented groups which are not weakly integral (Breuillard). So this notion is an obstruction.
 

Self-similar groups are groups of automorphisms of infinite rooted trees obeying a simple but powerful rule. Under this rule, groups with exotic properties can be generated from very basic starting data, most famously the Grigorchuk group which was the first example of a group with intermediate growth.

Nekrashevych introduced a groupoid and a C*-algebra for a self-similar group action on a tree as models for some underlying noncommutative space for the system. Our goal is to compute the K-theory of the C*-algebra and the homology of the groupoid. Our main theorem provides long exact sequences which reduce the problems to group theory. I will demonstrate how to apply this theorem to fully compute homology and K-theory through the example of the Grigorchuk group.

This is joint work with Benjamin Steinberg.

Let G be a free group of rank N, let f be an automorphism of G and let Fix(f) be the corresponding subgroup of fixed points. Bestvina and Handel showed that the rank of Fix(f) is at most N, for which they developed the theory of train track maps on free groups. Different arguments were provided later on by Sela, Paulin and Gaboriau-Levitt-Lustig. In this talk, we present a new proof which involves the Linnell division ring of G. We also discuss how our approach relates to previous ones and how it gives new insight into variations of the problem.

While the brain has long been conceptualized as a network of neurons connected by synapses, attempts to describe the connectome using established models in network science have yielded conflicting outcomes, leaving the architecture of neural networks unresolved. Here, we analyze eight experimentally mapped connectomes, finding that the degree and the strength distribution of the underlying networks cannot be described by random nor scale-free models. Rather, the node degrees and strengths are well approximated by lognormal distributions, whose emergence lacks a mechanistic model in the context of networks. Acknowledging the fact that the brain is a physical network, whose architecture is driven by the spatially extended nature of its neurons, we analytically derive the multiplicative process responsible for the lognormal neuron length distribution, arriving to a series of empirically falsifiable predictions and testable relationships that govern the degree and the strength of individual neurons. The lognormal network characterizing the connectome represents a novel architecture for network science, that bridges critical gaps between neural structure and function, with unique implications for brain dynamics, robustness, and synchronization.

Upper bounds on the number of incidences between points and lines, tubes, and other geometric objects, have many applications in combinatorics and analysis. On the other hand, much less is known about lower bounds. We prove a general lower bound for the number of incidences between points and tubes in the plane under a natural spacing condition. In particular, if you take $n$ points in the unit square and draw a line through each point, then there is a non-trivial point-line pair with distance at most $n^{-2/3+o(1)}$. This quickly implies that any $n$ points in the unit square define a triangle of area at most $n^{-7/6+o(1)}$, giving a new upper bound for the Heilbronn's triangle problem.

Joint work with Alex Cohen and Cosmin Pohoata.

Let X be a n by n unitary matrix, drawn at random according to the Haar measure on U_n, and let m be a natural number. What can be said about the distribution of X^m and its eigenvalues? 

The density of the distribution \tau_m of X^m can be written as a linear combination of irreducible characters of U_n, where the coefficients are the Fourier coefficients of \tau_m. In their seminal work, Diaconis and Shahshahani have shown that for any fixed m, the sequence (tr(X),tr(X^2),...,tr(X^m)) converges, as n goes to infinity, to m independent complex normal random variables (suitably normalized). This can be seen as a statement about the low-dimensional Fourier coefficients of \tau_m. 

In this talk, I will focus on high-dimensional spectral information about \tau_m. For example: 

(a) Can one give sharp estimates on the rate of decay of its Fourier coefficients?

(b) For which values of p, is the density of \tau_m  L^p-integrable? 

Using works of Rains about the distribution of X^m, we will see how Item (a) is equivalent to a branching problem in the representation theory of certain compact homogeneous spaces, and how (b) is equivalent to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements.

Based on joint works with Julia Gordon and Yotam Hendel and with Nir Avni and Michael Larsen.

A question we get asked all the time! We'll also be discussing the numerous ways our identities as Mathematicians are shaped by being a minority. Free lunch provided.

Gauging is a systematic way to construct a model with non-invertible symmetry from a model with ordinary group-like symmetry. In 2+1d dimensions or higher, one can generalize the standard gauging procedure by stacking a symmetry-enriched topological order before gauging the symmetry. This generalized gauging procedure allows us to realize a large class of non-invertible symmetries. In this talk, I will describe the generalized gauging of finite group symmetries in 2+1d lattice models. This talk will be based on my ongoing work with L. Bhardwaj, S.-J. Huang, S. Schäfer-Nameki, and A. Tiwari.

The Camassa–Holm equation, which is nonlinear one-dimensional nonlinear PDE which is completely integrable and has  applications in several areas, has received considerable attention. We will discuss recent work regarding the Camassa—Holm equation with transport noise, more precisely, the equation $u_t+uu_x+P_x+\sigma u_x \circ dW=0$ and $P-P_{xx}=u^2+u_x^2/2$. În particular, we will show existence of a weak, global, dissipative solution of the Cauchy initial-value problem on the torus.  This is joint work with L. Galimberti (King’s College), K.H. Karlsen (Oslo), and P.H.C. Pang (NTNU/Oslo).

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of $\mathbb{Q}_p^{ab}$, the maximal extension of the $p$-adic numbers $\mathbb{Q}_p$ with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of $\mathbb{Q}_p$) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).