Week 18.02.2024 – 24.02.2024

In this talk I will present a solution of the so-called Gaussian level set percolation problem on random graphs. It addresses the question whether a multivariate Gaussian defined on the vertices of a random graph (with inverse covariance matrix defined, e.g., in terms of a weighted graph Laplacian) exhibits a macroscopic contiguous cluster of sites on which the Gaussian exceeds a given level h, or whether on the contrary all such clusters are finite. Because of the correlations encoded in the multivariate Gaussian, the problem is considerably more complicated than the case of independent Bernoulli percolation, and a full solution has, to the best of my knowledge, not been available in the literature. It turns out that the solution of the problem requires control over the microscopic heterogeneity of the percolation problem, i.e. , over distributions of node dependent percolation probabilities on random graphs.

The basic operations of free probability - additive free convolution, multiplicative free convolution, and free compression - describe how the eigenvalues of large random matrices interact under the basic matrix operations, such as addition, multiplication, and taking minors.

In this talk we discuss how these free probability operations can be formulated in terms of an “entropic optimal transport” problem – an optimal transport problem but with an entropy penalty for the coupling measure.

Our proof of this formulation uses the quadrature formulas of Marcus, Spielman and Srivastava, which relate the expected characteristic polynomial of matrices under random unitary vs symmetric conjugation. The approach involves an asymptotic analysis of the quadrature formulas using a large deviation principle on the symmetric group.

This is joint work with Octavio Arizmendi (CIMAT).

One is sometimes faced with the task of assigning finite values to divergent integrals in a consistent and meaningful way. Since differential forms and integrals play a central role in geometry, how to think about divergent integrals in geometric terms? The goal of this talk will be to answer this question in the case of logarithmic divergences (such as the integral of 1/x near x=0).
The idea is to pass from manifolds to slightly more general objects called manifolds with log corners. They are the differential geometer’s version of logarithmic varieties in algebraic geometry. A key ingredient of our construction is a new notion of morphism between manifolds with log corners (or logarithmic varieties) which is more flexible than the obvious one and faithfully records the geometric information needed to regularize divergent integrals.
This is joint work with Erik Panzer and Brent Pym.

A few mathematicians had considerable influence during the covid-19 epidemic. Some mathematicians have focussed on designing and implementing mathematical models which only consider a single illness. Applied statisticians know that it is critical to first decide what the question is: "Minimise deaths from Covid-19?" or "Minimise deaths due to Covid-19 and our decisions this year?" or "Minimise the impact of Covid-19 on well-being over ten years?" The ethical status of an expert who gives a simple answer to the first question, without uncertainty or alternatives, will be examined.

Mathematical predictions were used to justify lockdowns even in countries where people would starve as a consequence. Some publications by influential mathematics groups were directly misleading or disparaging of African scientists. Some statisticians have tried to estimate the damage to children's education and wellbeing, and illness and deaths due lockdown.

I argue that such mathematical modelling cannot be justified within virtue, deontological, utilitarian or care ethics, though Zoroaster or Nietzsche might be invoked. It is always necessary to consider the wider context, and the probable consequences of actions, as explained in the International Statistics Institute Code of Professional Ethics. Assessment of the validity of model assumptions, data quality, adequacy of the fit of models and accuracy of predictions is essential, and essentially statistical.

In this talk, we shall talk about two invariants associated with complete Nevanlinna-Pick (CNP) spaces. One of the invariants is an operator-valued multiplier of a given CNP space, and another invariant is a positive real number. These two invariants are called characteristic function and curvature invariant, respectively. The origin of these concepts can be traced back to the classical theory of contractions by Sz.-Nagy and Foias. We will begin by delving into this classical theory, gradually leading into our main subject matter.

You are invited to a lunch panel discussion and Q&A with early carreer researchers in mathematics. During this event, our panel members are going to discuss opportunities and challenges that a female mathematician has to face nowdays and how these might affect a career path. Information about the Women in Number Theory and Geometry spring retreat and the work of Piscopia local community at King's College London will also be provided.

Registration: https://www.eventbrite.co.uk/e/women-in-maths-panel-discussion-tickets-803873827257?aff=oddtdtcreator

Lunch will be provided.

It is joint work with Caroline Chessang, Tom Cowling and Ruth Keogh.