Faculty of Physics University of Warsaw > Events > Seminars > "Theory of Duality" (KMMF) Seminar

"Theory of Duality" (KMMF) Seminar

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Piotr Minakowski (Otto von Guericke University Magdeburg)

Error Estimates for Physics-Inspired Neural Network Solutions of Partial Differential Equations

We develop an error estimator for neural network approximations of PDEs.The proposed approach is based on the dual weighted residual method (DWR). It is destined to serve as stopping criterion, that guarantees the accuracy ofthe solution independently of the design of the neural network training. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Enderalp Yakaboylu (MPI Garching)

Molecular Impurities as a Realization of Anyons on the Two-Sphere

tudies on experimental realization of two-dimensional anyons in terms of quasiparticles have been restricted, so far, to only anyons on the plane. It is known, however, that the geometry and topology of space can have significant effects on quantum statistics for particles moving on it. In this talk, I will show that the emerging fractional statistics for particles restricted to move on the sphere, instead of on the plane, arises naturally in the context of quantum impurity problems. In particular, I will demonstrate a setup in which the lowest-energy spectrum of two linear bosonic molecules immersed in a quantum many particle environment can coincide with the anyonic spectrum on the sphere. This paves the way towards experimental realization of anyons on the sphere using molecular impurities. Finally, I will present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as q-deformed Lie algebras. In particular, I will exemplify this approach by considering a quantum rotor interacting with a bath of bosons. . To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Katarzyna Grabowska (KMMF)

Information geometry in the language of Lie groupoids and Lie algebroids

Information geometry studies statistical-probabilistic models equipped with the structure of a differentiable manifold. We can think of such a model as a family of probabilistic distributions with parameters that play the role of coordinates on a manifold. The structure of a statistical manifold is encoded in the function called the divergence that provides the manifold with the metric, the skewness tensor and connections. During the talk I will try to show that all the above structures can be naturally introduced and investigated in the language of Lie groupoids and Lie algebroids. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Li Ben (Ningbo University)

Affine invariant maps for log-concave functions

Affine invariant points and maps for sets were introduced by Grunbaum to study the symmetry structure of convex sets. I will introduce these notions to a functional setting. We will show some typical examples for affine invariant points. Moreover, I will briefly connect this notion to recent progress on functionalization of convex geometry. This is a joint work with Schuett and Werner. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Christiaan Jozef Farielda Van De Ven (University of Trento)

Strict deformation quantization: a C*-algebraic approach to the classical limit of quantum systems

The concept of a strict deformation quantization provides a mathematical formalism that describes the transition from a classical theory to a quantum theory in terms of deformations of (commutative) Poisson algebras (representing the classical theory) into non-commutative C* -algebras (characterizing the quantum theory). In the first part of this talk we introduce the definitions, give several examples and show how quantization of the closed unit 3-ball B^3 in R^3 is related to quantization of its smooth boundary (i.e. the two-sphere S^2). In the second part we apply this formalism to prove the existence of the 'classical limit' of mean-field quantum spin systems and Schrodinger operators yielding a probability measure on the pertinent phase space. We moreover discuss the concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by taking a limit in the relevant semiclassical parameter. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Theotime Girardot (Universite Grenoble Alpes)

Semiclassical limit for almost fermionic anyons

In two-dimensional space there are possibilities for quantum statistics contin-uously interpolating between the bosonic and the fermionic one. Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions. In the first part of the talk we will see how such particles arise theoretically and experimentally.In the second part we will study a limit situation where the statistics/magnetic interaction is seen as a “perturbation from the fermionic end”. We vindicate a mean-field approximation, proving that the ground state of a gas of anyons is described to leading order by a semi-classical, Vlasov-like, energy functional. The ground state of the latter displays anyonic behavior in its momentum distribution. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Alberto Ibort (UC3M)

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

The dynamical evolution of classical and quantum states will be reviewed in the groupoidal frame of Schwinger’s picture of Quantum Mechanics. A natural relation between the classical Hamiltonian description and the quantum one will be discussed. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Daniel Peralta and Alberto Enciso (ICMAT)

Knotted vortex structures in incompressible fluids: deterministic and probabilistic aspects

In these lectures we shall report on some recent results on the geometry of the vortex structures of the stationary solutions of the incompressible Euler equations. In particular, we will consider the following problems: do there exist steady Euler flows exhibiting vortex tubes of arbitrary knot type? What is the typical stream line complexity of a Beltrami flow? Both deterministic and probabilistic aspects ofthese questions will be analyzed. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Julien Sabin (Ecole Polytechnique)

Weyl laws for singular potentials

Weyl laws give a precise description of the high energy behaviour of eigenvalues and eigenfunctions of Schrödinger operators describing trapped quantum particles (non-interacting fermions). In this talk I will recall these laws, their applications and their history. I will also describe a recent work done in collaboration with Rupert Frank (Caltech/Munich) that explains how these laws are modified when introducing a singular external potential. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Zohreh Ravanpak (IMPAN)

How to construct discrete mechanics on octonions?

The geometric description of the Euler-Lagrange equations of a mechanical system determined by a Lagrangian function relies on the velocity phase space TM of a configuration manifold M. In discrete mechanics, the starting point is to replace TM by M \times M, taking two nearby points as the discrete analogue of a velocity vector. Discrete mechanics has been developed on Lie groups and Lie groupoids as well. My talk is about the generalization of the discrete mechanics on Lie groups to non-associative objects, smooth loops, generalizing Lie groups. This shows that the associativity assumption is not crucial for mechanics and opens new perspectives. The motivating example is octonions, I will show how to construct the discrete Lagrangian and Hamiltonian mechanics on unitary octonions. To attend our seminar, please use the Zoom link: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Chiara Saffirio (University of Basel)

Semiclassical limit from the Hartree-Fock equation to the Vlasov-Poisson system

We consider the semiclassical limit from the Hartree-Fock equation to the Vlasov equation with general singular interaction potentials, including the Coulomb and gravitational interactions. We show that the convergence holds in the strong topologies of Schatten norms for a large class of initial data and provide explicit bounds (based on a joint work with L. Laflèche). To attend the seminar, please use the following Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Manuel de Leon (ICMAT-CSIC and Real Academia de Ciencias)

Contact Hamiltonian systems: applications to thermodynamics

In the first part of this lecture we will introduce the notion of contact Hamiltonian systems and their dissipative properties. In the second part we will apply this formalism to the study of thermodynamic systems. Indeed, by means of the Jacobi structure associated with a contact structure, we use the so-called evolution vector field to propose a new characterization of isolated thermodynamic systems with friction, a simple but important class of thermodynamic systems which naturally satisfy the first and second laws of thermodynamics, i.e. total energy preservation of isolated systems and non-decreasing total entropy, respectively. We also consider more general kinds of thermodynamic systems, which are described by at least two thermal variables and exchange heat between its components. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Marcin Lis (University of Viena)

Free boundary dimers: random walk representation and scaling limit

The standard dimer model is a uniform probability measure on the space ofperfect matchings of a graph, i.e., sets of edges such that each vertex is incident on exactly one edge. It was introduced as a model of adsorption of diatomic molecules on a surface of a crystal, and its correlations exhibit free fermionic statistics. In two dimensions, one can define an associated height function which naturally models a ‘’uniform'' random surface (with specified boundary conditions). Moreover the model can be solved exactly which in particular means that its correlations are given by the entries of the inverse Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon who proved, already 20 years ago, that the scaling limit of the height function in bounded domains approximated by the square lattice with vanishing mesh is the Dirichlet (or zero boundary conditions) Gaussian free field (or the continuum free boson). This was the first mathematically rigorous example of conformal invariance in planar statistical mechanics. In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched. This is the so-called free boundary dimer model. This modification complicates the classical analysis in several ways and I will discuss how to circumvent the arising obstacles.In the end, the main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane approximated by the square lattice with vanishing meshis the Neumann (or free boundary conditions) Gaussian free field. This is based joint work with Nathanael Berestycki (Vienna) and Wei Qian (Paris). To attend, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Anton Alexeev (University of Geneva)

Coadjoint orbits of Virasoro algebra and equivariant localization

In the first part of the talk, I'll review the classical theory of coadjoint orbits including the Kirillov-Kostant-Souriau symplectic structure, Duistermaat-Heckman oscillating integrals and equivariant localization. In the second part, I'll talk about coadjoint orbits of Virasoro algebra, the Schwarzian action and infinite dimensional orbital integrals. If time permits, we will touch upon recent applications of coadjoint orbits to ideas of holography in physics. The talk is based on a joint work with S. Shatashvili. To attend our online seminar, use the link: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Mirosław Lachowicz (MIMUW)

Równania różniczkowo-całkowe i piękne katastrofy

Integro-differential Equations and Beautiful Catastrophes

Pokażę, że wybuchy rozwiązań, traktowane często przez matematyków, jako coś złego, mogą w rzeczywistości opisywać różnego typu samoorganizację - ,,pozytywna'' (np. wyzdrowienie), lub ,,negatywną'' (polaryzacja społeczeństwa). Matematycznie jest to teoria równań różniczkowo-całkowych zastosowana do zagadnień z nauk społecznych (ustalanie się opinii), ekonomii (zagadnienie ,,cytryn i wisienek''), biologii (denaturacja DNA), medycyny (gojenie się zerwanych ścięgien) i rozmieszczenie osób w windzie. Aby brać udział na seminar, prosimy o skorzystanie z następnego link do Zooma: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

I am going to show, that blow-ups of solutions, that usually are treated as something very bad, can in fact describe some self-organization phenomena, "positive" (like healing) or "negative" (like society polarization). Mathematically it is going to be the theory of integro-differential equations that is applied to processes in Social Sciences (opinion formation), Economics ("lemons and cherries" theory), Biology (DNA denaturation), Medicine (tendon healing process - collagen remodelling) and the redistribution in a lift. To attend the seminar, please use the Zoom link: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Marius Lemm (EPFL)

Spectral Gaps in Quantum Spin Systems

Quantum spin systems are many-body models which are of wide interest in modern physics and at the same time amenable to rigorous mathematical analysis. A central question about a quantum spin system is whether its Hamiltonian exhibits a spectral gap above the ground state. The existence of such a spectral gap has far-reaching consequences, e.g., for the ground state correlations. In this talk, we survey recent progress on deriving spectral gaps for frustration-free quantum spin systems in dimensions greater than 1, including in the antiferromagnetic models of Affleck-Kennedy-Lieb-Tasaki (AKLT). To attend the seminar, use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Christian Brennecke (Harvard University)

Bose-Einstein Condensation beyond the Gross-Pitaevskii Regime

In this talk, I will consider Bose gases in a box of volume one that interact through a two-body potential with scattering length of the order $N^{-1+\kappa}$, for $\kappa >0$. For small enough $\kappa \in (0;1/43)$, slightly beyond the Gross-Pitaevskii regime (\kappa=0), I will outline a proof of Bose-Einstein condensation for low-energy states that provides bounds on the expectation and on higher moments of the number of excitations. The talk is based on joint work with A. Adhikari and B. Schlein. To attend the seminar, use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Radosław Adamczak (MIMUW)

Random G-circulant matrices. Spectrum of random convolution operators on large finite groups

I will describe the asymptotic behaviour of spectral measures of random G-circulant matrices, i.e., matrices corresponding to convolutions with random functions on a finite (not necessarily Abelian) group, when the order of the group tends to infinity. If time permits I will also mention asymptotic freeness for collections of independent random matrices and central limit theorems for linear eigenvalue statistics. I will conclude with some open problems. To join our online seminar, please make use of the following Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Thomas Strobl (University Lyon 1)

Singular Riemannian foliations and Gauge Theories beyond group actions

In the first part of the talk we will explain the notion of a singularRiemannian foliation (SRF) as a mathematical generalization of isometricgroup actions on a Riemannian manifold (M,g). We will relate this inparticular to constrained dynamical systems on T*M, where the definingcondition between the foliation on M and the metric g appears as acompatibility condition of first class constraints with the naturalHamiltonian flow generated by g. In the second part of the talk we will show how to construct gaugetheories beyond the standard use of group actions. A simple prototype ofsuch a gauge theory is the Poisson sigma model, which we will brieflyrecall in this context. However, this theory is defined in twodimensions only and, more importantly, is topological (thus in somesense "unphysical"). We show how the use of SRFs permits to constructgauge theories in arbitrary spacetime dimensions with propagatingdegrees of freedom. Replacing the notion of a quadratic Lie algebra,used to define standard Yang-Mills theories, by quadratic Liealgebroids, we will be led to what we call Curved Yang-Mills-Higgs GaugeTheories as a generalization of the Yang-Mills-Higgs sector of the Standard Model of particle physics. To join our online seminar, use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Torben Heinrich Krüger (University of Copenhagen)

Non-selfadjoint random matrices: spectral statistics and applications

The empirical spectral distribution of a non-hermitian random matrix concentrates around a deterministic probability distribution on the complex plane as its dimension increases. Despite the inherent spectral instability of such models, this approximation is valid all the way down to local scales just above the typical eigenvalue spacing distance. We will give an overview over some basic questions and techniques associated with the study of spectra for non-hermitian random matrices. Furthermore, we will present recent results for matrices with correlated entries and their application to systems of randomly coupled differential equations that are used to model a wide range of disordered dynamical systems ranging from neural networks to food webs. To join our online seminar, use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Ewelina Zatorska (Imperial College London)

On the Existence of Solutions to the Two-Fluids Systems

In this talk I will present the recent developments in the topic of existence of solutions to the two-fluid systems. The compensated compactness technique of P.-L. Lions and E. Feireisl for single-component fluids has certain limitations, distinctly in the context of multi-component flow models. A particular example of such model is two-fluids Stokes system with single velocity field and two densities, and with an algebraic pressure law closure. The first result that I will present is the existence of weak solutions for such system, using compactness criterion introduced recently by D. Bresch and P.-E. Jabin. I will also outline an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation. In the last part of my talk I will relate to a couple of more recent results: existence of solutions to one-dimensional system, non-uniqueness of solutions to inviscid system, and I will comment on issues around weak-strong uniqueness. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Wojciech de Roeck (KU Leuven)

The many-body Thouless pump

In the last decade, there has been quite some work on proofs of quantization of Hall conductance (in interacting systems). In particular, the work of Hastings and Michalakis provided a breakthrough perspective.In a few papers with Bachmann, Bols and Fraas, we developed a related approach that seems somehow shorter. I will try to discuss all this, realizing of course that the audience is not familiar with the subject. Our work also provides an explicit route from Hamiltonian models to topological quantum field theories and I will try to explain this as well. Finally, I could talk about more recent work on generalizing the concept of Thouless pumps to charges that are not related to a U(1) symmetry. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Jakub Zieliński (ICM)

Modelowanie epidemii. Przegląd różnych metod matematycznych

Epidemic models. An overview of different mathematical methods

W trakcie seminarium przedstawione zostaną najważniejsze klasy modelistosowanych w epidemiologii: modele wykorzystujące równania różniczkowe,wzmianka o modelach opartych o metody grafowe, modele agentowe. Wdrugiej części omówiony zostanie model agentowy opracowany w ICM UW w celu opisu epidemii wirusa SARS-CoV-2. Seminarium się odbędzie na ZOOM korzystając z identyfikatora: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

In this talk we will present the most important types of models applied in epidemiology: models using differential equations, models based on graph methods, and agent models. During the second part of the talk, an agent model developed by the ICM UW so as to describe the epidemic caused by the virus SARS-CoV-2 is to be analysed. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Andrew Bruce (University of Luxembourg)

On the supermanifold description of Pre-Courant algebroids

Loosley, a Courant algebroid is a vector bundle with a Loday-Leibniz bracket and a nondegenerate bilinear form on its space of sections together with some compatibility conditions. Following Roytenburg it is known that Courant algebroids have a neat supermanifold formulation as 'symplectic Lie 2-algebroids'. Without details, we have a graded symplectic supermanifold and an odd Hamiltonian that is homological, i.e., {\theta, \theta} =0. In this talk, I will show how pre-Courant algebroids, so 'Courant algebroids' without the Jacobi identity, have a very similar formulation as 'symplectic almost Lie 2-algebroids'. In particular we condition on the odd Hamiltonian is relaxed to be {{\theta, \theta},f} =0 for weight/degree zero functions. We will explore some of the consequences of this reformulation, including how to describe pre-Courant algebroids with an additional compatible non-negative grading. Examples of what we refer to as weighted pre-Courant algebroids include Courant algebroids in the category of vector bundles.As a side remark, Courant algebroids and their relatives have made a resurgence in theoretical physics via double field theory.Based on Andrew James Bruce & Janusz Grabowski, Pre-Courant algebroids, Journal of Geometry and Physics 142 (2019) 254-273. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09

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Marcin Zając (KMMF)

Hamilton-Jacobi theory for locally conformal symplectic manifolds

Locally conformal symplectic (lcs) manifolds have a veryinteresting geometry and provide a natural generalisation of asymplectic structure. It turns out that the generic example of a lcsmanifold is given by a cotangent bundle of a manifold, equipped withcertain closedone-form being a pull-back of a one-from on a base manifold. In my talkI will present main features of dynamics on locally conformal symplecticmanifolds and present a geometrical version of Hamilton-Jacobi theory forthis kind of structure. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09