Seminars from until

Although practised as an art and science for ages, cryptography had to wait until the mid-twentieth century before Claude Shannon gave it a strong mathematical foundation. However, Shannon's approach was rooted in his own information theory, itself inspired by the classical physics of Newton and Einstein. When quantum theory is taken into account, new vistas open up both for codemakers and codebreakers. Is this a blessing or a curse for the protection of privacy? We shall discuss quantum cryptography, from its humble origins more than a half-century ago to its current blooming, and speculate about prospects for the future. No prior knowledge in cryptography or quantum theory will be assumed.

Biography

Brassard is renowned for his groundbreaking contributions to the field of quantum cryptography, including pioneering work in quantum teleportation, quantum entanglement distillation, quantum pseudo-telepathy, and the classical simulation of quantum entanglement. In collaboration with Charles H. Bennett, Brassard invented the BB84 protocol for quantum cryptography in 1984, which he later extended to include the Cascade error correction protocol. This protocol efficiently detects and corrects noise caused by eavesdropping on quantum cryptographic signals.

In recognition of his outstanding achievements, Brassard has received numerous prestigious awards. He was awarded the Prix Marie-Victorin, the highest scientific award of the government of Quebec, in 2000, and was elected as a Fellow of the International Association for Cryptologic Research in 2006. In 2010, he was awarded the Gerhard Herzberg Canada Gold Medal, Canada's highest scientific honour. Brassard was also elected a Fellow of the Royal Society of Canada and the Royal Society of London in 2013. In December of that year, he was named an Officer in the Order of Canada by the Governor-General of Canada, the Right Honourable David Johnston. In 2018, he received the Wolf Prize in Physics, and in 2019 he was awarded both the BBVA Foundation Frontiers of Knowledge Award in Basic Science and the Micius Quantum Prize.

Most recently, in September 2022, Brassard was awarded the Breakthrough Prize in Fundamental Physics, the world's largest science prize.

We present a full picture of the out of equilibrium joint fluctuations for current and occupation time in the symmetric exclusion process in dimension one. The main tools developed for that are a Kipnis-Varadhan type inequality necessary to handle the occupation time and a multiple point space time correlation estimate necessary to prove the tightness for the current. Curiously, as a corollary we obtain that, in equilibrium, current and occupation time are independent for any time (but they are not independent seen as processes).

Talk based on a joint work with D. Erhard (UFBA) and T. Xu (UFBA).

Even though mathematics may seem universal, this is not the case. Constructive mathematics, whose main proponent was Brouwer, offers a radically different perspective compared to classical mathematics rooted in the logic of Aristotle. In this talk, I shall discuss various definitions of computable reals and computable functions. We shall see that a definition of computable reals that appears very natural make multiplication by 3 uncomputable.

Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

The talk is based on joint work in progress with Anuk Dayaprema (UW-Madison).

Quandles are algebraic structures that play a prominent role in knot theory and also form a class of set-theoretic solutions of the Yang-Baxter equation. After a brief introduction to quandles, I will focus on recent developments and open problems, including the Hayashi conjecture and the isomorphism problem for principal quandles.

We will discuss a conjecture of Zygmund concerning maximal operators defined on a family of axis parallel rectangles in the Euclidean space. If the historical version of the problem has been disproved by Soria, we will see that the idea behind Zygmund's conjecture may still be true.

In particular, a certain reformulation of the problem has been solved in the Euclidean plane by Stokolos but it remains open in higher dimensions. In the past fews years, different authors (among which D'Aniello, Hagelstei, Oniani, Moonens, Rey, Stokolos etc.) have established sharp weak type estimates in specific settings and their work lend weight to a certain reformulation of Zygmund's conjecture.

We will discuss this problem and in particular, I would like to focus on a specific family of rectangles that exhibits a product structure.

Imagina que alguém está a tocar um tambor. Será possível determinar a forma do tambor, exclusivamente através do som? A área da matemática que trata deste tipo de perguntas chama-se geometria espectral. Nesta palestra, vamos explorar a ideia de ouvir o som produzido por objectos geométricos, começando por determinar o som de cordas e tambores rectangulares, com objetivo final de determinar o som da esfera. No processo, serão introduzidas algumas noções básicas de geometria diferencial e estudadas algumas aplicações, nomeadamente à física e ao processamento de imagem.

In this talk, I will introduce a few related microscopic models for active matter. The models we consider are lattice gases, meaning that the active particles jump stochastically on a lattice. Their active nature is represented by a drift in their stochastic jumps, whose direction can evolve in time as particles interact with eachother. I will discuss how, with this type of lattice gases, one can model the behavior of active matter, and recover the emergence of Vicsek's alignment phase transition as well as Motility Induced Phase Separation (MIPS), both classical phenomena for active matter. Both have been well documented by the physics community, however mathematical results remain scarce. Notably, using the mathematical theory of hydrodynamic limit, one can prove the emergence of both phenomena mathematically, even for models with purely local interactions, without any mean-field type assumptions. I will talk about recent results on phase separation occuring in a non gradient active gas, and how even small proportion of active particles can induce phase separation. Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. Based on JW with Mourtaza Kourbane Houssène, Julien Tailleur, Thierry Bodineau, James Mason, Maria Bruna, Robert Jack.

Causal representation learning (CRL) aims at learning causal factors and their causal relations from high-dimensional observations, e.g. images. In general, this is an ill-posed problem, but under certain assumptions or with the help of additional information or interventions, we are able to guarantee that the representations we learn are corresponding to some true underlying causal factors up to some equivalence class.

In this talk I will first present CITRIS, a variational autoencoder framework for causal representation learning from temporal sequences of images, in systems in which we can perform interventions. CITRIS exploits temporality and observing intervention targets to identify scalar and multidimensional causal factors, such as 3D rotation angles. In experiments on 3D rendered image sequences, CITRIS outperforms previous methods on recovering the underlying causal variables. Moreover, using pretrained autoencoders, CITRIS can even generalize to unseen instantiations of causal factors.

While CRL is an exciting and promising new field of research, the assumptions required by CITRIS and other current CRL methods can be difficult to satisfy in many settings. Moreover, in many practical cases learning representations that are not guaranteed to be fully causal, but exploit some ideas from causality, can still be extremely useful. As examples, I will describe some of our work on exploiting these "causality-inspired" representations for adapting policies across domains in RL and to nonstationary environments, and how learning a factored graphical representations (even if not necessarily causal) can be beneficial in these (and possibly other) settings.

Causal discovery is an active research field that aims to uncover the underlying causal mechanisms that drive the relationship between a collection of variables and which has applications in many areas, including medicine, biology, economics, and social sciences. In principle, identifying causal relationships requires interventions. However, intervening is often impossible, impractical, or unethical, which has stimulated much research on causal discovery from purely observational data or mixed observational-interventional data. In this talk, after overviewing the causal discovery field, I will discuss some recent advances, namely on causal discovery from data with latent interventions and on what is the quintessential causal discovery problem: distinguishing the cause from the effect on a pair of dependent variables.

Consistent Kaluza-Klein truncations to maximal supergravities in $D=2$ spacetime dimensions remain "terra incognita". Using techniques based on exceptional geometry, I will present the scalar potential of all gauged supergravities that admit a consistent embedding in ten or eleven dimensions. This provides the first general expression for a multitude of theories with an interesting structure of vacua, covering potentially many new $AdS_2$ cases. As a concrete example, I will discuss the consistent truncation of IIA supergravity on the eight-sphere to $SO(9)$ gauged maximal supergravity. Fluctuations around its supersymmetric $SO(9)$-invariant vacuum holographically describe the dynamics of interacting $D0$-branes.

Recently, there has been dramatic progress in nonequilibrium thermodynamics, with diverse applications in biological and chemical systems. The central quantity of interest in the field is “entropy production” (EP), which reflects the increase of the entropy of a system and its environment. Major questions of interest include (1) quantitative tradeoffs between EP and performance measures like speed and precision, (2) inference of EP from data, and (3) decomposition of EP into contributions from different sources of dissipation. In this work, we study the thermodynamics of nonequilibrium processes by considering the information geometry of fluxes. Our approach can be seen as a dynamical generalization of existing work on the information geometry of probability distributions considered at a given instant in time. It is applicable to a broad range of nonequilibrium processes, including nonlinear ones that exhibit oscillations and/or chaos, and it has implications for thermodynamic tradeoffs, thermodynamic inference, and decompositions of EP. As one application, we derive a universal decomposition of EP into “excess” and “housekeeping” contributions, representing contributions from nonstationarity and cyclic fluxes respectively.

Joint work with Andreas Dechant, Kohei Yoshimura, Sosuke Ito. arXiv:2206.14599

This will be a talk touching on a few themes within analysis and differential equations. We will address the problem of estimating the operator norm of certain embeddings between spaces of entire functions (called Paley-Wiener spaces). The quest for such operator norms can be equivalently thought as a Fourier uncertainty principle for bandlimited functions. We provide precise asymptotics in general, and identify sharp constants in certain particular cases via techniques from reproducing kernel Hilbert spaces. Applications to sharp higher order Poincaré inequalities and other related extremal problems will be discussed. The talk will be accessible to a broad audience.