Past

Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. In the classical setup, the degree of a finite Galois extension of "nice" fields splits up neatly into the product of two well-understood numbers (ramification index and inertia degree) that encode how the base field changes. In the general case, however, a third factor called the defect (or ramification deficiency) can pop up. The defect is a mysterious phenomenon and the main obstruction to several long-standing open problems, such as obtaining resolution of singularities. The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many adjustments" fails when the defect is non-trivial. I will discuss my previous and ongoing work in ramification theory that allows us to understand and treat the defect.

We frame dynamic persuasion in a partial observation stochastic control game with an ergodic criterion. The receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the receiver through a device designed by the sender that generates the observation process. 

The commitment of the sender is enforced and an exogenous information process outside the control of the sender is allowed. We develop this approach in the case where all dynamics are linear and the preferences of the receiver are linear-quadratic.

We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the receiver’s value function. An extension to the case of persuasion of a mean field of interacting receivers is also provided. We illustrate this approach in two applications: the provision of information to electricity consumers with a smart meter designed by an electricity producer; the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. In the first application, we link the benefits of information provision to the mispricing of electricity production. In the latter, we show that when firms declare a high level of best-in-class target, the information provided by stringent accounting rules offsets the Nash equilibrium effect that leads firms to increase pollution to make their target easier to achieve.

This is a joint work with Prof. René Aïd, Prof. Giorgia Callegaro and Prof. Luciano Campi.

In this seminar I will begin by giving an overview of some problems in stochastic simulation and uncertainty quantification. I will then outline the Multilevel Monte Carlo for situations in which accurate simulations are very costly, but it is possible to perform much cheaper, less accurate simulations.  Inspired by the multigrid method, it is possible to use a combination of these to achieve the desired overall accuracy at a much lower cost.

 One of the main entries in the AdS/CFT dictionary is a relation between the bulk on-shell partition function with specified boundary conditions and the generating function of correlation functions of primary operators in the boundary CFT. In this talk, I will show how to construct a similar relation for gravity in 4d asymptotically flat spacetimes. For simplicity, we will restrict to the leading infrared sector, where a careful treatment of soft modes and their canonical partners leads to a non-vanishing on-shell action. I will show that this action localizes to a codimension-2 surface and coincides with the generating function of 2d CFT correlators involving insertions of Kac-Moody currents. The latter were previously shown, using effective field theory methods, to reproduce the leading soft graviton theorems in 4d. I will conclude with comments on the implications of these results for the computation of soft charge fluctuations in the vacuum. 

The global COVID19 pandemic has highlighted the lethality and morbidity associated with infectious respiratory diseases. These diseases can lead to devastating syndrome known as the acute respiratory distress syndrome (ARDS) where bacterial/viral infections cause excessive lung inflammation, pulmonary edema, and severe hypoxemia (shortness of breath). Although ARDS patients require artificial mechanical ventilation, the complex biofluid and biomechanical forces generated by the ventilator exacerbates lung injury leading to high mortality. My group has used mathematical and computational modeling to both characterize the complex mechanics of lung injury during ventilation and to identify novel ways to prevent injury at the cellular level. We have used in-vitro and in-vivo studies to validate our mathematical predictions and have used engineering tools to understand the biological consequences of the mechanical forces generated during ventilation. In this talk I will specifically describe how our mathematical/computational approach has led to novel cytoskeletal based therapies and how coupling mathematics and molecular biology has led to the discovery of a gene regulatory mechanisms that can minimize ventilation induced lung injury. I will also describe how we are currently using nanotechnology and gene/drug delivery systems to enhance the lung’s native regulatory responses and thereby prevent lung injury during ARDS.

In Bayesian inverse problems, it is common to consider several hyperparameters that define the prior and the noise model that must be estimated from the data. In particular, we are interested in linear inverse problems with additive Gaussian noise and Gaussian priors defined using Matern covariance models. In this case, we estimate the hyperparameters using the maximum a posteriori (MAP) estimate of the marginalized posterior distribution. 

However, this is a computationally intensive task since it involves computing log determinants.  To address this challenge, we consider a stochastic average approximation (SAA) of the objective function and use the preconditioned Lanczos method to compute efficient function evaluation approximations. 

We can therefore compute the MAP estimate of the hyperparameters efficiently by building a preconditioner which can be updated cheaply for new values of the hyperparameters; and by leveraging numerical linear algebra tools to reuse information efficiently for computing approximations of the gradient evaluations.  We demonstrate the performance of our approach on inverse problems from tomography. 

We investigate an interacting particle model to simulate a foraging colony of ants, where each ant is represented as a so-called active Brownian particle. Interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of the pheromone field. We show how the empirical measure of the interacting particle system converges to a solution of a mean-field limit (MFL) PDE for some subset of the model parameters. We situate the MFL PDE as a non-gradient flow continuity equation with some other recent examples. We then demonstrate that the MFL PDE for the ant model has two distinctive behaviors: the well-known Keller--Segel aggregation into spots and the formation of lanes along which the ants travel. Using linear and nonlinear analysis and numerical methods we provide the foundations for understanding these particle behaviors at the mean-field level. We conclude with long-time estimates that imply that there is no infinite time blow-up for the MFL PDE.

A topological quantum field theory (TQFT) is a functor from a category of bordisms to a category of vector spaces. Classifying low-dimensional TQFTs often involves considering presentations of bordism categories in terms of generators and relations. In this talk, we will introduce these concepts and outline a program for obtaining such presentations using Morse–Cerf theory.

Schwarzian Theory is a quantum field theory which has attracted a lot of attention in the physics literature in the context of two-dimensional quantum gravity, black holes and AdS/CFT correspondence. It is predicted to be universal and arise in many systems with emerging conformal symmetry, most notably in Sachdev--Ye--Kitaev random matrix model and Jackie--Teitelboim gravity.

In this talk we will discuss our recent progress on developing rigorous mathematical foundations of the Schwarzian Field Theory, including rigorous construction of the corresponding measure, calculation of both the partition function and a natural class of correlation functions, and a large deviation principle.

The stable uniqueness theorem for KK-theory asserts that a Cuntz-pair of *-homomorphisms between separable C*-algebras gives the zero element in KK if and only if the *-homomorphisms are stably homotopic through a unitary path, in a specific sense. This result, along with its group equivariant analogue, has been crucial in the classification theory of C*-algebras and C*-dynamics. In this talk, I will present a unitary tensor category analogue of the stable uniqueness theorem and explore its application to a duality in tensor category equivariant KK-theory. To make the talk approachable even for those unfamiliar with actions of unitary tensor categories or KK-theory, I will introduce the relevant definitions and concepts, drawing comparisons with the case of group actions. This is joint work with Kan Kitamura and Robert Neagu.

Simplicial volume is a homotopy invariant of manifolds introduced by Gromov to study their metric and rigidity properties. One of the strongest vanishing results for simplicial volume of closed manifolds is in presence of amenable covers with controlled multiplicity. I will discuss some conditions under which this result can be extended to manifolds with boundary. To this end, I will follow Gromov's original approach via the theory of multicomplexes, whose foundations have been recently laid down by Frigerio and Moraschini.

In a graph $H$ whose edges are coloured (not necessarily properly) a rainbow copy of a graph $G$ is a (not necessarily induced) subgraph of $H$ that is isomorphic to $G$ and whose edges are all coloured differently. In this talk I will explain why the problem of finding such rainbow copies is interesting, survey what we know, concentrating mainly on the case where $G$ is a Hamilton cycle, and then tell you a bit about a new result about finding rainbow Hamilton cycles resiliently in random graphs (which is joint work with Peter Allen and Liana Yepremyan).

All atmospheric phenomena, from daily weather patterns to the global climate system, are invariably influenced by atmospheric flow. Despite its importance, its complex behaviour makes extracting informative features from its dynamics challenging. In this talk, I will present a network-based approach to explore relationships between different flow structures. Using three phenomenon- and model-independent methods, we will investigate coherence patterns, vortical interactions, and Lagrangian coherent structures in an idealised model of the Northern Hemisphere stratospheric polar vortex. I will argue that networks built from fluid data retain essential information about the system's dynamics, allowing us to reveal the underlying interaction patterns straightforwardly and offering a fresh perspective on atmospheric behaviour.

Smooth generic representations of $GL_n$ over a $p$-adic field $F$, i.e. representations admitting a nondegenerate Whittaker model, are an important class of representations, for example in the setting of Rankin-Selberg integrals. However, in recent years there has been an increased interest in non-generic representations and their degenerate Whittaker models. By the theory of Bernstein-Zelevinsky derivatives we can associate to each smooth irreducible representation of $GL_n(F)$ an integer partition of $n$, which encodes the "degeneracy" of the representation. By using these "highest derivative partitions" we can define a stratification of the category of smooth complex representations and prove the surprising fact that all of the strata categories are equivalent to module categories over commutative rings. This is joint work with David Helm.

In the talk, I will start by recalling some basics of optimal transport and how it can be used to define Ricci curvature lower bounds for singular spaces, in a synthetic sense. Then, I will present some joint work with De Luca-De Ponti and Tomasiello,  where we show that some singular spaces,  naturally showing up in gravity compactifications (namely, Dp-branes),  enter the aforementioned setting of non-smooth spaces satisfying Ricci curvature lower bounds in a synthetic sense.  Time permitting, I will discuss some applications to the Kaluza-Klein spectrum.

On arbitrary Carnot groups, the only hypoelliptic Hodge-Laplacians on forms that have been introduced are 0-order pseudodifferential operators constructed using the Rumin complex.  However, to address questions where one needs sharp estimates, this 0-order operator is not suitable. Indeed, this is a rather difficult problem to tackle in full generality, the main issue being that the Rumin exterior differential is not homogeneous on arbitrary Carnot groups. In this talk, I will focus on the specific example of the free Carnot group of step 3 with 2 generators, where it is possible to introduce different hypoelliptic Hodge-Laplacians on forms. Such Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups

A successful strategy to handle problems involving primes is to approximate them by a more 'simple' function. Two aspects need to be balanced. On the one hand, the approximant should be simple enough so that the considered problem can be solved for it. On the other hand, it needs to be close enough to the primes in order to make it an admissible to replacement. In this talk I will present how one can construct general approximants in the context of the Circle Method and will use this to give a different perspective on Goldbach type applications.

We show that linear reflection groups in the sense of Vinberg are often Zariski dense in PGL(n). Among the applications are examples of low-dimensional closed hyperbolic manifolds whose fundamental groups virtually embed as Zariski-dense subgroups of SL(n,Z), as well as some one-ended Zariski-dense subgroups of SL(n,Z) that are finitely generated but infinitely presented, for all sufficiently large n. This is joint work with Jacques Audibert, Gye-Seon Lee, and Ludovic Marquis.

We consider the problem of parametric and non-parametric statistical inference for systems of weakly interacting diffusions and of their mean field limit. We present several parametric inference methodologies, based on stochastic gradient descent in continuous time, spectral methods and the method of moments. We also show how one can perform fully nonparametric Bayesian inference for the mean field McKean-Vlasov PDE. The effect of non-uniqueness of stationary states of the mean field dynamics on the inference problem is elucidated.

During the 14th to the 16th centuries CE, a succession of Indian scholars, collectively referred to as the Kerala school, made remarkable contributions in the fields of mathematics and astronomy. Mādhava of Saṅgamagrāma, a gifted mathematician and astronomer, is considered the founder of this school, and is perhaps best known for discovering an infinite series for pi, among other achievements. Subsequently, Mādhava's lineage of disciples, consisting of illustrious names such as Parameśvara, Dāmodara, Nīlakaṇṭha, Jyeṣṭhadeva, Śaṅkara Vāriyar, Citrabhānu, Acyuta Piṣaraṭi etc., made numerous important contributions of their own in the fields of mathematics and astronomy. Later scholars of the Kerala school flourished up to the 19th century. This talk will provide a historical overview of the Kerala school and highlight its important contributions.

Electron cryomicroscopy (cryo-EM) is an imaging technique widely used in structural biology to determine the three-dimensional structure of biological molecules from noisy two-dimensional projections with unknown orientations. As the typical pipeline involves processing large amounts of data, efficient algorithms are crucial for fast and reliable results. The stochastic gradient descent (SGD) algorithm has been used to improve the speed of ab initio reconstruction, which results in a first, low-resolution estimation of the volume representing the molecule of interest, but has yet to be applied successfully in the high-resolution regime, where expectation-maximization algorithms achieve state-of-the-art results, at a high computational cost. 
In this work, we investigate the conditioning of the optimisation problem and show that the large condition number prevents the successful application of gradient descent-based methods at high resolution. 
Our results include a theoretical analysis of the condition number of the optimisation problem in a simplified setting where the individual projection directions are known, an algorithm based on computing a diagonal preconditioner using Hutchinson's diagonal estimator, and numerical experiments showing the improvement in the convergence speed when using the estimated preconditioner with SGD. The preconditioned SGD approach can potentially enable a simple and unified approach to ab initio reconstruction and high-resolution refinement with faster convergence speed and higher flexibility, and our results are a promising step in this direction.