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

Seminarium KMMF "Teoria Dwoistości"

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Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Daniel Wysocki (KMMF)

Gradations, Grassmann algebras, and modified classical Yang–Baxter equations

A coboundary Lie bialgebra is a Lie algebra g equipped with a map δ : v ∈ g → [v, r]_S ∈Λ^2 g, where [·, ·]_S is the algebraic Schouten bracket on the Grassmann algebra Λg and r ∈ Λ^2 g is a solution of the modified classical Yang–Baxter equation (MCYBE), i.e.[v, [r, r]_S ]_S = 0 for any v ∈ g. The classification and properties of solutions of the MCYBE are well-studied mostly for semisimple Lie algebras or when dim g ≤ 3. To tackle non-semisimple and higher-dimensional cases, one needs new tools. In this talk, I will discussthe use of gradations on g and Λg in finding solutions and studying the structure of theMCYBE. Several examples will be presented to illustrate this approach.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Karol Trzeszczkowski (WF UW)

Układy całkowalne na sieci typu plastra miodu

Integrable systems on honeycomb lattices

Wprowadzę słuchaczy w teorię równania Kortewega-de Vries (KdV). Wychodząc od zasady nieliniowej superpozycji pokażę na czym polega dyskretyzacja równania KdV na sieci $Z^2$ i rozszerzenie tego zagadnienia na wyższe wymiary. Następnie pokażę jak przenieść ten problem na bardziej skomplikowane sieci - kwazi-regularną sieć rombiczną i sieć plastra miodu i zaprezentuję dziwaczne rozwiązania typu kink występujące na tych sieciach. Przedstawię w ten sposób najnowsze wyniki dotyczące równań HexaKdV i wymiernego addytywnego równania typu Tody na sieci plastra miodu.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Marek Pilch (KMMF)

Modelling of limitations of bulk heterojunction architecture in organic solar cells

Polymer solar cells are considered as very promising candidates for development of photovoltaics of the future. They are cheap and easy to fabricate, however, up to now, they possess fundamental drawback: low effectiveness. In the most popular BHJ (bulk heterojunction) architecture the actual long-standing top effciency is about 12%. One ask the question how fundamental this limitation is, as certain theoretical considerations suggest that it should be two times higher. We describe our simulations on possible influence of so called 'geometric factor' on total efficiency, sketch possible influence of some other factors, and argue that limitations of BHJ architecture are inevitable. We speculate also on perspectives of development of organic solar cells using other ideas.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Wojciech Kryński (IM PAN)

The Camassa-Holm equation and geometry of peakons

Multipeakons are special solutions to the Camassa-Holm equation. They can be described in terms of a geodesic flow on a Riemannian manifold. Singular points of the underlying metric correspond to collisions of the multipeakons. I shall investigate the metric near its singular points in order to analyse the dissipative prolongations of the multipeakons after a collision time.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Tomasz Smołka (KMMF)

Jądro ciepła

Heat kernel

Znalezienie metody rozwiązywania równania przewodnictwa cieplnego jest jednym z kluczowych osiągnięć, dające podwaliny współczesnej analizy funkcjonalnej. Zamierzam omówić klasyczne metody badania równań parabolicznych, bazujące na znajdowaniu jądra całkowego dla półgrupy ciepła. Jądrem ciepła operatora L nazywamy jądro całkowe półgrupy ciepła exp(-tL). W szczególności opowiem o podejściu wykorzystującym w konstrukcji funkcje Synge'a.

Method of solving the thermal conductivity equation is one of the key achievements of modern functional analysis. I will discuss the classical methods of studying parabolic equations, based on finding the integral kernel of the heat semi-group. The heat kernel of an operator L is the integral kernel of the heat semi-group exp (-tL). In particular, I will talk about an approach which uses the Synge functions in the construction.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Marian Wiatr (KMMF)

Funkcje eliptyczne

Elliptic functions

My talk will be a short introduction to the theory of elliptic functions, which will contain basic motivations, definitions and facts. I would like too to define Wierestrass function, give some of its properties and formulas which express every elliptic function by means of the Weierstrass function.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

don Giovanni Moreno (KMMF)

O pewnym skalarnym RRCz trzeciego rzędu, którego grupa symetrii zawiera Aff(3)

Korzystając z oprogramowania do rachunku symbolicznego można stosunkowo łatwo wyznaczyć wyraźną postać jedynego skalarnego RRCz trzeciego rzędu w dwóch niezależnych zmiennych, którego grupa symetrii zawiera Aff(3). Niestety, taka postać nic nie mówi o strukturze samego równania, oprócz faktu, że równanie jest Aff(3)-niezmiennicze. Będę pokazywał, że odpowiednim geometrycznym podejściem pozwala dojść do tego samego równania o wiele łatwiej, odkrywając po drodze kilka nowych i nieoczekiwanych właściwości równania, których trudno mogłyby pokazywać same obliczenia.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

prof. Henryk Żołądek (MIMUW)

First-order linear differential equations with involutive delay and hypergeometric functions

We present an alternative approach to functions satisfying second-order linear ordinary differential equations. It turns out that many of them satisfy a first-order ordinary differential equation with an involution. The involution acts on the argument as well as on parameters. Basic examples involve the hypergeometric functions and their descendants.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Carlos Perez (IFT, WF UW)

An introduction to tensor models

Proposed by Ambjørn, Durhuus and Jonsson as a generalisation of matrix models, tensor models were further propelled by Gurau, who found their large-N expansion, and collaborators. This talk will be devoted to some field theory techniques for tensor models. To wit: A powerful method, by Grosse and Wulkenhaar, that has led to solvable and solved models, consists in combining the U(N)-Ward identity for matrix ensemble of size N, with their Dyson-Schwinger equations. In this talk, I will implement this idea in order to derive the loop equations of (certain class of) quartic tensor models.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Piotr Mormul (MIMUW)

Twierdzenie Weierstrassa o rozkładzie funkcji całkowitej w iloczyn nieskończony (i jego związki z twierdzeniem Mittag-Lefflera o rozkładaniu funkcji meromorficznej w szereg ułamków prostych)

Chodzi o przedstawienie (czy też: zapisanie) funkcji całkowitej eksponujące jej miejsca zerowe - na wzór klasycznego zapisu [Eulera] sinusa w postaci iloczynu czynników: z oraz (1 - z^2/n^2). (Ten zapis doprowadził Eulera do zsumowania szeregu \sum 1/n^2.) Otóż miejsca zerowe sinusa bardzo przejrzyście, liniowo oddalają się od 0. Dla funkcji o takich - lub jeszcze szybciej oddalających się od 0 - zerach, rozkłady w iloczyn nieskończony były już znane w drugiej połowie XVIII wieku; tym bardziej były już znane Cauchy'emu. Duży problem stanowiły funkcje, których zera rosły co do modułu wolno, np jak |z|^{1/N}. Odpowiednie iloczyny nieskończone stawały się wtedy rozbieżne. Weierstrass w swoich berlińskich wykładach w latach 1860-ch podał rozstrzygający pomysł. Wprowadził - tylko i aż - czynniki uzbieżniające iloczyny nieskończone - tzw. czynniki pierwsze Weierstrassa - które przy tym nowych zer nie dorzucały. Udowodnię jego twierdzenie, a następnie je zastosuję: wyprowadzę ten słynny rozkład sinusa, wraz z kilkoma niespodziewanymi wnioskami. W drugiej części, na ważnym przykładzie funkcji sin(z) - z*cos(z) pokażę, jak twierdzenie Weierstrassa ściśle się splata z trochę późniejszym twierdzeniem Mittag-Lefflera o rozkładaniu funkcji meromorficznej w szereg ułamków prostych. Doprowadzi nas to do zsumowania szeregu \sum 1/(\lambda_n)^2, \lambda_n - dodatnie pierwiastki równania tg(x) = x, BEZ rachunku residuów. (Ten szereg, choć w istocie klasyczny, był sporym wyzwaniem na Wydziale MIM w połowie lat 1970-ch.)

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Leszek Kołodziejczyk (MIMUW)

Reverse mathematics

By Gödel's incompleteness theorem, for any reasonable (and reasonably strong) theory there exists an undecidable sentence, i.e. a statement in the language of the theory which the theory can neither prove nor disprove. Gödel's undecidable sentences concerned logical phenomena such as provability and consistency, but it gradually became clear that undecidability applies also to statements with a more evident mathematical meaning.For typical theories that attempt to axiomatize "all of mathematics", such as Zermelo-Fraenkel set theory, known undecidable statements tend to concern matters rather distant from the everyday experience of most mathematicians: the continuum hypothesis, the existence of very large cardinal numbers, and so on. Eventually logicians began to ask what sort of axioms are needed to prove basic theorems about more mundane mathematical objects such as e.g. natural and real numbers, continuous functions or separable metric spaces. I will talk about a research programme known as reverse mathematics, which was initiated almost 50 years in an attempt to answer such questions. I will be assuming that the topic is completely alien to most of the seminar audience, and I will attempt to make the talk at least somewhat understandable.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

prof. Jerzy Kijowski (CFT PAN)

The essence of gravity theory. The Einstein's greatest mistake

The affine formulation of General Relativity Theory is presented. From this point of view the cosmological constant is not an option but a necessary element of the field dynamics. Moreover, it is obvious why the Einstein's attempt (1925) to unify gravity with electromagnetism, based on the concept of an asymetric metric tensor "g+f", could not work, while a different, entirely new unification pattern looks natural. Finally, there is a room for dark matter, not just by "adding new terms" to the existing theory but by taking into account the natural geometric structure of spacetime.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Wojciech Kamiński (IFT WF UW)

Asymptotics of 6j and 10j symbols

6j and 10j symbols of SU(2) group have remarkable behaviour in the regime of large spins. I will describe (attempts of) derivation by geometric quantization.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Tomasz Cieślak (IM PAN)

Hamiltonian aspects of multipeakon dynamics

I will introduce the problem of evolution of multipeakons. Its connection to the Camassa-Holm equation will be presented. Next, some results concerning collisions of multipeakons as well as their evolution after collisions will be summarized.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Piotr Waluk (KMMF)

Na początku był chaos

Everything began from chaos

The Friedman-Lamaitre-Robertson-Walker metric of a homogeneous and isotropic universe is generally agreed upon today as the cosmological model. However, the question of why the observable universe is so homogeneou and isotropic still remains an open problem. To answer it, one must try to analyze the possible behaviour of space-time near a cosmological singularity. One of the attempts at such studies was the Balinskii-Khalatnikov-Lifshitz conjecture, which postulates that near the singularity time derivatives of the metric dominate heavily over spatial derivaives and matter fields. This in turn suggests consideration of spatially homogeneous, but anisotropic space-time models as good approximations.I will begin my talk by discussing the general structure of such space-times and showing that their properties are almost entirely determined by the Lie algebra of their spatial Killing fields. This allows them to be classified by the Bianch classification of the three-dimensional Lie algebras. Several examples of particular physical interest will be examined in detail - although still relatively simple, some of these solutions of the Einstein equations display very untrivial properties, such as modelling an isotropisation process or even undergoing chaotic oscillations when approaching the singularity.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

A.M. Ishkanyan (Inst..Phys. Research NAS of Armenia)

Solutions of the Schrodinger equations in terms of the Heun functions

We review the cases for which the 1D stationary Schrödinger equation is solved in terms of the general and (multi-)confluent Heun functions. We present the possible choices for coordinate transformation that provide energy-independent potentials that are proportional to an energy-independent continuous parameter and have a shape independent of that parameter. In contrast to the hypergeometric case, no Heun potential can in general be transformed into another one by specifications of the involved parameters.We show that there exist in total 29 independent Heun potentials. There are eleven independent potentials that admit the solution in terms of the general Heun functions, for nine independent seven-parametric potentials the solution is given in terms of the single-confluent Heun functions, there are three independent double-confluent and five independent bi-confluent Heun potentials (the six-parametric Lemieux-Bose potentials), and one tri-confluent Heun potential (the general five-parametric quartic oscillator).There are several independent potentials that present distinct generalizations of either a hypergeometric or a confluent hypergeometric classical potential, some potentials possess sub-cases of both hypergeometric types, and others possess particular conditionally integrable ordinary or confluent hypergeometric sub-potentials. We present several examples of explicit solutions for the latter potentials.We show that there exist other exactly or conditionally integrable sub-potentials the solution for which is written in terms of simpler special functions. However, these are solutions of different structure. For instance, there are sub-potentials for which each of the two fundamental solutions of the Schrödinger equation is written in terms of irreducible combinations of hypergeometric functions. Several such potentials are derived with the use of deformed Heun equations. A complementary approach is the termination of the hypergeometric series expansions of the solutions of the Heun equations.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Szymon Łęski (Samsung Corp.)

Language modeling (with or without neural networks)

Natural Language Processing, that is, processing large amounts of natural language data using computers, is both challenging and relevant for practical applications. The topic of the seminar is language modeling. In this task the goal is to "fill in the blank" in a sentence - in other words, to predict the probability distribution of words in a given context. Language modeling is important not only because of immediate applications (eg. in optical character recognition or in predictive keyboards), but also because such models are good starting points for other language processing tasks. We will cover both statistical (count-based) and neural language models. Specifically, I will focus on how discrete objects (words and sentences) can be represented to facilitate natural language processing tasks. I will also overview the challenges of developing commercial language models in the industry setting.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Krzysztof A. Meissner (IFT, WF UW)

Infinite dimensional symmetries and the Standard Model

Why are 48 fermionic degrees of freedom (6 quarks in 3 colors and 6 elptons) in the Standard Model? This fact may have surprising explanation by connecting the Standard Model to maximal (N=8) supergravity that necessarily has 48 fermionic degrees of freedom. In gauged supergravity both particle physics and gravity are in one unified picture. The scheme points to a mathematically virtually unexplored infinite group of symmetries E10.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Paweł Kasprzak (KMMF)

Taft Hopf algebra, its coideals and (co)integrals on them

Taft algebras are finite dimensional algebras distantly related to the CCR-algebra. They admit a Hopf algebra structure and they were introduced by Taft in order to show that the antipode of a finite dimensional Hopf algebra can be of an arbitrary (even) order. Remarkably every Taft Hopf algebra admits infinitely many coideal subalgebras, which is in contrast with the case of group algebras associated with finite groups and Hopf algebras of functions on finite groups (in the latter two cases they correspond to subgroups of a given group). In my talk I will: i) describe coideals of Taft Hopf algebras; ii) explain the concept of (co)integrals on coideal subalgebras of a Hopf algebras and describe them in case of the Taft Hopf algebra; iii) give a sketch of the theory of (co)integrals on coideal subalgebras of general Hopf algebra. Partially based on joint work with A. Chirvasitu and P. Szulim.

Taft algebras are finite dimensional algebras distantly related to the CCR-algebra. They admit a Hopf algebra structure and they were introduced by Taft in order to show that the antipode of a finite dimensional Hopf algebra can be of an arbitrary (even) order. Remarkably every Taft Hopf algebra admits infinitely many coideal subalgebras, which is in contrast with the case of group algebras associated with finite groups and Hopf algebras of functions on finite groups (in the latter two cases they correspond to subgroups of a given group). In my talk I will i) describe coideals of Taft Hopf algebras; ii) explain the concept of (co)integrals on coideal subalgebras of a Hopf algebras and describe them in case of the Taft Hopf algebra; iii) give a sketch of the theory of (co)integrals on coideal subalgebras of general Hopf algebra. Partially based on joint work with A. Chirvasitu and P. Szulim.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Rafał Roman Suszek (KMMF)

Contractible κ-symmetric supergerbes on homogeneous spaces of Lie supergroups

The geometrodynamics of topologically charged extended objects, in particular that of loops and paths, in the homogeneous space AdS_5 x |S^5 of the supersymmetry Lie supergroup SU(2,2|4) has long been known to play an important role in modern attempts, based on the so-called AdS/CFT correspondence, at understanding the non-perturbative quantum mechanics of realistic strongly coupled systems with gauge symmetry, such as, e.g., the quark-gluon plasma. An important feature of the dynamics is an asymptotic transition into its fairly well-understood counterpart on the super-Minkowski space under the İnönü-Wigner contraction su(2,2|4)/(so(4,1) x so(5)) -> siso(9,1|32)/so(9,1). A rigorous treatment of the gauge field coupling to the topological charge carried by these objects, leading through a supersymmetry-equivariant Dirac-type geometrisation of the corresponding class in the Cartan-Eilenberg cohomology of SU(2,2|4), paves the way to a geometric quantisation of the dynamics and a systematic construction of supersymmetric defects central to the AdS/CFT correspondence, and so offers hope for an in-depth elucidation of the higher geometry behind the holographic principle.In my talk, I shall recapitulate the construction of the so-called super-σ-model on a homogeneous space of a supersymmetry Lie supergroup G associated with a distinguished super-cocycle χ in the Cartan-Eilenberg cohomology of the latter, and the ensuing non-linear realisation of supersymmetry, as well as its gauged linearised variant - Siegel's κ-symmetry. I shall also discuss at length a recently proposed scheme of equivariant geometrisation of χ that employs integrable supercentral extensions of the Lie superalgebra of G induced by the super-cocycle through the standard correspondence between the Cartan-Eilenberg cohomology of the Lie (super)group and the Chevalley-Eilenberg cohomology of its Lie (super)algebra. An intricate topological interpretation of the ensuing supersymmetry extension in terms of the Kostelecký-Rabin winding charge shall be given, and due emphasis shall be laid upon the issues of (weak) κ-equivariance of the geometrisation and its compatibility with the İnönü-Wigner contraction on the base supermanifold. The general construction shall be illustrated with the examples of the super-1-gerbe associated with the super-3-cocycle of Metsaev and Tseytlin, and the super-0-gerbe associated with Zhou's super-2-cocycle.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Stanisław Głazek (IFT WF UW)

Zderzenia strun gluonowych

Collisions of gluon strings

Szacuje się, że zderzenia strun gluonowych mogą powodować azymutalna zmienność krotności i pływu eliptycznego rzędu kilku procent w peryferycznym rozpraszaniu proton-protonw LHC.

It is estimated that collisions of gluon strings can cause azimuthal variation on the order of one to ten percent in the multiplicity and elliptic flow in peripheral proton-proton scattering at LHC energies.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Jeremy Faupin (University of Lorraine, Metz)

Dissipative quantum systems: scattering theory and spectral singularities

In this talk, we will consider an abstract pseudo-hamiltonian given by a dissipative operator of the form H=H_V-iC^*C, where H_V=H_0+V is self-adjoint and C is a bounded operator. Such operators are frequently used to study scattering theory for dissipative quantum systems. We will recall conditions impliying the existence of the wave operators associated to H and H_0, and we will see that they are assymptotically complete if and only if H has no spectral singularities in its essential spectrum. In mathematical physics, spectral singularities have been considered in many different contexts. We will review several possibilities equivalent definitions of a spectral singularity. For dissipative Schrodinger operators, a spectral singularity corresponds to a real resonance, or, equivalently, to a point of the positive real axis where the scattering matrix is not invertible. The talk is based on two articles. The first ons is joint work with Jurg Frohlich and the second one is joint work with Francois Nicoleau.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Jacek Wojtkiewicz (KMMF)

Riemann $\zeta$ Function, Distribution of Primes, and Riemann Hypothesis: p.II: Selected interrelations between RH and physics

Some selected connections between Riemann Hypothesis and physics (mainly statistical mechanics) will be raised and shortly discussed.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Jacek Miękisz (MIMUW)

From Hilbert to Shechtman and beyond - a brief history of quasicrystals and a long story of unsolved problems

The first part of the presentation is a colloquium-style introduction to the mathematics of quasicrystals as seen from the theoretical physicist point of view. We will discuss simple examples of optimization problems in kissing numbers, packing spheres, non-periodic tilings, and classical-lattice gas models of statistical mechanics. We will also formulate the main open problem - the existence of non-periodic Gibbs measures for finite-range Hamiltonians. In the second part of the talk, we will discuss various connections between ergodic theory, substitution dynamics, dynamical systems of finite type, and classical lattice gas-models. I will outline a research project supported by NCN Harmonia grant "Mathematical models of quasicrystals". We are looking for young people to collaborate within the framework of the grant.

Zapraszamy do sali 2.23, ul. Pasteura 5 o godzinie 10:15

Yafet Sanchez Sanchez (Max Planck Institute)

Propagators on low regularity spacetimes

In this talk, I will present some motivations and technical difficulties that appear when one tries to do Quantum Field Theory in spacetimes with limited regularity. I will focus particularly on the causal propagator and the two-point function of the quasi-free states and its relationship with adiabatic states. Because this is a work in progress only preliminary results and general strategies will be discussed.

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