Past

Creating networks of statistical dependencies between brain regions is a powerful tool in neuroscience that has resulted in many new insights and clinical applications. However, recent interest in higher-order interactions has highlighted the need to address beyond-pairwise dependencies in brain activity. Multivariate information theory is one tool for identifying these interactions and is unique in its ability to distinguish between two qualitatively different modes of higher-order interactions: synergy and redundancy. I will present results from applying the O-information, the partial entropy decomposition, and the local O-information to resting state fMRI data. Each of these metrics indicate that higher-order interactions are widespread in the cortex, and further that they reveal different patterns of statistical dependencies than those accessible through pairwise methods alone. We find that highly synergistic subsystems typically sit between canonical functional networks and incorporate brain regions from several of these systems. Additionally, canonical networks as well as the interactions captured by pairwise functional connectivity analyses, are strongly redundancy-dominated. Finally, redundancy/synergy dominance varies in both space and time throughout an fMRI scan with notable recurrence of sets of brain regions engaging synergistically. As a whole, I will argue that higher-order interactions in the brain are an under-explored space that, made accessible with the tools of multivariate information theory, may offer novel insights.

We give an exponential improvement on the upper bound for the $r$-colour diagonal Ramsey number for all $r$. The proof relies on geometric insights and offers a simplified proof in the case of $r=2$.

Joint Work with: Paul Ballister, Béla Bollobás, Marcelo Campos, Simon Griffiths, Rob Morris, Julian Sahasrabudhe and Marius Tiba.

Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.

In this talk I will discuss asymmetric orbifolds and will focus on their application to toroidal compactifications of heterotic string theory. I will consider theories in 6 and 4 dimensions with 16 supercharges and reduced rank. I will present a novel formalism, based on the Leech lattice, to construct ‘islands’ without vector multiplets.

We propose a density functional theory of Thomas-Fermi-(von Weizsacker) type to describe the response of a single layer of graphene to a charge some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers. We further provide conditions under which those minimizers are unique. The associated Euler-Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. For a class of special potentials, we also establish a precise universal asymptotic decay rate, as well as an exact charge cancellation by the graphene sheet. In addition, we discuss the existence of nodal minimizers which leads to multiple local minimizers in the TFW model. This is a joint work with Cyrill Muratov (University of Pisa).

We show that any finite coloring of an amenable group contains 'many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables.  This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem.  Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.

In this talk, I will describe recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. These are Q-factorial terminal Fano varieties, and they are the key players in the Minimal Model Program. In this work, we ask and answer if machine learning can determine if a toric Fano variety has terminal singularities. We build a high-accuracy neural network that detects this, which has two consequences. Firstly, it inspires the formulation and proof of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk.

Multiparameter persistence is an area of topological data analysis that synthesises the geometric information of a topological space via filtered homology. Given a topological space and a function on it, one can consider a filtration given by the sublevel sets of the space induced by the function and then take the homology of such filtration. In the case when the filtering function assumes values in the real plane, the homological features of the filtered object can be recovered through a "curved" grid on the plane called the extended Pareto grid of the function. In this talk, we explore how the computation of the biparameter matching distance between regular filtering functions on a regular manifold depends on the extended Pareto grid of these functions. 
 

What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, and a current student will offer tips and advice based on their experience.

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. (Students who attended the Part C/OMMS induction event will find significant overlap between the advice offered there and this session!)

Let F be a p-adic field. In this talk I'll study the Om(F)-distinction of some specific principal series representations  of Glm(F). The main goal is to give a computing method to see if those representations are distinguished or not so we can also explicitly find a non zero  Om(F)-equivariant linear form. This linear form will be given by the integral of the representation's matrix coefficient over Om(F).
 

After explaining on what specific principal series representations I'm working and why I need those specificities, I'll explain the different steps to compute the integral of my representation's matrix coefficient over Om(F). I'll explicitly give the obtained result for the case m=3. After that I'll explain an asymptotic result we can obtain when we can't compute the integral explicitly.

Immunological health relies on a balance between the ability to mount an immune response against potential pathogens and tolerance to self. However, how we keep that balance in health and what goes wrong in disease is not well understood. Here, I will describe combination of novel experimental and computational approaches using multi-omics datasets, imaging and functional experiments to dissect the role and defects in immune cells across several disease areas in cancer and autoimmunity. We show how shared mechanisms that are disrupted across diseases, including cellular, migration, immuno-surveillance, regulation and activation, as well as the immunological features associated with better prognosis and immunomodulation.

Given a locally compact topological group, there is a correspondence between idempotent probability measures and compact subgroups. An analogue of this correspondence continues into the model theoretic setting. In particular, if G is a stable group, then there is a one-to-one correspondence between idempotent Keisler measures and type-definable subgroups. The proof of this theorem relies heavily on the theory of local ranks in stability theory. Recently, we have been able to extend a version of this correspondence to the abelian setting. Here, we prove that fim idempotent Keisler measures correspond to fim subgroups. These results rely on recent work of Conant, Hanson and myself connecting generically stable measures to generically stable types over the randomization. This is joint work with Artem Chernikov and Krzysztof Krupinski.

If $n$ is congruent to 0 or 4 modulo 6, there are infinitely many primes of the form $x^2 + ny^2$ with both $x$ and $y$ prime. (Joint work with Mehtaab Sawhney, Columbia)

The backward error analysis is an important part of the perturbation theory and it is particularly useful for the study of the reliability of the numerical methods. We focus on the backward error for nonlinear eigenvalue problems. In this talk, the matrix-valued function is given as a linear combination of scalar functions multiplying matrix coefficients, and the perturbation is done on the coefficients. We provide theoretical results about the backward error of a set of approximate eigenpairs. Indeed, small backward errors for separate eigenpairs do not imply small backward errors for a set of approximate eigenpairs. In this talk, we provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the matrix coefficients of the matrix-valued function have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the perturbed matrix coefficients are also given. This is a joint work with Leonardo Robol (University of Pisa).

Please attend if you would like to give a talk in the Logic Advanced Class this term.

How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? 

This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…

I will explain the recent techniques developed with co-authors to obtain fine estimates about the large values of the Riemann zeta functions on the critical line. An emphasis will be put on the ideas originating from statistical mechanics and large deviations that may be of general interest for a stochastic analysis audience. No number theory knowledge will be assumed!

In this talk, based on a joint work with Ilijas Farah, I will present an application of an old continuous selection theorem due to Michael to the study of II1 factors. More precisely, I'll show that if two strongly continuous paths (or loops) of projections (p_t), (q_t), for t in [0,1], in a II1 factor are such that every p_t is subequivalent to q_t, then the subequivalence can be realized by a strongly continuous path (or loop) of partial isometries. I will then use an extension of this result to solve affirmatively the so-called trace problem for factorial W*-bundles whose base space is 1-dimensional.

I will present joint work with Alessandro Fazzari in which we prove precise conditional estimates for the third (non-absolute) moment of the logarithm of the Riemann zeta function, beyond the Selberg central limit theorem, both for the real and imaginary part. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which captures the interaction of a prime power with Montgomery's pair correlation function. This conjecture can be proved on a certain subrange unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.