Past

Multivariate point processes are used to model event-type data in a wide range of domains. One interesting application is to model the emission of electric impulses of biological neurons. In this context, the point process model needs to capture the time-dependencies and interactions between neurons, which can be of two kinds: exciting or inhibiting. Estimating these interactions, and in particular the functional connectivity of the neurons are problems that have gained a lot of attention recently. The general nonlinear Hawkes process is a powerful model for events occurring at multiple locations in interaction. Although there is an extensive literature on the analysis of the linear model, the probabilistic and statistical properties of the nonlinear model are still mainly unknown. In this paper, we consider nonlinear Hawkes models and, in a Bayesian nonparametric inference framework, derive concentration rates for the posterior distribution.  We also infer the graph of interactions between the dimensions of the process and prove that the posterior distribution is consistent on the graph adjacency matrix.

I will discuss some of Mazur's work about congruences between Eisenstein series and cusp forms, and then end with an application to class groups of fields $\mathbb{Q}(N^{1/p})$, where $N$ and $p$ are primes.  I will only assume some algebraic number theory.  In particular, nothing about modular forms will be assumed.
 

In this talk I will consider model-independent pricing problems in a stochastic interest rates framework. In this case the usual tools from Optimal Transport and Skorokhod embedding cannot be applied. I will show how some pricing problems in a fixed-income market can be reformulated as Weak Optimal Transport (WOT) problems as introduced by Gozlan et al. I will present a super-replication theorem that follows from an extension of WOT results to the case of non-convex cost functions.
This talk is based on joint work with M. Beiglboeck and G. Pammer.

 In this talk I will discuss possible extensions of the euclidean quantitative isoperimetric inequality to compact Riemannian manifolds. 
This is joint work with O. Chodosh (Stanford) and M. Engelstein (University of Minnesota).

Artin groups are a family of groups generalizing braid groups. The Tits conjecture, which was proved by Crisp-Paris, states that squares of the standard generators generate an obvious right-angled Artin subgroup. In a joint work with Kevin Schreve, we consider a larger collection of elements, and conjecture that their sufficiently large powers generate an obvious right-angled Artin subgroup. In the case of the braid group, regarded as a mapping class group of a punctured disc, these elements correspond to Dehn twist around the loops enclosing multiple consecutive punctures. This alleged right-angled Artin group is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for some classes of Artin groups. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.

In this talk, I will work on a smooth projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the projective space P^3 or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. This is joint work with Richard Thomas. 

I will discuss various operator algebras in supersymmetric quantum field theories in various dimensions. The operator algebras are induced and classified by generalised topological twists. Omega deformation plays an important role in connecting different sectors. This talk is based on previous works and a work in progress with Junya Yagi.

One of the very unique properties of AdS_3 spacetimes is that we can introduce a finite number of massless higher spin fields without yielding an inconsistent theory. In this talk, we would like to comment on what the spectrum of these theories looks like: from the known contribution of the light spectrum, that corresponds to the vacuum character of the W_N algebra, we can use modular invariance to constraint the heavy spectrum of the theory. However, in doing so, we find negative norm states, inconsistent with unitarity. We propose a possible cure by adding light states that can be interpreted as massive particles with a conical defect associated to them, and study what scenario we are left with. The results that we will revisit are those presented in 2009.01830. 

In this session we will discuss how interviewing and being interviewed has changed now that interviews are conducted online. We will have a panel comprising Marya Bazzi, Mohit Dalwadi, Sam Cohen, Ian Griffiths and Frances Kirwan who have either experienced being interviewed online and have interviewed online and we will compare experiences with in-person interviews. 

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

The puzzle-shaped cells that appear in the epidermis of many plants are a striking example of a complex cell shape. Since shape in an organism is often thought to be closely related to its function, it suggests that these unusual shapes must have some functional benefit to the plant. We 
propose that the creation of these complex shapes is an effective strategy to reduce mechanical stress in the cell wall. Although the 
formation of these shapes requires highly anisotropic and non-uniform growth at the sub-cellular level, it appears to be triggered by 
isotropic growth at the organ level. Analysis of cell shape over multiple species is consistent with the idea that the puzzle is in 
response to a developmental constraint, and that the mechanism is like to be conserved among higher plants.

One of the fundamental examples of geometric representation theory is the Springer correspondence which parameterises the irreducible representations of the Weyl group of a lie algebra in terms of nilpotent orbits of the lie algebra and irreducible representations of the equivariant fundamental group of said nilpotent orbits. If you don’t like geometry this may sound entirely mysterious. In this talk I will hopefully offer a gentle introduction to the subject and present a preprint by Lusztig (2020) which gives an entirely algebraic description of the springer correspondence.

Daniel M. Anderson

Department of Mathematical Sciences, George Mason University

Applied and Computational Mathematics Division, NIST

Binary and multicomponent alloy solidification occurs in many industrial materials science applications as well as in geophysical systems such as sea ice. These processes involve heat and mass transfer coupled with phase transformation dynamics and can involve the formation of mixed phase regions known as mushy layers.  The understanding of transport mechanisms within mushy layers has important consequences for how these regions interact with the surrounding liquid and solid regions.  Through linear stability analyses and numerical calculations of mathematical models, convective instabilities that occur in solidifying ternary alloys will be explored.  Novel fluid dynamical phenomena that are predicted for these systems will be discussed.

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.

In this talk, we present simple ideas to combine nonparametric approaches based on positive definite kernels with deep learning models. There are many good reasons for bridging these two worlds. On the one hand, we want to provide regularization mechanisms and a geometric interpretation to deep learning models, as well as a functional space that allows to study their theoretical properties (eg invariance and stability). On the other hand, we want to bring more adaptivity and scalability to traditional kernel methods, which are crucially lacking. We will start this presentation by introducing models to represent graph data, then move to biological sequences, and images, showing that our hybrid models can achieves state-of-the-art results for many predictive tasks, especially when large amounts of annotated data are not available. This presentation is based on joint works with Alberto Bietti, Dexiong Chen, and Laurent Jacob.

Steklov eigenproblems and their variants (where the spectral parameter appears in the boundary condition) arise in a range of useful applications. For instance, understanding some properties of the mixed Steklov-Neumann eigenfunctions tells us why we shouldn't use coffee cups for expensive brandy. 

In this talk I'll present a high-accuracy discretization strategy for computing Steklov eigenpairs. The strategy can be used to study questions in spectral geometry, spectral optimization and to the solution of elliptic boundary value problems with Robin boundary conditions.

--

A link for the 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.

A graded unipotent group U is a unipotent group with a 1PS of automorphisms C^* -- > Aut(U), such that the this 1PS acts on the Lie(U) with all weights positive. Let \hat U be the semi-direct product of U with this 1PS. Let \hat U act linearly on (X,L), a projective variety with a very ample line bundle. With the condition `semistability coincides with stability', and after suitable twist of rational characters, the \hat U-linearisation has a projective geometric quotient, and the invariants are finitely generated. This is a result from \emph{Geometric invariant theory for graded unipotent groups and applications} by G Bérczi, B Doran, T Hawes, F Kirwan, 2018.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_NzU0ODY5MTUtMzUz…

Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. It is a generalisation of an older invariant, the Teichmüller polynomial, defined by McMullen in 2002.

The aim of my talk is to demonstrate that veering triangulations provide a convenient setup for computations. More precisely, I will use fairly easy arguments to obtain a fairly strong statement which generalises the results of McMullen relating the Teichmüller polynomial to the Alexander polynomial.

I will not assume any prior knowledge on the Alexander polynomial, the Teichmüller polynomial or veering triangulations.

This talk will focus on the following two related problems:
    (1) What is the minimum number of edges in a graph containing all $n$-vertex planar graphs as subgraphs? A simple construction of Babai, Erdos, Chung, Graham, and Spencer (1982) has $O(n^{3/2})$ edges, which is the best known upper bound.
    (2) What is the minimum number of *vertices* in a graph containing all $n$-vertex planar graphs as *induced* subgraphs? Here steady progress has been achieved over the years, culminating in a $O(n^{4/3})$ bound due to Bonamy, Gavoille, and Pilipczuk (2019).
    As it turns out, a bound of $n^{1+o(1)}$ can be achieved for each of these two problems. The two constructions are somewhat different but are based on a common technique. In this talk I will first give a gentle introduction to the area and then sketch these constructions. The talk is based on joint works with Vida Dujmović, Louis Esperet, Cyril Gavoille, Piotr Micek, and Pat Morin.

Averages of multiplicative eigenvalue statistics of Haar distributed unitary matrices are Toeplitz determinants, and asymptotics for these determinants are now well understood for large classes of symbols, including symbols with gaps and (merging) Fisher-Hartwig singularities. Similar averages for Haar distributed orthogonal matrices are Toeplitz+Hankel determinants. Some asymptotic results for these determinants are known, but not in the same generality as for Toeplitz determinants. I will explain how one can systematically deduce asymptotics for averages in the orthogonal group from those in the unitary group, using a transformation formula and asymptotics for certain orthogonal polynomials on the unit circle, and I will show that this procedure leads to asymptotic results for symbols with gaps or (merging) Fisher-Hartwig singularities. The talk will be based on joint work with Gabriel Glesner, Alexander Minakov and Meng Yang.

In this talk I will review the work that has been done by me, N. Gromov, V. Kazakov, G. Korchemsky and G. Sizov on the analysis of fishnet Feynman graphs in a particular scaling limit of $\mathcal N=4$ SYM, a theory dubbed $\chi$FT$_4$. After introducing said theory, in which the Feynman graphs take a very simple fishnet form — in the planar limit — I will review how to exploit integrable techniques to compute these graphs and, consequently, extract the anomalous dimensions of a simple class of operators.

Let $p$ be a prime. Minkowski (1887) gave a bound for the order of a finite $p$-subgroup of the linear group $\mathsf{GL}(n,\mathbf Z)$ as a function of $n$, and this necessarily holds for $p$-subgroups of $\mathsf{GL}(n,\mathbf Q)$ also. A few years ago, Serre asked me whether some analogous result might be obtained for subgroups of $\mathsf{GL}(n,\mathbf C)$ using the methods I employed to obtain optimal bounds for Jordan's theorem.

Bounds can be so obtained and I will explain how but, while Minkowski's bound is achieved, no linear bound (as Serre initially suggested) can be achieved. I will discuss progress on this problem and the issues that arise in seeking an ideal form for the solution.