# Seminários de até

Segunda feira

## QM3 Matéria Quântica & Matemática

, University of Cambridge.

Resumo

Chiral fermions have the property that their left-handed and right-handed components transform differently under some symmetry. Folklore suggests that it is impossible to give such fermions a mass without breaking this symmetry. I'll show, through a number of examples, why this folklore is wrong. In particular, I'll show how one generation of fermions in the Standard Model can get a mass without the need for a Higgs boson that breaks electroweak symmetry.

Terça feira

## Geometria em Lisboa

, Università di Trento.

Resumo

Character varieties parametrise representations of the fundamental group of a curve. In general these moduli spaces are singular, therefore it is customary to slightly change the moduli problem and consider smooth analogues, called twisted character varieties. In this setting, the P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In a joint work with M. Mauri we establish (and in some cases formulate) analogous P=W phenomena in the singular case.

In particular we show that the P=W conjecture holds for character varieties which admit a symplectic resolution, namely in genus 1 and arbitrary rank and in genus 2 and rank 2. In the talk I will first mention basic notions of non abelian Hodge theory and introduce the P=W conjecture for smooth moduli spaces; then I will explain how to extend these phenomenas to the singular case, showing the proof our results in a specific example.

Quinta feira

## Lisbon WADE — Webinar em Análise e Equações Diferenciais

, Western Washington University.

Resumo

We study almost minimizer for functionals that yield a free boundary, as in the work of Alt-Caffarelli and Alt-Caffarelli-Friedman. The almost minimizing property can be understood as the defining characteristic of a minimizer in a problem that explicitly takes noise into account. In this talk, we discuss the regularity of almost minimizers to energy functionals with variable coefficients. This is joint work with Guy David, Max Engelstein & Tatiana Toro.

Sexta feira

## Matemática, Física & Aprendizagem Automática

, Georgia Institute of Technology.

Resumo

Cyber-physical systems (CPS) comprise interacting digital, analog, physical, and human components engineered for function through integrated physics and logic. Incorporating intelligence in CPS, however, makes their physical components more exposed to adversaries that can potentially cause failure or malfunction through actuation attacks. As a result, augmenting CPS with resilient control and design methods is of grave significance, especially if an actuation attack is stealthy. Towards this end, in the first part of the talk, I will present a receding horizon controller, which can deal with undetectable actuation attacks by solving a game in a moving horizon fashion. In fact, this controller can guarantee stability of the equilibrium point of the CPS, even if the attackers have an information advantage. The case where the attackers are not aware of the decision-making mechanism of one another is also considered, by exploiting the theory of bounded rationality. In the second part of the talk, and for CPS that have partially unknown dynamics, I will present an online actuator placement algorithm, which chooses the actuators of the CPS that maximize an attack security metric. It can be proved that the maximizing set of actuators is found in finite time, despite the CPS having uncertain dynamics.

Sexta feira

## Teoria Quântica do Campo Topológica

, Stellenbosch University.

Resumo
The classical (resp. quantum) 6j symbols are real numbers which encode the associator information for the tensor category of representations of SU(2) (resp. the quantum group of SU(2) at level k). They form the building blocks for the Turaev-Viro 3-dimensional TQFT. I will review the intriguing asymptotic formula for these symbols in terms of the geometry of a Euclidean tetrahedron (in the classical case) or a spherical tetrahedron (in the quantum case), due to Ponzano-Regge and Taylor-Woodward respectively. There is a wonderful integral formula for the square of the classical 6j symbols as a group integral over SU(2), and I will report on investigations into a similar conjectural integral formula for the quantum 6j symbols. In the course of these investigations, we observed and proved a certain reciprocity formula for the Wigner derivative for spherical tetrahedra. Joint with Hosana Ranaivomanana.

Segunda feira

## Teoria de Cordas

Daniel Waldram, Imperial College London.

Resumo

“G-algebroids” are natural extension of Lie and Courant algebroids that give a unified picture of the symmetries that underlie generalised and exceptional geometry as well as new “non-exact” versions. We analyse their structure in the exceptional case, and translate the problem of finding maximally supersymmetric consistent truncations to a relatively simple algebraic condition. We then show how Poisson-Lie U-duality is encoded in this framework and prove, in particular, that it is compatible with the supergravity equations of motion.

Segunda feira

## QM3 Matéria Quântica & Matemática

, Universität Tübingen.

Resumo

Open quantum systems composed of atoms interacting with light exhibit behaviour that is akin to that of associative memories [1]. This means that they possess stationary states that can be interpreted as stored memory patterns, which are retrieved when the initial state is inside the basin of attraction of a given pattern [2]. The corresponding pattern retrieval dynamics can be observed in actual experimental settings. In these experiments atoms are confined within an optical cavity whose photons mediate long-range interactions [3]. Stored patterns are encoded in the atom-light coupling constants. This setting offers an interesting opportunity for studying quantum generalisations of associative memories and stored (quantum) patterns in this context [4]. Moreover, it allows to systematically construct scenarios in which quantum effects might be beneficial, e.g., for speeding up the pattern retrieval process [5]. I will talk about recent research of my group on this subject, which builds a bridge between classic machine learning concepts, such as the Hopfield Neural Network, and most recent experimental manifestations of synthetic quantum matter.

[1] E. Fiorelli et al., Physical Review Letters 125, 070604 (2020)

[2] F. Carollo and I. Lesanovsky, arXiv:2009.13932 (2020)

[3] V.D. Vaidya et al., Physical Review X 8, 011002 (2018)

[4] P. Rotondo et al., Journal of Physics A 51, 115301 (2018)

[5] E. Fiorelli et al., Physical Review A 99, 032126 (2019)

Terça feira

## Geometria em Lisboa

Alberto Abbondandolo, Ruhr Universität Bochum.

Resumo

The prototypical question in metric systolic geometry is to bound the length of a shortest closed geodesic on a closed Riemannian manifold by the volume of the manifold. This question has been extensively studied for non simply connected manifolds, but in the recent years there has been some progress also for simply connected manifolds, on which closed geodesics cannot be found simply by minimizing the length. This progress involves extending systolic questions to Reeb flows, a class of dynamical systems generalising geodesic flows. On the one hand, this extension and the use of symplectic techniques provide some answers to classical questions within metric systolic geometry. On the other hand, new questions arise from the more general setting and relate seemingly distant fields such as the study of rigidity properties of symplectomorphisms and the integral geometry of convex bodies. I will give a non-technical panoramic view of some of these recent developments.

Quarta feira

## Matemática Aplicada e Análise Numérica

Henrique Oliveira, Instituto Superior Técnico, Department of Mathematics and CMAGSD.

Resumo

In this talk we overview the mathematical continuous and discrete models in use in Mathematical epidemiology. We analyse the evolution of COVI-19 in Portugal.

Quinta feira

## Lisbon WADE — Webinar em Análise e Equações Diferenciais

Online

Julio D. Rossi, Universidad de Buenos Aires.

Resumo

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized $p-$Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game.

Our goal in this talk is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Ampère equation.

Joint work with P. Blanc (Jyvaskyla), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh).

Sexta feira

## Matemática, Física & Aprendizagem Automática

, Universitat de València.

Resumo

Most problems in Earth sciences aim to do inferences about the system, where accurate predictions are just a tiny part of the whole problem. Inferences mean understanding variables relations, deriving models that are physically interpretable, that are simple parsimonious, and mathematically tractable. Machine learning models alone are excellent approximators, but very often do not respect the most elementary laws of physics, like mass or energy conservation, so consistency and confidence are compromised. I will review the main challenges ahead in the field, and introduce several ways to live in the Physics and machine learning interplay that allows us (1) to encode differential equations from data, (2) constrain data-driven models with physics-priors and dependence constraints, (3) improve parameterizations, (4) emulate physical models, and (5) blend data-driven and process-based models. This is a collective long-term AI agenda towards developing and applying algorithms capable of discovering knowledge in the Earth system.

Sexta feira

## Teoria Quântica do Campo Topológica

Online

, International Institute of Physics.

Terça feira

## Geometria em Lisboa

Carolina Araujo, Instituto de Matemática Pura e Aplicada.

Resumo

Fano manifolds, i.e., complex projective manifolds having positive first Chern class, play a key role in higher dimensional algebraic geometry. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and describe special geometric features of these manifolds.

Sexta feira

## Matemática, Física & Aprendizagem Automática

, Google Research, Brain team, Paris.

Sexta feira

## Teoria Quântica do Campo Topológica

, University of Heidelberg.

Segunda feira

## Teoria de Cordas

Syo Kamata, National Centre for Nuclear Research, Warsaw.

Resumo

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with $N$ minima on $S^{1}$. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of $N$-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with 't Hooft anomaly for even $N$ and global inconsistency for odd $N$. By using Delabaere Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

This talk is based on:

1. On exact-WKB analysis, resurgent structure, and quantization conditions, N.Sueishi, S.K, T.Misumi, and M.Ünsal, arXiv:2008.00379.
2. Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$, N.Sueishi, S.K, T.Misumi, and M.Ünsal, arXiv.2103.06586

Sexta feira

## Matemática, Física & Aprendizagem Automática

, University of Washington.

Segunda feira

## QM3 Matéria Quântica & Matemática

, Max-Planck-Institute of Quantum Optics.

Sexta feira

## Matemática, Física & Aprendizagem Automática

, Imperial College London.

Resumo

This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. We show that this class leads to mixed-integer formulations for trained ReLU neural networks. The approach balances model size and tightness by partitioning node inputs into a number of groups and forming the convex hull over the partitions via disjunctive programming. At one extreme, one partition per input recovers the convex hull of a node, i.e., the tightest possible formulation for each node. For fewer partitions, we develop smaller relaxations that approximate the convex hull, and show that they outperform existing formulations. Specifically, we propose strategies for partitioning variables based on theoretical motivations and validate these strategies using extensive computational experiments. Furthermore, the proposed scheme complements known algorithmic approaches, e.g., optimization-based bound tightening captures dependencies within a partition.

This joint work with Calvin Tsay, Jan Kronqvist, Alexander Thebelt is based on two papers (https://arxiv.org/abs/2102.04373, https://arxiv.org/abs/2101.12708).

Sexta feira

## Teoria Quântica do Campo Topológica

, Dalhousie University and Perimeter Institute.

Resumo

Each fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter.

Segunda feira

## Teoria de Cordas

Sven Krippendorf, LMU Munich.

Resumo

Determining whether a dynamical system is integrable is generally a difficult task which is currently done on a case by case basis requiring large human input. Here we propose and test an automated method to search for the existence of relevant structures, the Lax pair and Lax connection respectively. By formulating this search as an optimization problem, we are able to identify appropriate structures via machine learning techniques. We test our method on standard systems of classical integrability and find that we can distinguish between integrable and non-integrable deformations of a system. Due to the ambiguity in defining a Lax pair our algorithm identifies novel Lax pairs which can be easily verified analytically.

Segunda feira

## QM3 Matéria Quântica & Matemática

, Beijing Computational Science Research Center.

Terça feira

## Geometria em Lisboa

, University of Toronto.

Resumo

Together with Jihyeon Jessie Yang, we are resurrecting an old idea of Raoul Bott for using large torus actions to construct canonical bases for unitary representations of compact Lie groups. Our methods are complex analytic; we apply them to families of Bott-Samelson manifolds parametrized by C^n. Our construction requires the vanishing of higher cohomology of sheaves of holomorphic sections of certain line bundles over the total spaces of such families; this vanishing is conjectural, hence the question mark in the title.

Sexta feira

## Teoria Quântica do Campo Topológica

, Mathematical Institute, University of Oxford.

Resumo

This is a report on joint work in progress with L. Katzarkov, M. Kontsevich, and P. Pandit. The Homological Mirror Symmetry conjecture is stated as an equivalence of triangulated categories, one coming from algebraic geometry and the other from symplectic topology. An enhancement of the conjecture also identifies stability conditions (in the sense of Bridgeland) on these categories. We adopt the point of view that triangulated (DG/A-infinity) categories are non-commutative spaces of an algebraic nature. A stability condition can then be thought of as the analog of a Kähler class or polarization. Many, often still conjectural, constructions of stability conditions hint at a richer structure which we think of as analogous to a Kähler metric. For instance, a type of Donaldson and Uhlenbeck-Yau theorem is expected to hold. I will discuss these examples and common features among them, leading to a tentative definition.

Segunda feira

## Teoria de Cordas

Ingmar Saberi, University of Heidelberg.

Resumo

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated).

Quarta feira

## Probabilidades e Estatística

, Department of Statistical Science, University College London.

Quarta feira

## Webinar de Teoria de Valores Extremos da UL

, Department of Statistical Science, University College London.

Instituto Superior Técnico
Av. Rovisco Pais, Lisboa, PT