A modern perspective on symmetry in quantum theories identifies the topological invariance of a symmetry operator within correlation functions as its defining property. In addition to suggesting generalised notions of symmetry, this viewpoint enables a calculus of topological defects, which has a strong category-theoretic flavour, that leverages well-established methods from topological quantum field theory. Focusing on finite symmetries, I will delve during these lectures into a realisation of this program in the context of one-dimensional quantum lattice models. Concretely, I will present a framework for systematically investigating lattice Hamiltonians, elucidating their symmetry operators, defining duality/gauging transformations and computing the mapping of topological sectors through such transformations. Moreover, I will comment on the classification of gapped symmetric phases for generalised symmetry and the construction of the corresponding order/disorder parameters. I will provide explicit treatments of familiar physical systems from condensed matter theory, shedding light on celebrated results and offering resolutions to certain open problems. Time permitting, I will briefly touch upon generalisations to higher dimensions and implications to numerical simulations.   

A modern perspective on symmetry in quantum theories identifies the topological invariance of a symmetry operator within correlation functions as its defining property. In addition to suggesting generalised notions of symmetry, this viewpoint enables a calculus of topological defects, which has a strong category-theoretic flavour, that leverages well-established methods from topological quantum field theory. Focusing on finite symmetries, I will delve during these lectures into a realisation of this program in the context of one-dimensional quantum lattice models. Concretely, I will present a framework for systematically investigating lattice Hamiltonians, elucidating their symmetry operators, defining duality/gauging transformations and computing the mapping of topological sectors through such transformations. Moreover, I will comment on the classification of gapped symmetric phases for generalised symmetry and the construction of the corresponding order/disorder parameters. I will provide explicit treatments of familiar physical systems from condensed matter theory, shedding light on celebrated results and offering resolutions to certain open problems. Time permitting, I will briefly touch upon generalisations to higher dimensions and implications to numerical simulations.   

We start with reminding  basic molecular structures (Crick dogma, genetic code etc.) in living  entities and classical  examples of the mathematical thought in genetic (Darwin, Mendel, Morgan, Sturtevant, trees of sequences…) and the traditional discussion/controversy  on the nature of Life. Then we present a mathematical counterpart to the question “What is Life?”, indicate possible role of mathematics in the future bioengineering and conclude with the current  and projected numerical data on the human role in ecology of Earth.

We start with reminding  basic molecular structures (Crick dogma, genetic code etc.) in living  entities and classical  examples of the mathematical thought in genetic (Darwin, Mendel, Morgan, Sturtevant, trees of sequences…) and the traditional discussion/controversy  on the nature of Life. Then we present a mathematical counterpart to the question “What is Life?”, indicate possible role of mathematics in the future bioengineering and conclude with the current  and projected numerical data on the human role in ecology of Earth.

We start with reminding  basic molecular structures (Crick dogma, genetic code etc.) in living  entities and classical  examples of the mathematical thought in genetic (Darwin, Mendel, Morgan, Sturtevant, trees of sequences…) and the traditional discussion/controversy  on the nature of Life. Then we present a mathematical counterpart to the question “What is Life?”, indicate possible role of mathematics in the future bioengineering and conclude with the current  and projected numerical data on the human role in ecology of Earth.

We start with reminding  basic molecular structures (Crick dogma, genetic code etc.) in living  entities and classical  examples of the mathematical thought in genetic (Darwin, Mendel, Morgan, Sturtevant, trees of sequences…) and the traditional discussion/controversy  on the nature of Life. Then we present a mathematical counterpart to the question “What is Life?”, indicate possible role of mathematics in the future bioengineering and conclude with the current  and projected numerical data on the human role in ecology of Earth.

Seminar on  Quantum and Modularity ResurgenceIn this talk, I will review the exact WKB approach used in the QFT/ODE correspondence related to the NS phase of the Ω-background. By AGT correspondence, those QFT’s are related to c = ∞ CFTs. In particular, I will focus on the Stokes graph, also known as the spectral network in physics. The spectral network plays an essential role in the exact WKB and the nonabelianization of SL(N,C) flat connections. We find that the very same structure also exists at the self dual phase of the Ω-background, which is the c = 1 Liouville CFT by AGT correspondence. I will introduce our work on nonabilianization of the Virasoro conformal blocks using Heisenberg conformal blocks with the key ingredient of spectral networks. This is analogous to the nonabelianization of SL(2, C) flat connections by the GL(1, C) connections. This is a joint work in progress with Andrew Neitzke.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Einstein’s path to the discovery of General Relativity, from 1907 to November 1915, will be described. A particular emphasis will be given to the multi-pronged character of Einstein’s strategy.Supported by the « 2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects », awarded to Thibault Damour

Numerical General Relativity is the art of solving Einstein’s Field Equations with computational methods. These lectures will review the formalism currently employed for astrophysical simulations in strong-gravity, notably comprising compact binaries, gravitational collapse and gravitational waves. They will cover the following topics: 3+1 formulation of Einstein’s equations, the initial data problem, the evolution problem, the gauge choice and the extraction of gauge-invariant quantities. A selection of breakthrough results for gravitational-wave astronomy will also be presented.Supported by the « 2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects », awarded to Thibault Damour

Numerical General Relativity is the art of solving Einstein’s Field Equations with computational methods. These lectures will review the formalism currently employed for astrophysical simulations in strong-gravity, notably comprising compact binaries, gravitational collapse and gravitational waves. They will cover the following topics: 3+1 formulation of Einstein’s equations, the initial data problem, the evolution problem, the gauge choice and the extraction of gauge-invariant quantities. A selection of breakthrough results for gravitational-wave astronomy will also be presented.Supported by the « 2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects », awarded to Thibault Damour

Seminar on  Quantum and Modularity ResurgenceThese are the first talks of the series of seminars on Resurgence and Quantum Modularity, and they are meant to be an introduction to the main topics of the series, namely resurgence and quantum modular forms. The first talk will introduce the notion of Borel and Laplace transforms, resurgent structures and the median resummation. The second talk will go through modular forms and their asymptotics leading to mock and then quantum modular forms.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Exponential Networks are a useful tool to count BPS states in local Calabi-Yau 3-folds. In this talk we apply Exponential Networks to D0-D4 bound states in flat space. This leads us to an explicit correspondence between torus fixed points of the Hilbert scheme of points on ${mathbb C}^2$ and anomaly free finite webs attached to the quadratically framed pair of pants. This can be viewed as an A-model description of the ADHM construction.