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

"Theory of Duality" (KMMF) Seminar

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room 2.23, Pasteura 5 at 10:15

Paweł Ciosmak (MIMUW)

Yang-Mills theory in dimension 2

Yang-Mills theory had a great effect on the development of differential and algebraic geometry, leading to the formulation of many invariants of the manifolds in low dimensions. This theory is special in dimension 2, where the partition function can be evaluated using combinatorial techniques. The result of this evaluation can be connected with the variety of representations of the fundamental group of the base manifold, computing its naturally defined symplectic volume.

room 2.23, Pasteura 5 at 10:15

Agnieszka Świerczewska-Gwiazda (ZRFM MIMUW)

Well-posedness and energy conservation for some compressible fluid models

In the recent years a significant attention has been directed again to Euler system, which was derived more than 250 years ago. The system describes the motion of an inviscid fluid. The main attention has been directed to incompressible fluids. Nevertheless, also the system of compressible fluids is an emerging topic, however still very far from a complete understanding. The classical results of Scheffer and Schnirelman pointed out the problem of non-uniqueness of distributional solutions to incompressible Euler system. However the crucial step appeared to be an application of methods arising from differential geometry, namely the celebrated theorem by Nash and Kuiper. This brought Camillo De Lellis and Laszlo Szekelyhidi Jr. in 2010 to the proof of existence of bounded (even Holder continuous) nontrivial compactly supported in space and time solutions of the Euler equations (obviously not conserving physical energy!). Without a doubt this result is a first step towards the conjecture of Lars Onsager, who in his 1949 paper about the theory of turbulence asserted the existence of such solutions for any Holder exponent up to 1/3.The talk is based on several recent joint results with Jose A. Carrillo, Eduard Feireisl, Piotr Gwiazda and Emil Wiedemann and concerns various notions of solutions to compressible Euler equations and some systems of a similar structure. Firstly we shall concentrate on weak solutions and discuss the issue of non-uniqueness and the non-conservation of the energy. We show the existence of infinitely many global-in-time weak solutions for any bounded initial data by adapting the method of convex integration, used to the incompressible Euler system by De Lellis and Szekelyhidi. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality). We give sufficient conditions on the regularity of solutions to the compressible isentropic Euler systems in order for the energy to be conserved.

room 2.23, Pasteura 5 at 10:15

Sławomir Klimek (Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis)

Unbounded derivations on noncommutative spaces and spectral triples

In this talk I will describe natural unbounded derivations on simple examples of noncommutative spaces: the quantum disk and the quantum torus. In particular I will answer the question whether such derivations come from operators with compact parametrix and thus can be used to define spectral triples (joint work with M. McBride, K. Sakai, and S. Rathnayake).

room 2.23, Pasteura 5 at 10:15

Rafał R. Suszek (KMMF WFUW)

On supergerbes for the Green-Schwarz supermembranes

The theory of bundle gerbes with connection has long been known to provide us with the most adequate geometric and cohomological tools for a rigorous canonical description, rigid- and gauge-symmetry as well as duality analysis and equivariant geometric quantisation of the loop mechanics on metric target manifolds determined by a natural generalisation of the standard lagrangean model of propagation of a charged massive point-like particle in background gravitational and electromagnetic fields, known as the two-dimensional bosonic σ-model. Up to now, no extension of the grebe-theoretic description to σ-models with supersymmetry has been worked out although proposals of the relevant geometric structures (involving, i.a., Chern–Simons 2-gerbes based on spin groups) exist. This is a most unsatisfactory situation also from the physical vantage point as it is among supersymmetric σ-models that we find those describing (super)string theory on anti-de Sitter (AdS) metric target spaces, and the latter has proven instrumental in the study of realistic systems of strongly interacting fields with colour, such as the quark–gluon plasmas, beyond the perturbative paradigm.

In my talk, I shall present an original proposal of the gerbe theory behind the so-called Green–Schwarz super-σ-models that capture the dynamics of extended solitonic objects — the superparticle, the superstring and various supermembranes — on supertargets with the structure of a homogeneous space of a super-Lie group, with emphasis on the super-Minkowski space. The proposal is based on geometrisation of elementary Chevalley–Eilenberg cocycles on the relevant super-Lie algebras through the construction of so-called extended superspaces (due to de Azcárraga et al.) and yields supersymmetry-equivariant n-gerbes with connection over the supertargets. The latter are also quasi-equivariant with respect to the local supersymmetry of the respective super-σ-models (Siegel's κ-symmetry) responsible for the removal of the spurious fermionic degrees of freedom. Time permitting, I shall give an outlook on the super n-gerbe-theoretic approach to loop dynamics on the AdS backgrounds of Мецаев and Цейтлин.

room 2.23, Pasteura 5 at 10:15

Christopher J. Fewster (University of York)

An analogue of the Coleman—Mandula theorem for QFT in curved spacetimes

The Coleman—Mandula (CM) theorem states that the Poincaré and internal symmetries of a Minkowski spacetime quantum field theory cannot combine nontrivially in an extended symmetry group. In this talk, I will describe some of the background to the CM theorem and then establish an analogous result for a general class of quantum field theories in curved spacetimes. Unlike the CM theorem, our result is valid in dimensions n ≥ 2 and for free or interacting theories. It makes use of a general analysis of symmetries induced by the action of a group G on the category of spacetimes. Such symmetries are shown to be canonically associated with a cohomology class in the second degree nonabelian cohomology of G with coefficients in the global gauge group of the theory. The main result proves that the cohomology class is trivial if G is the universal cover S of the restricted Lorentz group. Among other consequences, it follows that the extended symmetry group is a direct product of the global gauge group and S, all fields transform in multiplets of S, fields of different spin do not mix under the extended group, and the occurrence of noninteger spin is controlled by the centre of the global gauge group.

room 2.23, Pasteura 5 at 10:15

Junya Yagi (IFT, WF UW)

Integrable lattice models and supersymmetric gauge theories

I discuss a new approach to integrable lattice models and supersymmetric gauge theories, which was motivated by string theory considerations. This approach provides a unified perspective on various important notions, such as the Belavin model, the Jimbo–Miwa–Okado model, the Bazhanov–Sergeev model, Felder's elliptic quantum groups, the vertex-face correspondence, the elliptic modular double, and an elliptic lift of the relation between the chiral Potts model and the six-vertex model. Furthermore, it relates these notions to four-dimensional supersymmetric gauge theories and their surface operators.

room 2.23, Pasteura 5 at 10:15

Maciej Borodzik (ZUD MIMUW)

Solid angles and Seifert hypersurfaces

We show an explicit way to provide an explicit construction of a Seifert hypersurface for a codimension-2 link in R^{n+2}. The construction involves calculating solid angles of some surfaces. This reveals connections of our construction with magnetic potential studied already by Maxwell. This is a joint project with Supredee Dangskul (Edinburgh/Bangkok) and Andrew Ranicki (Edinburgh).

room 2.23, Pasteura 5 at 10:15

Maciej Karczmarczyk (KMMF, WF UW)

On quantization of Gaussians

room 2.23, Pasteura 5 at 10:15

prof. Daniel Wójcik (IBB)

On some physical and mathematical challenges in modern brain studies

The brain, which allows us to reflect on ourselves and the world, is a complex physical system. From complex subcellular biochemical processes, through nonlinear physics of neuronal membrane, complex cell morphology, communication in non-trivial networks, to behavior, every level of brain functioning is bringing different challenges.In my lecture I will focus on three problems we have been working on. First, I will say a few words about the problem of information coding in the brain and the language used to describe the activity of the nervous system (I call it the kinematics of spike trains). Then, I will discuss the problem of the reference system in the brain, or brain atlases. Whenever we want to localize a phenomenon in the brain we must use some reference system. Given substantial inter-specimen variability this poses a challenge. I will discuss how we handle this.Finally, I will discuss the problem of measurement, the relation between the activity of the cells and the measurement on the example of extracellular electric potential measurement. I will discuss the problems of inference from limited and imprecise measurements.The lecture will be more popular than formal yet I will not shy away from mathematics. My main goal will be to inspire: show that biology in general and neuroscience in particular are great fields for a theoretician lending a wealth of interesting and difficult problems of practical and conceptual importance.

room 2.23, Pasteura 5 at 10:15

prof. Tomasz Taylor (Fulbright Professor - WF UW)

Twistors and amplitudes

I will discuss the structure of perturbative scattering amplitudes in Yang—Mills theory, in particular their superconformal properties. These amplitudes can be Fourier/Penrose transformed from momentum space to twistor space where they are supported on certain holomorphic curves.

room 2.23, Pasteura 5 at 10:15

(KMMF)

Torsion in Khovanov homology

We recall the definition of the Khovanov homology of links and mention its modification to odd Khovanov homology. We discuss the fact that in Khovanov homology of links, presence of ℤ_2-torsion is a very common phenomenon. We outline the several reasons for this observation (e.g., Hochschild homology of the algebra ℤ[x]/(x^2), crucial in Khovanov homology, has only ℤ_2 torsion). For other than ℤ_2 torsion only a finite number of examples of knots with ℤ_n-torsion were known, none for n>8. In this talk, we show that there are infinite families of links whose Khovanov homology contains ℤ_n-torsion for 2 < n < 9 and ℤ_(2^s)-torsion for s < 24. We also introduce 4-braid links with ℤ_3-torsion which are counterexamples to the PS braid conjecture. We mention that our construction also provides an infinite family of knots with ℤ_5-torsion in reduced Khovanov homology and ℤ_3-torsion in odd Khovanov homology. Most of our examples are twisted torus knots and links.

room 2.23, Pasteura 5 at 10:15

prof. Józef Przytycki (MIMUW, George Washington University)

Introduction to Khovanov homology

We offer gentle introduction to Khovanov homology. We startfrom chain complexes and classical examples of homology (forsimplicial complexes, groups and quandles). We introduce the notion of(pre)simplicial sets (and modules) and (pre)cubic sets (and modules).We show how to construct geometric realization of (pre)simplicial and(pre)cubic sets. We give an elementary definition of Khovanov homologyof links and discuss geometric realization in extreme and almostextreme grading.

room 2.23, Pasteura 5 at 10:15

Jochen Zahn (Institut für Theoretische Physik, Universität Leipzig)

The Nambu-Goto string as an effective field theory and its semi-classical limit

The Nambu-Goto string, being purely geometrical and exhibiting diffeomorphism invariance, can be seen as a toy model for (quantum) gravity. I present an effective field theory approach to its quantization, based on perturbation theory around non-trivial classical solutions. It employs the BRST formalism (to deal with diffeomorphism invariance) and renormalization methods developed for QFT on curved space-times. It can be seen as the string theory analog of perturbative quantum gravity. Using rotating classical solutions as the starting point of perturbation theory, one can compute semi-classical corrections to the energy. Several deviations from standard quantum strings are found. For example, the theory is consistent, as an effective field theory, for any dimension of the target space. Furthermore, the semi-classical limit of standard quantum strings does not coincide with that of our perturbative approach. The origin of these deviations will be briefly discussed. The talk is partly based on joint work with D. Bahns and K. Rejzner.

room 2.23, Pasteura 5 at 10:15

Kenji Yajima (Gakashuin University, Tokio)

L^p-boundedness of wave operators for the Schrödinger operator with point interactions

We prove that wave operators for the three-dimensional Schrödinger operators with multi-center local point interactions are bounded in L^p for all 1 < p < 3 but not for other p's for arbitrary centers and non-zero strengths. This is joint work with G. Dell'Antonio, A. Michelangeli and R. Scandone.

room 2.23, Pasteura 5 at 10:15

Javier de Lucas Araujo (KMMF, WF UW)

Geometric structures and Lie systems: towards the multisymplectic case

This talk is aimed at presenting an on-going research on Lie systems possessing Vessiot—Guldberg Lie algebras of Hamiltonian vector fields relative to different types of geometric structures, e.g., Poisson, k-symplectic, and multisymplectic structures. As a main novelty, I will focus upon the case of multisymplectic structures, namely non-degenerate and closed k-forms. I will present some techniques for constructing such Lie systems, and I will study a reduction process for them. This suggests us to use a quite general type of multisymplectic momentum map and a multisymplectic reduction procedure for multisymplectic volume forms. Results will be illustrated with examples of physical and mathematical interest.

room 2.23, Pasteura 5 at 10:15

prof. Jan Dereziński (KMMF, WF UW)

Almost homogeneous Schroedinger operators

First I will describe a certain natural holomorphic family of closed operators with interesting spectral properties. These operators can be fully analyzed using just trigonometric functions. Then I will discuss 1-dimensional Schroedinger operators with a 1/x^2 potential with general boundary conditions, which I studied recently with S.Richard. Even though their description involves Bessel and Gamma functions, they turn out to be equivalent to the previous family.Some operators that I will describe are homogeneous--they get multiplied by a constant after a change of the scale. In general, their homogeneity is weakly broken--scaling induces a simple but nontrivial flow in the parameter space. One can say (with some exaggeration) that they can be viewed as "toy models of the renormalization group".

room 2.23, Pasteura 5 at 10:15

Tomek Smołka (KMMF)

Electromagnetic and gravitational Hopfions

Hopfions are a family of field solutions which have non-trivial topological structure. Their connections with Hopf fibration will be presented. I will focus on two physical applications of Hopfions: electromagnetism and linear gravitation. Using Hopfion solution,I will discuss problem of energy in linear gravitation.

room 2.23, Pasteura 5 at 10:15

Piotr Stachura (Katedra Zastosowań Matematyki SGGW)

Nierówność Bogoliubova, jej uogólnienia i przykład zastosowania

Bogolyubov inequality, its generalizations and some application

Omówię klasyczną nierówność Bogoliubowa, jej postać w temperaturze 0 oraz uogólnienie na nieskończenie wymiarową przestrzeń Hilberta. Jeżeli czas pozwoli, zaprezentuję przykład jej zastosowania dla układu rotorów w temperaturze 0.

I will present the classical Bogolyubov inequality, its form at temperature zero and generalization for the case of infinite dimensional on-site Hilbert space. Time permitting, I will present an example of its application to the interacting rotor system at temperature zero.

room 2.23, Pasteura 5 at 10:15

Michał Jachura (KOKiFA, IFT WFUW)

From two-photon interference to quantum holography

Recent developements in single-photon-sensitive cameras have substantially stimulated the exploration of quantum phenomena such as ghost imaging or EPR-correlations. It also resulted in the first recording of Hong-Ou-Mandel effect being one of the most direct manifestation of quantum nature of light. In further experiments we observed that the excellent spatial resolution of the camera enables to restore sub-shot noise sensitivity in quantum-enhanced interferometry. Finally we showed that the spatial wavefunction of the unknown photon can be fully recovered from the spatial pattern resulting from its two-photon interference with the reference photon in a scheme similar to optical holography. In my talk I am going to discuss the results of aforementioned experiments and concisely present current research directions related to single photon spatial structure

room 2.23, Pasteura 5 at 10:15

prof. Paweł Nurowski (CFT PAN)

How the green light was given for gravitational wave search

The recent detection of gravitational waves by the LIGO/VIRGO team is anincredibly impressive achievement of experimental physics. It is also atremendous success of the theory of General Relativity. It confirms theexistence of black holes; shows that binary black holes exist; that they maycollide and that during the merging process gravitational waves are produced.These are all predictions of General Relativity theory in its fully nonlinearregime.The existence of gravitational waves was predicted by Albert Einstein in 1916within the framework of linearized Einstein theory. Contrary to common belief,even the very definition of a gravitational wave in the fully nonlinearEinstein theory was provided only after Einstein's death. Actually, Einsteinhad arguments against the existence of nonlinear gravitational waves (theywere erroneous but he did not accept this), which virtually stoppeddevelopment of the subject until the mid 1950s. This is what we refer to asthe Red Light for gravitational waves research.In the following years, the theme was picked up again and studied vigorouslyby various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson andAndrzej Trautman, where the theoretical obstacles concerning gravitationalwave existence were successfully overcome, thus giving the `Green Light'for experimentalists to start designing detectors, culminating in therecent LIGO/VIRGO discovery.In this lecture we tell the story of this theoretical breakthrough. Particularattention will be given to the fundamental 1958 papers of Trautman, whichseem to be lesser known outside the circle of General Relativity experts.

room 2.23, Pasteura 5 at 10:15

Krzysztof Andrzejewski (Uniwersytet Łódzki)

Symetria konforemna w układach nierelatywistycznych

Conformal symmetry in non-relativistic systems

Obfitość realizacji symetrii konforemnej budzi coraz większe zainteresowanie. Aby lepiej zrozumieć znaczenie i konsekwencje z niej płynące, warto badać jej odpowiedniki w świecie nierelatywistycznym. Na seminarium omówię pojęcie symetrii konforemnej w przypadku nierelatywistycznym i jego dynamiczne interpretacje.

The notion of conformal symmetry in the non-relativistic systems as well as its dynamical interpretation will be presented and discussed.

room 2.23, Pasteura 5 at 10:15

prof. Konrad Banaszek (IFT, WF UW)

Optical communication in the low-power regime

Quantum theory of electromagnetic fields imposes fundamental limits on the capacity of optical communication channels. We will present exemplary keying techniques and discuss their efficiency in the regime of low average input power.

room 2.23, Pasteura 5 at 10:15

Michał Bejger (CAMK)

Bezpośrednia detekcja fal grawitacyjnych. Idea detekcji i źródła astrofizyczne

Direct detection of gravitational waves. Detection principle and astrophysical sources

Niedawne bezpośrednie detekcje fal grawitacyjnych otwierają zupełnienowe okno obserwacyjnena Wszechświat i obiecują postęp w eksperymentalnym weryfikowaniuteorii grawitacji.Podczas wykładu opowiem o idei detekcji, najczulszych obecnieurządzeniach zdolnychwykrywać fale grawitacyjne (laserowych interferometrach LIGO i Virgo)oraz najbardziejobiecujących astrofizycznych źródłach fal grawitacyjnych.

Recent direct detections of gravitational waves herald a new era inthe observationalastronomy and experimental verifications of the theories of gravity. Iwill talk about theprinciples of detection of gravitational waves, the currentstate-of-art laser interferometricdetectors (LIGO/Virgo) and the most promising astrophysical sources ofgravitational waves.

room 2.23, Pasteura 5 at 10:15

Mikołaj Korzyński (CFT PAN)

Black-hole lattices and the backreaction problem in general relativity

The problem of backreaction in GR concerns the question how the field equations of the modern theory of gravity, i.e. general relativity (GR), behave under coarse-graining. These nonlinear partial differential equations, called Einstein's equations, relate the curvature of the spacetime metric tensor, composed of the second derivatives of the metric, with the stress-energy energy tensor representing the matter content of the spacetime. By coarse-graining I mean averaging out the complicated, fine details of the solution while keeping its large-scale structure. It is well known that linear field equations like the Maxwell's equations of electromagnetism preserve their form exactly under coarse-graining: if we impose them on the local electric and magnetic fields as well as charges and currents then exactly the same Maxwell equations hold for their averages over finite-size domains. For nonlinear PDE's, this is not true in general, but in GR we may demand any additional terms appearing in the averaged equations to be of the form of a correction to the averaged stress-energy energy tensor, called the backreaction. Coarse-graining is a necessary step in application of GR to any astrophysical situation, because it is in general impossible to solve the Einstein's equations exactly for physical objects while taking into account every fine detail of their matter distribution. However, the problem of applicability and the precise definition of coarse-graining in GR has been poorly researched in the literature.Black hole lattices, i.e. regular arrangements of black holes, offer an excellent example to study the backreaction effects. Instead of a uniform matter distribution we have localized, Schwarzschild-like sources with vacuum metric outside, arranged in a lattice with a discrete symmetry group. I will discuss recent results about solutions of this kind, including my own results concerning the continuum limit in the black-hole lattices.

room 2.23, Pasteura 5 at 10:15

Daniel Siemssen (KMMF, WF UW)

Feynman propagators on curved spacetimes

In this talk I will discuss various propagators important in classical and quantum physics of the free scalar field as described by the Klein–Gordon equation. The focus of this talk will be the Feynman propagator. I will show that on static spacetimes the Feynman propagator can be rigorously defined as the limit of the resolvent of the Klein-Gordon operator from above to zero. I will also outline how these results may be extended to some more general spacetimes, thus yielding a distinguished Feynman inverse in such cases.

room 2.23, Pasteura 5 at 10:15

Giovanni Moreno (IM PAN)

Geometry of completely exceptional nonlinear PDE's

The problem of describing solutions to a particular PDE is — at a first glance — totally unrelated to the problem of describing all the PDEs sharing a common feature. Such PDEs are usually collected into a class. For instance, we usually deploy the toolbox of functional analysis within the class of linear PDEs, and such a possibility is precisely the feature making this class so important. But there are other classes, whose definition (and existence) is not so obvious (and clear) as that of the class of the linear ones. For instance, everybody is able to provide a rigorous definition of a linear PDE, whereas defining the class of Monge—Ampère equations requires a deeper thinking. There exist, however, other classes of PDEs which do not possess an actual definition, but are more similar to collections of examples: these usually arise in Physics, where people tend to group PDEs according to the similitudes in their solving methods. The chief examples are provided by the various notions of integrable PDEs. In this seminar I will focus on the class of completely exceptional (nonlinear) PDEs, introduced in 1954 by P. Lax, by imposing a certain âlinear behavior" on their (potential) solutions. Postponing the discussion of the mathematical rigor of Lax's definition, it is truly remarkable that he was able to single out a certain class of PDEs by requiring their solutions to behave in a certain way, without the necessity of computing the solutions themselves. I will show that, by using an appropriate geometric framework, Lax's condition appears to be natural and, as such, absolutely rigorous. More precisely, I will construct a differential operator whose solutions are precisely the operators defining all the completely exceptional PDEs, thus showing that the two problems mentioned at the beginning are in fact one and the same. The existence of such an operator should not come as a surprise: for instance, linear operators are the zeros of the operator ''which takes second-order derivatives'', though nobody ever thinks of linear PDEs in this cumbersome way. This seminar is based on the recent publication ''Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution'', by Jan Gutt, Gianni Manno and myself.

room 2.23, Pasteura 5 at 10:15

Marcin Napiórkowski (KMMF, WF UW)

A mathematical physics perspective on spin wave theory

Spin wave theory suggests that low temperature properties of the Heisenberg model can be described in terms of noninteracting quasiparticles called magnons. In my talk I will review the basic concepts and predictions of spin wave approximation and report on recent rigorous results in that direction.

room 2.23, Pasteura 5 at 10:15

prof. Jerzy Kijowski (CFT PAN)

Higher-order curvature tensors, higher-order Bianchi identities

Trying to understand cosmological anomalies (dark energy, dark matter), many physicists consider various generalizations of the General Relativity Theory. E.g.: theories derived from a Lagrangian depending not only upon the curvature tensor, but also upon its higher covariant derivatives. Personally, I do not believe in the physical relevance of such theories. But, when analizing their mathematical structure, one discovers a beautiful "Terra Nova" of geometric constructions, which sheds also new light on the classical notion of curvature.