Wydział Fizyki UW > Badania > Seminaria i konwersatoria > Seminarium KMMF "Teoria Dwoistości"

Seminarium KMMF "Teoria Dwoistości"

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Zapraszamy do sali 1.03, ul. Pasteura 5 o godzinie 10:15

Maciej Zworski (UC Berkeley)

Mathematics of magic angles in 2D structures

Magic angles refer to a remarkable theoretical(Bistritzer--MacDonald, 2011) and experimental (Cao et al 2018)discovery, that two sheets of graphene twisted by certain (magic)angles display unusual electronic properties such assuperconductivity.Mathematically, this is related to having flat bands of nontrivialtopology for the corresponding periodic Hamiltonian and theirexistence was shown in the chiral model of twisted bilayer graphene(Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterizationof magic angles (Becker--Embree--Wittsten--Z, 2021, Galkowski--Z,2023) also produces complex values and the distribution of theirreciprocals looks remarkably like a distribution of scatteringresonances, with the real magic angles corresponding to anti-boundstates. I will provide a gentle introduction to the subject andhighlight some recent results.The talk is based on joint works with S Becker, M Embree, J Galkowski,M Hitrik, T Humbert, Z Tao, J Wittsten and H Zeng.

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

Adam Kaliński (FUW)

Unified approach to hypergeometric class functions

During the presentation, I will discuss a new approach to hypergeometricclass equations. Instead of analyzing each equation separately, we canexamine the general concept of Miller's Lie algebra and construct afamily of differential operators. By studying the algebraic propertiesof these operators, we can derive properties for the entire family ofequations.This approach provides a unified framework for understanding andderiving the properties of hypergeometric class equations. In the secondpart of the presentation, I will present some examples that illustratethis approach.The presentation is based on an unpublished paper by Professor JanDereziński.

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

Erik Skibsted (Aarhus University)

Stationary completeness: the N-body short-range case

For a general class of N-body Schrödinger operators with short-rangepair-potentials the wave and scattering matrices as well as the restricted wave operators are quantities given at non-threshold energies. From previous work by Yafaev they are known to exist and to depend weakly continuously on the energy parameter. In the present work we introduce the notion of stationary complete energies and show that all non-threshold energies are stationary complete. It is a consequence of this new result that the above scattering quantities depend strongly continuously on the energy parameter. Another new result is that the scattering matrix is unitary at any non-threshold energy. As side results we obtaina purely stationary proof of asymptotic completeness for N -body short-range systems as well as top-order (far distance) resolvent asymptotics.

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

Iwona Chlebicka (MIMUW)

Limiting Approximation in the Calculus of Variations

Critical points of variational functionals describe a widerange of real life phenomena. The typical approach to tackle them is viathe limits of families of approximate problems. I would like to discussapproximation in unconventional function spaces that is useful in thecalculus of variations, focusing on when and why it may be impossible. Iwill also show when it is feasible and how to use it.

Zapraszamy na spotkanie o godzinie 10:15

Boris Kolev (ENS-Paris-Saclay)

Poisson brackets in Hydrodynamics

Different Poisson structures have been proposed to give a Hamiltonianformulation to evolution equations issued from fluid mechanics. In thistalk, I will investigate the main brackets which appear in theliterature and discuss the difficulties which arise when one tries togive a rigorous meaning to these brackets. In the first part of my talk,I will recall basic material on Poisson brackets on a finite dimensionalmanifold and present the difficulties which arise when one extend theseconcepts to infinite dimensional manifolds. In the second part of mytalk, I will focus on the problem of defining a valid and usable bracketto study rotational fluid flows with a free boundary. I will discusssome results which have emerged in the literature to solve thedifficulties which arise and conclude that (up to my knowledge) the mainproblems are still open.To attend our online seminar please use the Zoom identificator:https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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

Michał Oszmaniec (CFT PAN)

Saturation and recurrence of quantum complexity in random local quantum dynamics

Quantum complexity is a measure of the minimal number of elementary operations required to approximately prepare a given state or unitary channel. Recently, this concept has found applications beyond quantum computing -- in studying the dynamics of quantum many-body systems and the long-time properties of AdS black holes. In this context Brown and Susskind conjectured that the complexity of a chaotic quantum system grows linearly in time up to times exponential in the system size, saturating at a maximal value, and remaining maximally complex until undergoing recurrences at doubly-exponential times. In this work we prove the saturation and recurrence of complexity in two models of chaotic time evolutions based on (i) random local quantum circuits and (ii) stochastic local Hamiltonian evolution. Our results advance an understanding of the long-time behaviour of chaotic quantum systems and could shed light on the physics of black hole interiors. From a technical perspective our results are based on a quantitative connection between spectral gaps of random walks on the unitary group and the property of approximate equidistribution, which turns out to be crucial for establishing saturation and recurrence.The talk is based on a joint work with Marcin Kotowski, Nick Hunter Jones and Michał Horodecki, preprint arXiv:2205.09734 (accepted for publication in Physical Review X).

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

Norbert Mokrzański (KMMF)

The Bogoliubov-Bose-Hubbard model: existence of superfluid-Mott insulator phase transition

We consider a variational approach to the Bose-Hubbard model based onthe Bogoliubov theory. We introduce the free energy functional for whichwe prove the existence of minimizers. By analyzing their structure weshow the existence of a superfluid--Mott insulator phase transition atpositive temperatures. We also show that this model does not exhibit aquantum phase transition. Joint work with Marcin Napiórkowski.

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.