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

Alfonso Giuseppe Tortorella (Universita di Firenze)

The BV-complex and the deformation problem of a coisotropic submanifold

In this talk, following works by Schaetz, I will review the construction of the BV-complex and its role in the coisotropic deformation problem. In a Poisson manifold, each coisotropic submanifold comes attached with two cohomological resolutions of its reduced Poisson algebra. The first one is the L-infinity-algebra introduced by Oh&Park in the symplectic case, and by Cattaneo&Felder in the Poisson case. As well-known this L-infinity-algebra controls the formal coisotropic deformation problem, and under a generically non-trivial necessary and sufficient condition the non-formal deformation as well. The second one is the BV-complex originally introduced by physicists dealing with Hamiltonian systems with symmetries. As proved by Schaetz, the BV-complex and the L-infinity-algebra are L-infinity quasi-isomorphic, and so they control equally well the formal coisotropic deformation problem. However, as proven by Schaetz, the BV-complex encodes also further information about the non-formal deformations, which generically is not captured by the L-infinity-algebra. If time permits I will start to describe how these results can be transferred into the more general framework of Jacobi manifolds.

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

Giovanni Moreno (University of Salerno)

Hypersurfaces in Lagrangian Grassmannians and geometric theory of nonlinear PDEs

I will begin by reviewing the simplest nontrivial case of a Lagrangian Grassmannian manifold, namely the three-dimensional L(2,4), stressing its remarkable isomorphism with the Lie quadric Q^3, mentioning also its "meta-symplectic counterpart", the four-dimensional L(2,5), which is a rather unexplored object. In spite of the low dimensionality of the objects involved, even at this level, it is possible to formulate rather tricky questions. Then I will switch to the PDE side, examining the standard framework for 2nd order 2D (nonlinear) PDEs based on the Lagrangian bundle of a 5D contact manifold (perhaps better known as "2nd order jet space"), with a particular emphasis on the geometric formulation of Cauchy problems, which, in turn, needs the notion of a characteristic Cahuchy surface. Coming back to the Lagrangian Grassimannians, I will show that the PDEs correspond, in fact, to their hypersurfaces, and that the presence of a lot of characteristics is a well-known phenomenon in Algebraic Geometry known as a "ruling". The advantage of such a bridge between the two disciplines is that is allows to recast known results in a more transparent way, and to formulate new ones as well, especially in the meta-symplectic context, i.e., that of 3rd order 2D (nonlinear) PDEs. As an example, I will demonstrate that 2D parabolic Monge-Ampere equations correspond to the so-called hyperplane sections of Q^3. To conclude with, I will bring in the dual variety of the generic Lagrangian Grassmannian L(n,2n), as well as its singular loci, which are in correspondence with remarkable classes of PDEs, like, e.g., the linearisable ones. In such a framework, a still open conjecture by E. V. Ferapontov, about the integrability of n-dimensional PDEs, may become more easy to work out. The content of this seminar is based on the project "GEOGRAL", which I'm going to carry out at IMPAN with a Marie Skłodowska-Curie fellowship, commencing September 1st. Due to the highly multidisciplinary character of GEOGRAL, collaborative efforts are mandatory, and I hope that this small introduction will stimulate the attention of local personnel potentially interested in joining in.

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

Artur Janda (Centrum Badań Kosmicznych)

The action principle, electrodynamics and causality

The classical electrodynamics may be cosidered as a gauge invariant theory of sections of the cotangent bundle over the spacetime manifold. Basic notions of causality applicable to a general theory of nonlinear electrodynamics of continuous media will be presented.

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

Tomasz Maciążek (CFT)

Critical points of the squared norm of the momentum map for complex projective spaces

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

Jerzy Lewandowski (IFT FUW)

Generally Relativistic classical and quantum particles

The canonical theory of a test particle in a background curved spacetime is a fantastic example of a Hamiltonian constrained system, that is a system whose Hamiltonian is a generator of the gauge transformations. On this example we will explain definitions of the physical observables and the procedure of the canonical quantization. Taking into account that the existence of a (specially) relativistic quantum mechanics is usually questioned, the results I am going to present in this talk may even go beyond the well known quantum physics. In particular, the issue of a quantum position and quantum time operators will be revisited and solved within the geometric framework.

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

Piotr Sułkowski (IFT FUW)

S-duality and quantum field theory

I will review the notion of S-duality in quantum field theory, i.e. the equivalence of pairs of quantum field theories, respectively with weak and strong interactions. I will present a historical development of this and related ideas, as well as modern perspectives, in particular a prediction of the existence of strongly coupled quantum field theories without lagrangian description.

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

Bogdan Balcerzak (Politechnika Łódzka)

On Dirac operators on Lie algebroids

Dirac type operators on Lie algebroids with respect to different geometric structures will be defined and discussed.

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

Antoni Pierzchalski (Uniwersytet Łódzki)

Generalized gradients

Uogólnione gradienty to nieredukowalne ze względu na działającą grupę składniki pochodnej kowariantnej. Zakodowane są w nich dane o strukturze geometrycznej rozmaitości, na której są badane. Pokazana zostanie ogólna konstrukcja gradientów oraz ich przykłady. Omówione będzie zachowanie na brzegu rozmaitości. Wspomniana także będzie konstrukcja w przypadku struktur ogólniejszych - algebroidów Liego.

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

Javier de Lucas (KMMF)

Definition, properties and applications of superdifferential equations

There exist several non-equivalent meanings of the term "superdifferential equation" in the literature. In this talk, we aim to describe them and to motivate that one of these meanings amounts to a certain type of differential equations on graded bundles. Next, we present some preliminary results on the geometric and algebraic properties of such differential equations.

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

Andrew Bruce (IMPAN)

On graded bundles in the category of Lie groupoids

Lie groupoids are to be found throughout differential geometry including the theory of group actions, foliations, Poisson geometry, connection theory and the study of singular spaces such as orbifolds. Over the years there has been much interest in 'categorified 'objects in the category of Lie groupoids; this has been spurred on by the original ideas of Mackenzie such as 'double Lie groupoids' etc. Very recently there has been a resurgence of interest in VB-groupoids and VB-algebroids, which are vector bundles in the category of Lie groupoids and Lie algebroids respectively. Part of this resurgence is motivated by the representation theory of Lie groupoids and algebroids. However, the original definitions of 'VB-objects' are very complicated and not obvious. Bursztyn, Cabrera and de Hoyo realised last year that VB-groupoids and VB-algebroids can be neatly described using regular actions of the multiplicative monoid of real numbers a la Grabowski and Rotkiewicz. In this talk I will highlight how, rather naturally, one can generalise and simplify the notion of 'VB-objects' by using homogeneity structures; that is smooth actions of the multiplicative monoid of reals. It is known, via Grabowski and Rotkiewicz that homogeneity structures always lead to what they called graded bundles; these are manifolds with a non-negative grading on their structure sheaf and give particularly nice examples of polynomial bundles. Graded bundles can be viewed as a very natural generalisation of a vector bundle. Thus, we can pass from VB-groupoids to weighted Lie groupoids which we understand as graded bundles in the category of Lie groupoids, or indeed vice-versa. We will highlight natural examples of this very rich geometric theory and briefly describe the Lie theory relating weighted Lie groupoids and weighted Lie algebroids. Time permitting I will also describe weighted Poisson-Lie groupoids and weighted Courant algebroids. This talk is based on joint work with K. Grabowska and J. Grabowski.

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

Alfonso Giuseppe Tortorella (KMMF)

Deformations of coisotropic submanifolds in abstract Jacobi manifolds

In this talk, using the Atiyah algebroid and ﬁrst order multi-diﬀerential calculus on non-trivial line bundles, we attach an L∞-algebra to any coisotropic submanifold S in an abstract (or Kirillov’s) Jacobi manifold. Our construction generalizes and uniﬁes analogous constructions in the symplectic case (Oh and Park), the Poisson case (Cattaneo and Felder), locally conformal symplectic case (Le and Oh). As a new special case, we attach an L∞-algebra to any coisotropic submanifold in a contact manifold, including Legendrian submanifolds. The L∞-algebra of a coisotropic submanifold S governs the (formal) deformation problem of S.

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

Tomasz Zawadzki (Uniwersytet Łódzki)

Conformal submersions with totally umbilical fibers

Conformal submersions are a natural generalisation of Riemannian submersions. Under an additional assumption of fibers being totally umbilical (which is a reasonable geometric condition, due to the conformal invariance of umbilicity), we examine those mappings, obtaining their existence conditions expressed in terms of curvatures of the domain and the image. We also present a number of examples of conformal submersions with umbilical fibers and discuss their relations with Riemannian submersions.

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

Tomasz Tylec (CFT PAN)

Quantum logics and generalised probability

Immediately after formulation of quantum mechanics there were attempts to “derive” it from some small set of plausible axioms. The two most successful approaches were: algebraic axiomatisation (started by Jordan, von Neumann and Wigner) and so called quantum logic approach, based on the partially ordered structure of “propositions” about physical system (started by Birkhoff and von Neumann). During many years of development the latter approach became mostly the theory of partially ordered structures with a little application in physics.Since 1990s some physicist working on quantum information theory become interested in the so-called no-signalling theories. These are very intuitive, in some sense minimal, models of compound systems. It seemed that such theories are more general than quantum mechanics.Our idea is to rigorously reformulate assumptions of non-signalling theories in the framework of quantum logics and then study its properties and true relation to quantum mechanics.

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

Katarzyna Grabowska (KMMF)

Dirac algebroids in action

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

Witold Respondek (KMMF)

Linearization of mechanical control systems

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

Tatiana Shulman (IMPAN)

Completely positive maps and zero-error quantum information theory

In quantum information theory for mathematical description of quantum channels one uses the notion of completely positive maps on matrix spaces. In the first part of the talk we will discuss general theory of completely positive maps and their connection with quantum channels. In the second part of the talk I will focus on some mathematical problems arising in zero-error quantum information theory, namely I will talk on various zero-error capacities of quantum channels and superactivation effect. This is ajoint work with M. Shirokov.

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

Michał Jóźwikowski (IMPAN)

On symmetries of differential equations

In the talk I will discuss symmetries of differential equations from the geometric viewpoint in the spirit of Alexandre M. Vinogradov. As a particular application I will discuss symmetries of a k-th order ODE

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

Piotr Stachura (KMMF)

Short and biased introduction to groupoids

The approach to groupoids started by S. Zakrzewski will be recalled. In particular various (but equivalent) definitions of morphisms and actions of groupoids will be presented with examples (mainly on algebraic level).

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

Bronisław Jakubczyk (IMPAN)

Regular control systems and Lagrange geometry (part II)

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

Bronisław Jakubczyk (IMPAN)

Regular control systems and Lagrange geometry

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

Mikołaj Rotkiewicz (KMMF)

On applications and generalizations of the canonical kappa-map

The canonical kappa involution for the iterated tangent bundle TTM has a surprising number of important applications and interesting generalizations. One can study the kappa map for M being a Lie groupoid. One can also look for generalizations to the higher order tangent bundles. These questions will be discussed in my talk which is partially based on my joint paper with Michał Jóźwikowski "Prototypes of higher algebroids with applications to variational calculus", arXiv:1306.3379.

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

Paweł Urbański (KMMF)

Symplectic groupoid actions and reductions (part II)

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

Paweł Urbański (KMMF)

Symplectic groupoid actions and reductions (part I)

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

Andrew James Bruce (IMPAN)

Weighted algebroids: theory and outlook for applications ( part II )

In this talk I will outline the theory of the recently discovered weighted Lie algebroids, which should be considered as a higher version of a Lie algebroid. We will then suggest how such structures can be employed in higher order Lagrangian mechanics. This is joint work with K. Grabowska and J. Grabowski.

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

Andrew James Bruce (IM PAN)