Past

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw.  We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.  
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting:  the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q.  Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above:  a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.  

Let us say that a theory $T$ in the language of set theory is $\beta$-consistent at $\alpha$ if there is a transitive model of $T$ of height $\alpha$, and let us say that it is $\beta$-categorical at $\alpha$ iff there is at most one transitive model of $T$ of height $\alpha$. Let us also assume, for ease of formulation, that there are arbitrarily large $\alpha$ such that $\mathrm{ZFC}$ is $\beta$-consistent at $\alpha$.

The sentence $\mathrm{VEL}$ has the feature that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-categorical at $\alpha$, for every $\alpha$. If we assume in addition that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-consistent at $\alpha$, then the uniquely determined model is $L_\alpha$, and the minimal such model, $L_{\alpha_0}$, is model of determined by the $\beta$-categorical theory $\mathrm{ZFC}+\mathrm{VEL}+M$, where $M$ is the statement "There does not exist a transitive model of $\mathrm{ZFC}$."

It is natural to ask whether $\mathrm{VEL}$ is the only sentence that can be $\beta$-categorical at $\alpha$; that is, whether, there can be a sentence $\phi$ such that $\mathrm{ZFC}+\phi$ is $\beta$-categorical at $\alpha$, $\beta$-consistent at $\alpha$, and where the unique model is not $L_\alpha$.  In the early 1970s Harvey Friedman proved a partial result in this direction. For a given ordinal $\alpha$, let $n(\alpha)$ be the next admissible ordinal above $\alpha$, and, for the purposes of this discussion, let us say that an ordinal $\alpha$ is minimal iff a bounded subset of $\alpha$ appears in $L_{n(\alpha)}\setminus L_\alpha$. [Note that $\alpha_0$ is minimal (indeed a new subset of $\omega$ appears as soon as possible, namely, in a $\Sigma_1$-definable manner over $L_{\alpha_0+1}$) and an ordinal $\alpha$ is non-minimal iff $L_{n(\alpha)}$ satisfies that $\alpha$ is a cardinal.] Friedman showed that for all $\alpha$ which are non-minimal, $\mathrm{VEL}$ is the only sentence that is $\beta$-categorical at $\alpha$. The question of whether this is also true for $\alpha$ which are minimal has remained open.

In this talk I will describe some joint work with Hugh Woodin that bears on this question. In general, when approaching a "lightface" question (such as the one under consideration) it is easier to first address the "boldface" analogue of the question by shifting from the context of $L$ to the context of $L[x]$, where $x$ is a real. In this new setting everything is relativized to the real $x$: For an ordinal $\alpha$, we let $n_x(\alpha)$ be the first $x$-admissible ordinal above $\alpha$, and we say that $\alpha$ is $x$-minimal iff a bounded subset of $\alpha$ appears in $L_{n_x(\alpha)}[x]\setminus L_{\alpha}[x]$.

Theorem. Assume $M_1^\#$ exists and is fully iterable. There is a sentence $\phi$ in the language of set theory with two additional constants, \r{c} and \r{d}, such that for a Turing cone of $x$, interpreting \r{c} by $x$, for all $a$

1. if $L_\alpha[x]\vDash\mathrm{ZFC}$ then there is an interpretation of \r{d}  by something in $L_\alpha[x]$ such that there is a $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$, and
2. if, in addition, $\alpha$ is $x$-minimal, then there is a unique $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$.

The sentence $\phi$ asserts the existence of an object which is external to $L_\alpha[x]$ and which, in the case where $\alpha$ is minimal, is canonical. The object is a branch $b$ through a certain tree in $L_\alpha[x]$, and the construction uses techniques from the HOD analysis of models of determinacy.

In this talk I will sketch the proof, describe some additional features of the singleton, and say a few words about why the lightface version looks difficult.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much (but deliver little) in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how topological methods can be used to study Nielsen transformations.

The classical Turán problem asks: given a graph $H$, how many edges can an $4n$-vertex graph have while containing no isomorphic copy of $H$? By viewing $(k+1)$-uniform hypergraphs as $k$-dimensional simplicial complexes, we can ask a topological version (first posed by Nati Linial): given a $k$-dimensional simplicial complex $S$, how many facets can an $n$-vertex $k$-dimensional simplicial complex have while containing no homeomorphic copy of $S$? Until recently, little was known for $k > 2$. In this talk, we give an answer for general $k$, by way of dependent random choice and the combinatorial notion of a trace-bounded hypergraph. Joint work with Jason Long and Bhargav Narayanan.

Non-Hermitian random matrices with complex eigenvalues represent a truly two-dimensional (2D) Coulomb gas at inverse temperature beta=2. Compared to their Hermitian counter-parts they enjoy an enlarged bulk and edge universality. As an application to ecology we model large scale data of the approximately 2D distribution of buzzard nests in the Teutoburger forest observed over a period of 20 y. These birds of prey show a highly territorial behaviour. Their occupied nests are monitored annually and we compare these data with a one-component 2D Coulomb gas of repelling charges as a function of beta. The nearest neighbour spacing distribution of the nests is well described by fitting to beta as an effective repulsion parameter, that lies between the universal predictions of Poisson (beta=0) and random matrix statistics (beta=2). Using a time moving average and comparing with next-to-nearest neighbours we examine the effect of a population increase on beta and correlation length.
 

Often in scattering applications it is advantageous to reformulate the problem as an integral equation, discretize, and then solve the resulting linear system using a fast direct solver. The computational cost of this approach is typically dominated by the work needed to compress the coefficient matrix into a rank-structured format. In this talk, we present a novel technique which exploits the bandlimited-nature of solutions to the Helmholtz equation in order to accelerate this procedure in environments where multiple frequencies are of interest.

This talk is based on joint work with Gunnar Martinsson (UT Austin).

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Coadmissible modules over Frechet-Stein algebras arise naturally in p-adic representation theory, e.g. in the study of locally analytic representations of p-adic Lie groups or the function spaces of rigid analytic Stein spaces. We show that in many cases, the category of coadmissible modules admits an exact and fully faithful embedding into the category of complete bornological modules, also preserving tensor products. This allows us to introduce derived methods to the study of coadmissible modules without forsaking the analytic flavour of the theory. As an application, we introduce six functors for Ardakov-Wadsley's D-cap-modules and discuss some instances where coadmissibility (in a derived sense) is preserved.

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian, and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension, so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional, and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogues of structures seen in related network models.

arXiv link: https://arxiv.org/abs/2010.07421

Motivated by the work of K.R. Rajagopal, the objective of the talk is to discuss the construction and analysis of numerical approximations to a class of models that fall outside the realm of classical Cauchy elasticity. The models under consideration are implicit and nonlinear, and are referred to as strain-limiting, because the linearised strain remains bounded even when the stress is very large, a property that cannot be guaranteed within the framework of classical elastic or nonlinear elastic models. Strain-limiting models can be used to describe, for example, the behavior of brittle materials in the vicinity of fracture tips, or elastic materials in the neighborhood of concentrated loads where there is concentration of stress even though the magnitude of the strain tensor is limited.

We construct a finite element approximation of a strain-limiting elastic model and discuss the theoretical difficulties that arise in proving the convergence of the numerical method. The analytical results are illustrated by numerical experiments.

The talk is based on joint work with Andrea Bonito (Texas A&M University) and Vivette Girault (Sorbonne Université, Paris).

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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 contact @email.

Self-similar fragmentation processes are random models for particles that are subject to successive fragmentations. When the index of self-similarity is negative the fragmentations intensify as the masses of particles decrease. This leads to a shattering phenomenon, where the initial particle is entirely reduced to dust - a set of zero-mass particles - in finite time which is what we call the extinction time. Equivalently, these extinction times may be seen as heights of continuous compact rooted trees or scaling limits of heights of sequences of discrete trees. Our objective is to set up precise bounds for the large time asymptotics of the tail distributions of these extinction times.

Coronary heart disease is characterised by the formation of plaque on artery walls, restricting blood flow. If a plaque deposit ruptures, blood clot formation (thrombosis) rapidly occurs with the potential to fatally occlude the vessel within minutes. Von Willebrand Factor (vWF) is a shear-sensitive protein which has a critical role in blood clot formation in arteries. At the high shear rates typical in arterial constrictions (stenoses), vWF undergoes a conformation change, unfolding and exposing binding sites and facilitating rapid platelet deposition. 

To understand the effect of  stenosis geometry and blood flow conditions on the unfolding of vWF and subsequent platelet binding, we developed a continuum model for the initiation of thrombus formation by vWF in an idealised arterial stenosis. In this talk I will discuss modelling proteins in flow using viscoelastic fluid models, the insight asymptotic reductions can offer into this complex system and some of the challenges of studying fast arterial blood flows. 

The propagation of high-frequency electromagnetic waves can be analyzed using the geometrical optics approximation. In the case of large but finite frequencies, the geometrical optics approximation is no longer accurate, and polarization-dependent corrections at first order in wavelength modify the propagation of light in an inhomogenous medium via a spin-orbit coupling mechanism. This effect, known as the spin Hall effect of light, has been experimentally observed. In this talk I will discuss recent work which generalizes the spin Hall effect to the propagation of light and gravitational waves in inhomogenous spacetimes. This is based on joint work with Marius Oancea and Jeremie Joudioux.

We discuss a novel backward Ito-Ventzell formula and an extension of the Aleeksev-Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the same, yielding what seems to be the first results of this type for this class of  anticipative models.

We illustrate the impact of these results in the context of diffusion perturbation theory, interacting diffusions and discrete time approximations.

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. This flow is also related to the classical continuous Heisenberg model in ferromagnetism and to the 1-D cubic Schrödinger equation. In this lecture I will first talk about the state of the art of the binormal flow conjecture, as well as about mathematical methods and results for the binormal flow. Then I will introduce a class of solutions at the critical level of regularity that generate singularities in finite time and describe some of their properties. These results are joint work with Luis Vega.

In 1966 Chen Jingrun showed that every large even integer can be written as the sum of two primes or the sum of a prime and a semiprime. To date, this weakened version of Goldbach's conjecture is one of the most remarkable results of sieve theory. I will talk about the big ideas which paved the way to this proof and the ingenious trick which led to Chen's success. No prior knowledge of sieve theory required – all necessary techniques will be introduced in the talk.

In rational conformal field theory, the sewing and factorization properties are probably the most important properties that conformal blocks satisfy. For special examples such as Weiss-Zumino-Witten models and minimal models, these two properties were proved decades ago (assuming the genus is ≤1 for the sewing theorem). But for general (strongly) rational vertex operator algebras (VOAs), their proofs were finished only very recently. In this talk, I will first motivate the definition of conformal blocks and VOAs using Segal's picture of CFT. I will then explain the importance of Sewing and Factorization Theorem in the construction of full rational conformal field theory.

The torus T of projective space also acts on the Hilbert
scheme of subschemes of projective space, and the T-graph of the
Hilbert scheme has vertices the fixed points of this action, and edges
the closures of one-dimensional orbits. In general this graph depends
on the underlying field. I will discuss joint work with Rob
Silversmith, in which we construct of a subgraph, which we call the
spine, of the T-graph of Hilb^N(A^2) that is independent of the choice
of field. The key technique is an understanding of the tropical ideal,
in the sense of tropical scheme theory, of the ideal of the universal
family of an edge in the spine.

Recent developments on the black hole information paradox have shown that Euclidean wormholes — so called “replica wormholes’’  — can dominate the von Neumann entropy as computed by a gravitational path integral, and that inclusion of these wormholes results in a unitary Page curve. This development raises some puzzles from the perspective of factorization, and has raised questions regarding what the gravitational path integral is computing. In this talk, I will focus on understanding the relationship between the gravitational path integral and the partition function via the gravitational free energy (more generally the generating functional). A proper computation of the free energy requires a replica trick distinct from the usual one used to compute the entropy. I will show that in JT gravity there is a regime where the free energy computed without replica wormholes is pathological. Interestingly, the inclusion of replica wormholes is not quite sufficient to resolve the pathology: an alternative analytic continuation is required. I will discuss the implications of this for various interpretations of the gravitational path integral (e.g. as computing an ensemble average) and also mention some parallels with spin glasses. 

In this session, William Durham from the Bank of England will give a presentation about working as a mathematician in the BoE, and will give advice on interviewing for non-academic jobs. He has previously provided mock interviews in our department for jobs aimed at mathematicians with PhDs, and is happy to conduct some mock interviews (remotely, of course) for individuals as well.

Please email Helen McGregor (@email) by Monday 22 February if you might be interested in having a mock interview with William Durham on 5 March.
 

 I will be reviewing our recent article arXiv:2101.02218 where we propose a simple method for detecting global (a.k.a. non-perturbative) anomalies for generic quantum field theories. The basic idea is to study how the symmetries are realized on the Hilbert space of the theory. I will present several elementary examples where everything can be solved explicitly. After that, we will use these results to make non-trivial predictions about strongly interacting theories.

For $G$ a finite group, $p$ a prime and $(K, \mathcal{O}_K, k)$ a $p$-modular system the group ring $\mathcal{O}_K G$ is an $\mathcal{O}_k$-order in the $K$-algebra $KG.$ Graduated $\mathcal{O}_K$-orders are a particularly nice class of $\mathcal{O}_K$-orders first introduced by Zassenhaus. In this talk will see that an $\mathcal{O}_K$-order $\Lambda$ in a split $K$-algebra $A$ is graduated if the decomposition numbers for the regular $A$-module are no greater than $1$. Furthermore will see that graduated orders can be described (not uniquely) by a tuple $n$ and a matrix $M$ called the exponant matrix. Finding a suitable $n$ and $M$ for a graduated order $\Lambda$ in the $K$-algebra $A$ provides a parameterisation of the $\Lambda$-lattices inside the regular $A$-module. Understanding the $\mathcal{O}_K G$-lattices inside representations of certain groups $G$ is of interest to those involved in the Langlands programme as well as of independent interest to algebraists.

We propose a mathematical model that unifies the psychiatric concepts of drug-induced incentive salience (IST), reward prediction error

(RPE) and opponent process theory (OPT) to describe the emergence of addiction within substance abuse. The biphasic reward response (initially

positive, then negative) of the OPT is activated by a drug-induced dopamine release, and evolves according to neuro-adaptative brain

processes.  Successive drug intakes enhance the negative component of the reward response, which the user compensates for by increasing the

drug dose.  Further neuroadaptive processes ensue, creating a positive feedback between physiological changes and user-controlled drug

intake. Our drug response model can give rise to qualitatively different pathways for an initially naive user to become fully addicted.  The

path to addiction is represented by trajectories in parameter space that depend on the RPE, drug intake, and neuroadaptive changes.

We will discuss how our model can be used to guide detoxification protocols using auxiliary substances such as methadone, to mitigate withdrawal symptoms.

If this is useful here are my co-authors:
Davide Maestrini, Tom Chou, Maria R. D'Orsogna

Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this talk, I will first give very a brief introduction to tensors and their decompositions. After that, an improved version named iAPD of the classical APD will be proposed and all the following four fundamental properties of iAPD will be discussed : (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in optimization; (iii) more importantly, it converges R-linearly for a generic tensor without any assumption; (iv) for almost all LROTA problems, iAPD reduces to APD after finitely many iterations if it converges to a local minimizer. If time permits, I will also present some numerical experiments.