Faculty of Physics University of Warsaw > Events > Seminars > The Trans-Carpathian Seminar on Geometry & Physics

The Trans-Carpathian Seminar on Geometry & Physics

(formerly Geometric Seminar)

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room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Stefan Rauch-Wojciechowski (Linköping University)

Global dynamics of a rolling and sliding disc, asymptotic solutions, stability

The problem of a disc rolling in a plane has been treated in classical works of P.Apple, D.J.Korteweg, E.J.Routh and S.A.Chaplygin. It is described by a dynamical system of 4 equations, has 3 integrals of motion and is integrable. When sliding is allowed there are 2 more dynamical variables, equations are dissipative, non-integrable and have energy as a monotonously decreasing function of time. The key for understanding the dynamics are asymptotic solutions, their stability properties and a La´Salle type theorem on asymptotic behavior of solutions. These results, together with with computer simulations of solutions starting in vicinity of the asymptotic solutions provide a basis for global understanding of what happens for different initial conditions. I shall explain formulation of the problem, present analytical results and tell how much we have learnt from numerical simulations.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Artur Giżycki (MiNI PW)

Representations of groups in relation to representations of transformation groupoids

I present the fundamentals of representation theory of groups and groupoids. I will define the induced representation, imprimitivity systems and theorems of Mackey and Landsman that allow to move between these concepts. I narrow our discussion to transitive transformation groupoid G ✕ K/G where G is locally compact group and K is its compact subgroup. I am going to show that representation of transformation groupoid is irreducible if and only if inducing representation of subgroup K is irreducible and several other theorems.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Paweł Ciosmak (MIMUW)

Partition function in Yang-Mills theory and the moduli space of flat connections in dimension 2

The moduli space of flat connections over a smooth manifold can be equipped in a very natural way with a symplectic structure. According to the theorem by Witten the symplectic volume of this space is given by a certain limit of the partition function in Yang-Mills theory. In dimension two this partition function can be calculated using a discrete approximation, which happen to give the exact value of the integral. I will present these topics and give examples when gauge group is SU(2) or SO(3)

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Paweł Urbański (KMMF)

Kaluza Klein theory contra geometry of affine values (part II)

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Paweł Urbański (KMMF)

Kaluza-Klein theory contra geometry of affine values

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andrzej Weber (MIMUW)

"Motivic" decomposition and C* action

According to the Grothendieck idea every algebraic variety is built in some sense from indecomposable pieces. These pieces are called motives.When a cohomology theory is applied to a variety (i.e. a certain type of functor from varieties to an abelian category), then the decomposition is visible as a decomposition into a direct sum. We will discuss an example of such decomposition coming from an action of C^* (for complex varieties).We confront this decomposition with the localization theorem in equivariant cohomology.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Jorge Jover (Univesidad de Zaragoza)

Dynamics on the Space of States: a Geometrical Description

I will present a picture of Quantum Mechanics based on the geometrical description of physical observables in terms of expectation value functions, generalizing thus the so called Ehrenfest picture of quantum systems. In particular, I will describe the set of pure and mixed states of a quantum system and its geometrical properties. The basic geometrical tools reproduce the algebraic structure of the set of observables on a Hilbert space. This formalism incorporate in a natural way the probabilistic description of Quantum Mechanics. The geometrical formalism allows to analyze from a new perspective the properties of quantum systems, such as dynamical equations, uncertainty relations, coherent states and the non-unitary evolution of open systems. I will focus in the application of the formalism to describe the Markovian evolution of quantum systems determined by the Kossakowski-Lindblad equation.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Norbert Poncin (UniLu)

Towards integration on certain supermanifolds

The aim of the talk is to describe a generalization of Superalgebra and Supergeometry to Z_2^n -gradings, n>1. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the involved Z_2^n -degrees. This Z_2^n -Supergeometry exhibits interesting differences with classical Supergeometry, provides a sharpened viewpoint, and has better categorical properties. Further, it is closely related to Clifford calculus: Clifford algebras have numerous applications in Physics, but the use of Z_2^n -gradings has never been investigated. More precisely, we discuss the geometry of Z_2^n -supermanifolds, give examples of such colored supermanifolds beyond graded vector bundles, and study the generalized Batchelor-Gawędzki theorem. However, the main focus is on the Z_2^n -Berezinian and on first steps towards the corresponding integration theory, which is related to an algebraic variant of the multivariate residue theorem.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Mikołaj Rotkiewicz (MIMUW)

On smooth monoid actions on manifolds (II)

Grabowski and Rotkiewicz showed that a smooth action of the monoid of multiplicative reals equips a manifold with the structure of a, so called, graded bundle, being a natural generalization of the notion of a vector bundle. In the light of this result it is interesting to ask what are the natural structures on manifolds related with actions of other natural monoids generalizing the multiplicative reals. We shall discuss this problem in a series of two lectures. In the first of these (M. Jóźwikowski, 23. March) we will recall the known results of Grabowski and Rotkiewicz and discuss actions of the monoid of second jets of real functions which preserve zero. In the second (M.Rotkiewicz, 30. March) we will discuss actions of the multiplicative complex numbers and actions of the multiplicative reals on supermanifolds.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Michał Jóźwikowski (IMPAN)

On smooth monoid actions on manifolds

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Luca Vitagliano (University of Salerno)

Generalized geometry in odd dimensions

eneralized complex structures were introduced by Hitchin and further studied by Gualtieri. They encompass symplectic structures and complex structures as extreme cases. In the general case, a generalized complex manifold can be seen as an even dimensional Poisson manifold equipped with additional structures. In the first part of the talk, we will review the definition of generalized complex structures and outline their relation with Poisson geometry, Lie algebroids and Lie groupoids. In the second part of the talk, we will propose “generalized contact bundles” as odd dimensional analogues of generalized complex manifolds. Finally, we will outline the relation between generalized contact bundles and Jacobi manifolds, Lie algebroids and Lie groupoids.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Giovanni Moreno (IMPAN)

Introduction to BV-BFV theories on manifolds with boundary (II)

Batalin-Vilkovisky (BV) theories are classical field theories where the target space is Z-graded. They are particularly well-suited for gauge symmetry reduction. A BV theory comprises an action functional, a symplectic form, and a homological vector field: from the action functional one obtains the Euler-Lagrange (EL) field equations, the homological vector field encodes the gauge symmetries of the theory, and the symplectic form captures their interrelationship. Under physically reasonable assumptions, these data allows for a nice cohomological description of the tangent space to the so-called EL-moduli space (the space of solutions to the EL equations, modulo gauge symmetries), at a smooth point.In this two-parts seminar (based on the paper “Classical BV Theories on Manifolds with Boundary”, by A.S. Cattaneo et al., Commun. Math. Phys., 2014), I will review the main features of the BV formalism on closed manifolds. Then I will switch to manifolds with boundary, and show that a BV theory induces a so-called Batalin-Fradkin-Vilkovisky (BFV) theory on the boundary, in a compatible way: the result is a particular case of a BV-BFV theory. In the BV-BFV context, the symmetry reduction, and the corresponding cohomological “infinitesimal” description of the EL-moduli space at a smooth point, go along the same conceptual lines as in the BV case, but the actual procedure is more delicate, since it has to take into account the additional boundary structures. The final output will be the symplectic EL-moduli space, which fulfils the expected gluing properties, indispensable for quantisation.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Giovanni Moreno (IMPAN)

Introduction to BV-BFV theories on manifolds with boundary (I)

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Giovanni Moreno (IMPAN)

Finite-dimensional symplectic formalism for higher-order field theories

In Mechanics, the cotangent space to the configuration space, understood as the space of initial data for (regular) Lagrangian theories, is equipped with a natural symplectic structure. Similarly, in field theory, one gets an infinite-dimensional symplectic space of boundary fields, containing the space of initial data as a Lagrangian submanifold: such spaces are at the heart of the so-called BV-BFV theories with boundary, which are enjoying a renewed interest. In this talk, I will review a recent work with J. Kijowski, where the symplectic structures behind higher-order fields theories have been studied in detail. The so-obtained framework, based on an obvious jet-theoretic analogy with Mechanics, represents probably the simplest geometric description of the dynamics of a Lagrangian field theory. The symplectic structures involved are defined ‘fibre-by-fibre’ and, in this sense, they can be treated as finite-dimensional.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Piotr Waluk (KMMF)

The Ricci flow

The Ricci flow is a technique for analyzing Riemannian manifolds by evolving their metric with respect to a certain differential equation. The method was first introduced by Hamilton in his paper form 1982, as means to solve a problem concerning classification of 3-dimensional compact manifolds. My talk will be a short introduction to the topic of the Ricci flow, explaining its basic ideas and illustrating them with some application examples.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Rafał R Suszek (KMMF)

"A geometrisation of the (T)QFT functor

I shall formulate and illustrate on a simple and physically meaningful example the general principle by which (generalised Cheeger-Simons) differential characters and related transport operators defined by geometrisations of de Rham classes on the configuration bundle of a field theory with topological charges realise - in a concrete and computable manner - Segal's ambitious dream of functorial quantisation within and beyond the topological category. Time permitting, I shall also discuss certain important consequences of that principle.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Javier de Lucas Arraujo (KMMF)

Lie systems and geometric phases

In the context of the Floquet theory, using a variation of parameter argument, we show that the logarithm of the monodromy of a real periodic Lie system with appropriate properties admits a splitting into two parts called dynamic and geometric phases. The dynamic phase is intrinsic and linked to the Hamiltonian of a periodic linear Euler system on the co-algebra. The geometric phase is represented as a surface integral of the symplectic form of a co-adjoint orbit

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Piotr Kopacz (Akademia Morska w Gdyni, Uniwersytet Jagielloński)

Finslerian versus variational approach to the solutions to Zermelo's problem with application to navigation

We consider the solutions to the Zermelo navigation problem on Riemannian manifolds, under perturbation represented by the weak vector field, in Finsler geometry with application of Randers metric. We compare it to the variational solutions via the Euler-Lagrange equations with non-restricted wind distribution. We focus on the river-type perturbation in the corresponding low-dimensional examples in the context of real applications in navigation. We also propose the geometric modi fication of the standard search patterns in case of acting vector field basing on the time-optimal paths.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Michał Jóżwikowski (IMPAN)

A contact approach to the sub-Riemannian geodesic problem

I will discuss the SR geodesic problem from the point of view developed by Witold Respondek and myself in arxiv:1509.01628. The most important points of this approach are: (i) the Hamiltonian formalism is obsolete (ii) geometric reasonings are elementary (iii) the emphasis is put on the flows rather than on Lie brackets. If time allows I will address issues concerning local optimality of solutions.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andrzej Dragan (FUW)

Ideal clocks - a convenient fiction

We show that no device built according to the rules of quantum field theory can measure proper time along its path. Highly accelerated quantum clocks experience the Unruh effect, which inevitably influences their time rate. This contradicts the concept of an ideal clock, whose rate should only depend on the instantaneous velocity.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Giovanni Moreno (IMPAN)

Invariant hypersurfaces in low-dimensional Lagrangian Grassmannians

It is an easy exercise to show that the two-dimensional Monge-Ampère equations are the only two-dimensional second-order PDEs that are invariant under the natural action of the affine group of the plane. In three dimensions, an analogous statement can be proved, though it requires much more computations. In four dimensions, computations are simply unendurable, and the necessity of a more conceptual approach to the problem begins to show. In this talk I will recall that hypersurfaces of Lagrangian Grassmannians and second-order PDEs are basically the same thing, so that the notion of the invariancy (with respect to a given Lie group G) of a (multidimensional) second-order PDE can be formulated in terms of the G-invariancy of the corresponding hypersurface of the Lagrangian Grassmannian. Via the Plucker embedding, hypersurfaces of Lagrangian Grassmannians can be embbeded in a projective space. Such a projective space turns out to be a natural G-module, so that repesentation theory can be used for finding all the (relative) G-invariants polynomials, whose zero loci corresponds to G-invariant hypersurfaces. Up to 3 independent variables, such a method reveals nothing new, and it is just another way to show that Monge-Ampère equations correspond precisely to GL(n)-invariant hypersurfaces. Surprisingly enough, for n=4, a new unexpected class of invariant second-order PDEs pops out, which is not made of Monge-Ampère equations.This talk is based on a joint work with D. Alekseevsky and G. Manno.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Mikołaj Rotkiewicz (MIMUW)

Metric double vector bundles vs. the linearization of graded vector bundles (part II)

The full linearization functor enables studying graded bundles in a framework of multi-vector bundles. I will explain an unexpected relation between the linearization of graded bundles (of degree 2) and symplectic and metric double vector bundles discussed recently by Jotz Lean. This will be preceded by a gentle introduction to a beautiful theory of double vector bundles.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Mikołaj Rotkiewicz (MIMUW)