Seminars from until

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. I will explain how logarithmic derivatives of the Fredholm determinants can be expressed in terms of a $2\times 2$ Riemann-Hilbert problem.

This Riemann-Hilbert representation can be used to derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal, and it reveals a remarkable connection with a family of unbounded solutions to the Korteweg-de Vries (KdV) equation and with an integro-differential version of the Painlevé II equation.

This talk is aimed at reviewing what is known about $G_2$-monopoles and motivate their study. After this, I will mention some recent results obtained in collaboration with Ákos Nagy and Daniel Fadel which investigate the asymptotic behavior of $G_2$-monopoles. Time permitting, I will mention a few possible future directions regarding the use of monopoles in $G_2$-geometry.

In March 11th of 2020, the World Health Organization declared the COVID-19 global public health emergency a pandemic [1]. Since the appearance of the first cases in Wuhan, China, several countries have employed the use of mathematical and statistical techniques to ascertain the course of the disease spread. The most common mathematical tool available to model such phenomena are systems of differential equations. The most notable are the SIR and SEIR model first developed by Kermack and McKendrick (1927). These models have been used to study an array of different epidemic questions. At the start of the pandemic, these models were employed to nowcast and forecast the national spread of SARS-CoV-2 in China. In [2] the authors create scenarios of transmissibility reduction and mobility reduction associated with the measures employed by the Chinese government. Similar models were also used to estimate the proportion of susceptible individuals in a population, i.e. how much is the case ascertainment in a given country [3]. This topic is very important since it has been shown that a high percentage of infected individuals do not develop symptoms [4] but are still able to infect others [5]. The main purpose of these modelling techniques has been to evaluate the impact of contagion mitigation measures, such as the closure of schools and lockdowns [6].

In Portugal, the team at the department of epidemiology Instituto Nacional de Saúde Doutor Ricardo Jorge, has been, since the start of the epidemic developing reports with an array of different statistical and mathematical procedures [7], in order to present a clear picture of the evolution of the epidemic, with the objective of supporting public health policy making. Part of this work involved building a SEIR-type model with heterogeneous mixing among age groups. This model was key to provide some evidence on the impact of the lockdown in Portugal from March 22th until May 4th. Using data from google mobility reports [8], the model showed that a decrease in transmission was expect after the implementation of the lockdown, which was not yet noticeable due to the delay between infection and case notification.

With the increase, as of late, of the daily incidence of COVID-19 cases and with the opening of schools, public health decision makers need to know what will be the expected impact on the Portuguese health system, and what non-pharmaceutical-interventions (NPI) can be adapted in order to compensate for such increase. Several epidemiologist state that higher and faster contact tracing might be the best and most efficient measure to compensate for such increase. The team is currently developing a new model that takes into account several NPIs, such as contact tracing, case ascertainment, mask usage, shielding of vulnerable (elderly) individuals, and closure/opening of schools, among others. The main objective is to provide possible scenarios for the magnitude of the impact of these measures.

Joint work with:

Maria Luísa Morgado, Departamento de Matemática, UTAD & CEMAT IST

Paula Patrício, Centro de Matemática e Aplicações & Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa

Baltazar Nunes, Instituto Nacional de Saúde Doutor Ricardo Jorge

References:

1. ECDC: Event Background-COVID-19, https://www.ecdc.europa.eu/en/novel-coronavirus/event-background-2019.

2. Wu, J. T., Leung, K., & Leung, G. M. (2020). Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study. The Lancet, 395(10225), 689–697. doi: 10.1016/s0140-6736(20)30260-9

3. Maugeri A, Barchitta M, Battiato S, Agodi A. Estimation of Unreported Novel Coronavirus (SARS-CoV-2) Infections from Reported Deaths: A Susceptible-Exposed-Infectious-Recovered-Dead Model. J Clin Med. 2020;9(5):1350. Published 2020 May 5. doi:10.3390/jcm9051350

4. Instituto Nacional de Saúde Dr. Ricardo Jorge (2020). Relatório de Apresentação dos Resultados Preliminares do Primeiro Inquérito Serológico Nacional COVID-19. Available: http://www.insa.min-saude.pt/wp-content/uploads/2020/08/ISN_COVID19_Relatorio_06_08_2020.pdf (acesso a 25/08/2020)

5. Huang L-S, Li L, Dunn L, He M. Taking Account of Asymptomatic Infections in Modeling the Transmission Potential of the COVID-19 Outbreak on the Diamond Princess Cruise Ship. medRxiv. 2020:2020.04.22.20074286.

6. Prem, K., Liu, Y., Russell, T., Kucharski, A. J., Eggo, R. M., Davies, N., … Klepac, P. (2020). The effect of control strategies that reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China. The Lancet Public Health. doi: 10.1101/2020.03.09.20033050

7. Nunes B, Caetano C, Antunes L, et al. Evolução do número de casos de COVID-19 em Portugal. Relatório de nowcasting. Inst. Nac. Saúde Doutor Ricardo Jorge. 2020; http://www.insa.min-saude.pt/category/areas-de-atuacao/epidemiologia/covid-19-curva-epidemica-e-parametros-de-transmissibilidade/

8. Relatórios de mobilidade da comunidade da COVID19. https://www.google.com/covid19/mobility/

Deep Learning is a powerful collection of techniques for statistical learning, which has shown dramatic applications in many different directions, including including the study of data sets of images, text, and time series. It uses neural networks, specifically convolutional neural networks (CNN's), to produce these results. What we have observed recently is that methods of topology can contribute to this effort, in diagnosing behavior within the CNN's, in the design of neural networks with excellent computational properties, and in improving generalization, i.e. the transfer of results of one neural network from one data set to another of similar type. We'll discuss topological methods in data science, as well as there application to this interesting set of techniques.

What might happen if we have points in a high dimensional space and one decided to project them into a random low dimensional space?

In this seminar, we will discuss this subject and will see some simple applications.

In cosmology, the universe is typically modelled by spatially homogeneous and isotropic solutions to Einstein’s equations. However, for large classes of matter models, such solutions are unstable in the direction of the singularity. For this reason, it is of interest to study the anisotropic setting.

The purpose of the talk is to describe a framework for studying highly anisotropic singularities. In particular, for analysing the asymptotics of solutions to linear systems of wave equations on the corresponding backgrounds and deducing information concerning the geometry.

The talk will begin with an overview of existing results. This will serve as a background and motivation for the problem considered, but also as a justification for the assumptions defining the framework we develop.

Following this overview, the talk will conclude with a rough description of the results.

The monster potentials were introduced by Bazhanov-Lukyanov-Zamolodchikov in the framework of the ODE/IM correspondence. They should in fact be in 1:1 correspondence with excited states of the Quantum KdV model (an Integrable Conformal Field Theory) since they are the most general potentials whose spectral determinant solves the Bethe Ansatz equations of such a theory. By studying the large momentum limit of the monster potentials, I retrieve that

1. The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.
2. The leading correction to such asymptotics is described by the roots of Wronskians of Hermite polynomials.

This allows me to associate to each partition of $N$ a unique monster potential with $N$ roots, of which I compute the spectrum. As a consequence, I prove up to a few mathematical technicalities that, fixed an integer $N$, the number of monster potentials with $N$ roots coincide with the number of integer partitions of $N$, which is the dimension of the level $N$ subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

This is joint work with Riccardo Conti (Group of Mathematical Physics of Lisbon University).

We will see how to represent the Donaldson-Futaki invariant as an intersection of equivariant closed forms. We will use it to express this invariant as the intersection on some specific subvarieties of the central fibre of the test configuration. As an application we provide a proof that for Kähler orbifolds the Donaldson-Futaki invariant is the Futaki invariant of the central fiber.

Neural network-based deep learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, deep learning has also acquired the reputation of being a set of “black box” type of tricks, without fundamental principles. This has been a real obstacle for making further progress in machine learning.

In this talk, I will try to address the following two questions:

1. How machine learning will impact computational mathematics and computational science?
2. How computational mathematics, particularly numerical analysis, can impact machine learning? We describe some of the most important progresses that have been made on these issues so far. Our hope is to put things into a perspective that will help to integrate machine learning with computational science.

Quantum higher Teichmüller theory, as described by Fock and Goncharov, endows a quantum character variety on a surface $S$ with a cluster structure. The latter allows one to construct a canonical representation of the character variety, which happens to be equivariant with respect to an action of the mapping class group of $S$. It was conjectured by the authors that these representations behave well with respect to cutting and gluing of surfaces, which in turn yields an analogue of a modular functor. In this talk, I will show how the quantum group and its positive representations arise in this context. I will also explain how the modular functor conjecture is related to the closedness of positive representations under tensor products as well as to the non-compact analogue of the Peter-Weyl theorem. If time permits, I will say a few words about the proof of the conjecture.

This talk is based on joint works with Gus Schrader.

In my talk I will discuss topologically twisted compactification of 6d (1,0) theories on 4-manifolds with background flavor symmetry bundles. The effective 2d theory generically has (0,1) supersymmetry and a residual flavor symmetry. Evaluation of its elliptic genus thus produces an invariant of the 4-manifold equipped with a principle bundle valued in the ring of (equivariant) modular forms. By further including torsion valued invariants of (0,1) 2d theories, one obtains an invariant of 4-manifolds valued in (equivariant) topological modular forms (TMF). I will describe basic properties of this map and present a few simple examples. I will also mention some byproduct results on 't Hooft anomalies of 6d (1,0) theories. The talk is based on a joint work with Gukov, Pei and Vafa.

The eigenstate thermalization hypothesis (ETH) is a cornerstone in our understanding of quantum statistical mechanics. The extent to which ETH holds for nonlocal operators (observables) is an open question. I will address this question using an analogy with random matrix theory. The starting point will be the construction of extremely non-local operators, which we call Behemoth operators. The Behemoths turn out to be building blocks for all physical operators. This construction allow us to derive scalings for both local operators and different kinds of nonlocal operators.

Inspired by the celebrated $C^0, C^2$ and $C^3$ a priori estimate of Calabi, Yau and others on Kähler Einstein metrics, we will present an expository report of a priori estimates on the constant scalar curvature Kähler metrics. With this estimate, we prove the Donaldson conjecture on geodesic stability and the properness conjecture on Mabuchi energy functional.

This is a joint work with Cheng JingRui.

I will give an overview of my work on topological methods in condensed matter physics almost 40 years ago. Include will be Homotopy and $\operatorname{TKN}^2$ integers, holonomy and Berry's phase and quarternions and Berry's phase for Fermions. If time allows, I'll discuss supersymmetry and pairs of projections.

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

Emergence of anomalous transport laws in deterministic interacting many-body systems has become a subject of intense study in the past few years. One of the most prominent examples is the unexpected discovery of superdiffusive spin dynamics in the isotropic Heisenberg quantum spin chain with at half filling, which falls into the universality class of the celebrated Kardar-Parisi-Zhang equation. In this talk, we will theoretically justify why the observed superdiffusion of the Noether charges with anomalous dynamical exponent $z=3/2$ is indeed superuniversal, namely it is a feature of all integrable interacting lattice models or quantum field theories which exhibit globally symmetry of simple Lie group $G$, in thermal ensembles that do not break $G$-invariance. The phenomenon can be attributed to thermally dressed giant quasiparticles, whose properties can be traced back to fusion relations amongst characters of quantum groups called Yangians. Giant quasiparticles can be identified with classical solitons, i.e. stable nonlinear solutions to certain integrable PDE representing classical ferromagnet field theories on certain types of coset manifolds. We shall explain why these inherently semi-classical objects are in one-to-one correspondence with the spectrum of Goldstone modes. If time permits, we shall introduce another type of anomalous transport law called undular diffusion that generally occurs amongst the symmetry-broken Noether fields in $G$-invariant dynamical systems at finite charge densities.

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents, though we do not know how accurate it is. I will show that the low-energy properties of the Kitaev honeycomb lattice model, such as the shape of Majorana zero modes or the deformations of the correlation length, are faithfully described in terms of Riemann-Cartan geometry. Moreover, I will present how effective axial gauge fields can couple to Majorana fermions, thus giving a unified picture between vortices and lattice dislocations that support Majorana zero modes.

Inverse problems in imaging range from tomographic reconstruction (CT, MRI, etc) to image deconvolution, segmentation, and classification, just to name a few. In this talk I will discuss approaches to inverse imaging problems which have both a mathematical modelling (knowledge driven) and a machine learning (data-driven) component. Mathematical modelling is crucial in the presence of ill-posedness, making use of information about the imaging data, for narrowing down the search space. Such an approach results in highly generalizable reconstruction and analysis methods which come with desirable solutions guarantees. Machine learning on the other hand is a powerful tool for customising methods to individual data sets. Highly parametrised models such as deep neural networks in particular, are powerful tools for accurately modelling prior information about solutions. The combination of these two paradigms, getting the best from both of these worlds, is the topic of this talk, furnished with examples for image classification under minimal supervision and for tomographic image reconstruction.