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Seminarium KMMF "Teoria Dwoistości"

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Zapraszamy do sali 1.02, ul. Pasteura 5 o godzinie 10:15

Tomasz Komorowski (IM PAN)

On the Conversion of Work into Heat: Microscopic Models and Macroscopic Equations

Nature has a hierarchical structure with macroscopic behavior arising from the dynamics of atoms and molecules. The connection between different levels of the hierarchy is however not always straightforward, as seen in the emergent phenomena, such as phase transition and heat convection. Establishing in a mathematical precise way the connection between the different levels is the central problem of rigorous statistical mechanics. One of the methods leading to such results is to introduce some stochasticity inside the system.We summarise some of the results obtained recently concerning the derivation of the macroscopic heat equation from the microscopic behaviour of a harmonic chain with a stochastic perturbation. We focus our attention on the emergence of macroscopic boundary conditions. The results have been obtained in collaboration with Joel Lebowitz, Stefano Olla, Marielle Simon.

Zapraszamy do sali 1.02, ul. Pasteura 5 o godzinie 10:15

Piotr Szańkowski (IF PAN)

Double or nothing: a Kolmogorov extension theorem for quantum mechanics

The multitime probability distributions that describe the sequential measurements of a quantum observable generally violate Kolmogorov's consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory representing the observable. We show that, nonetheless, they do result from the sampling of one pair of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario.[1] D. Lonigro, F. Sakuldee, Ł. Cywiński, D. Chruściński, and P. Szańkowski, Quantum 8, 1447 (2024)

Zapraszamy do sali 1.02, ul. Pasteura 5 o godzinie 10:15

Oleksii Kotov (University of Hradec Králové)

Kantor algebra, Leibniz algebras and vector fields on the path space

The topic of this talk is the Kantor algebra as a natural extension of the free Lie (super)algebra. It will be shown how the Kantor algebra encodes information about Leibniz algebras and homotopy Leibniz algebras, and how it is embedded in the Lie algebra of (super) vector fields on the path (super)space.

Zapraszamy na spotkanie o godzinie 10:15

Alberto Cattaneo (University of Zurich)

Graded geometry and generalized reduction

Several differential geometric structures (e.g., Poisson, Courant, generalized complex) may be encoded as graded symplectic structures with additional data. Reduction of coisotropic submanifolds respecting the data yields reduction procedures for the corresponding geometric structure. This viewpoint unifies existing reduction procedures and produces new ones.Zoom link: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

Zapraszamy do sali 1.02, ul. Pasteura 5 o godzinie 10:15

Maciej Łebek (IFT)

Chapman-Enskog theory and Navier-Stokes equations for nearly integrable quantum gases

The Navier-Stokes equations are paradigmatic equations describinghydrodynamics of an interacting system with microscopic interactionsencoded in transport coefficients. An extremely important problem ofnon-equilibrium statistical physics is to show how Navier-Stokesequations arise from microscopic dynamics of particles. In my talk, Iwill present a partial solution to this problem known under the name ofChapman-Enskog theory. In the first part, I will revisit the results forclassical gases governed by the Boltzmann equation. In the second part,I will show how this framework can be generalised to a new class ofsystems, which are weakly perturbed integrable quantum gases.

Zapraszamy na spotkanie o godzinie 10:15

(Universite de Lille)

Bruhat-Poisson structure of the restricted Grassmannian and the KdV hierarchy

Alice Barbara Tumpach

We will consider particular examples of Poisson manifolds, namely Banach Poisson-Lie groups related to the Korteweg-de-Vries hierarchy.We construct a Banach Poisson-Lie group structure on the unitary restricted group, as well as on a Banach Lie group consisting of (a class of) upper triangular bounded operators. We show that the restricted Grassmannian inherite a Bruhat-Poisson structure from the unitary restricted group, and that the action of the triangular Banach Lie group on it by ``dressing transformations'' is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.

Zapraszamy do sali 1.02, ul. Pasteura 5 o godzinie 10:15

Serena Cenatiempo (Gran Sasso Science Institute)

Ground State Energy of Dilute Bose Gases: The Case of Hard Spheres

In recent years, there has been substantial progress in the mathematical understanding of the thermodynamic properties of dilute Bose gases. In particular, the validity of a celebrated formula for the second-order asymptotic of the ground state energy of dilute bosons — first predicted by Lee, Huang, and Yang in 1957 — has been fully established in the case of integrable (non-negative) interactions. In the first part of my talk, I will present the main ideas behind the recent results, the relevant length scales of the problem, and the open questions that lie ahead. I will then discuss how a simple trial state, introduced by Bijl-Dingle-Jastrow in the 1950s, can be used to derive an upper bound for the ground state energy of a dilute Bose gas of hard sphere, which captures the Lee-Huang-Yang expansion up to the order of the sub-leading correction. An upper bound that establishes the Lee-Huang-Yang formula for hard spheres is, in fact, still missing.

Zapraszamy do sali 1.02, ul. Pasteura 5 o godzinie 10:15

Pavlos Kassotakis (KMMF)

On quadrirational Yang-Baxter and pentagon maps

The Yang-Baxter and the pentagon equation serve as importantequations in mathematical physics. They appear in two equallysignificant versions, the operator and the set-theoretical one. In thistalk, we focus on the set-theoretic versions of both equations, wheretheir solutions are known as Yang-Baxter maps and pentagon maps,respectively. First, we recall rational Yang-Baxter maps of a specifictype (quadrirational maps) and show their connection to discreteintegrable systems. Then, we propose a classification scheme forquadrirational solutions of the pentagon equation. That is, we give afull list of representatives of quadrirational maps that satisfy thepentagon equation, modulo an equivalence relation that is defined onbirational functions on $\mathbb{CP}^1 \times \mathbb{CP}^1$. Finally,we demonstrate how Yang-Baxter maps can be derived from quadrirationalpentagon maps.