Seminář z diferenciální geometrie PDF Tisk

Seminář se koná v učebně M5 (a/nebo online) Ústavu matematiky a statistiky, budova 08, areál Přírodovědecké fakulty, Kotlářská 2, Brno, vždy v 10:00.

Vedoucí semináře:

  • Prof. RNDr. Jan Slovák, DrSc.,
  • Prof. RNDr. Josef Janyška, DSc.,
  • Doc. Josef Šilhan, PhD. (programový vedoucí),

Přednášky v podzimním semestru 2023

  • 25. 9. 2023 (start at 10:00, lecture room M5 and MSTeams)
    Ioannis Chrysikos
    On the geometry of torsion-free SO*(2n)Sp(1)-structures and related bundle constructions

    In this talk, first we will focus on the 2nd order geometry of torsion-free SO*(2n)Sp(1)-structures, where SO*(2n) is the quaternionic real form of SO(2n, C).  This includes a curvature characterization of the so called ``quaternionic skew-Hermitian manifolds'' (torsion-free case), and other related results.
    We will also discuss the construction of SO*(2(n+1))-structures on the Swann bundle over a quaternionic skew-Hermitian manifold (M^{4n}, Q, \omega) with $n>1$.
    Time permitting, we will describe conditions related to their 1st-order integrability. This provides a generalization of the well-known construction of (pseudo) hyper-Kähler structures on the Swann bundle over a (pseudo) quaternionic Kählerian manifold  (M^{4n}, Q, g).
    The talk is based on a joint work with V. Cortés and J. Gregorovič.

Přednášky v jarním semestru 2023

  • 17. 4. 2023 (start at 10:00, lecture room M5 and MSTeams)
    Igor Zelenko
    Morse inequalities for ordered eigenvalues of generic families of self-adjoint matrices

    The topic of the talk was originally motivated by the Floquet-Bloch theory of Schroedinger equations with periodic potential and other problems in Mathematical Physics and also related to the topology of vector-valued quadratic maps,  motivated by second-order conditions in optimal control. The eigenvalue branches of families of self-adjoint matrices are not smooth at points corresponding to repeated eigenvalues (called diabolic points or Dirac points). Generalizing the notion of critical points as points for which the homotopical type of (local) sub-level set changes after the passage through the corresponding value, in the case of the generic family we give an effective criterion for a  diabolic point to be critical for those branches and compute the contribution of each such critical point to the Morse polynomial of each branch, getting the appropriate Morse inequalities as a byproduct of the theory. These contributions are expressed in terms of the homologies of Grassmannians. The talk is based on the joint work with Gregory Berkolaiko.
  • 20. 3. 2023 (start at 10:00, lecture room M5 and MSTeams)
    Jan Gregorovič
    First BGG operators on homogeneous conformal geometries

    I will talk about first BGG operators and their solutions on homogeneous conformal geometries. I will describe an invariant calculus that allows us to find solutions explicitly using only algebraic computations. I will discuss applications to holonomy reductions and conserved quantities of conformal circles.

  • 6. 3. 2023 (start at 10:00, lecture room M5)
    Henrik Winther
    Minimal Projective Orbits of Semi-simple Lie Groups

    Let \(G\) be a Lie group \(G\) with representation \( \rho\) on a real simple \(G\) -module \( \mathbb{V} \). We will call the orbits of the induced action of \( \rho\) on the projectivization \( P\mathbb{V} \) the projective orbits, and projective orbits of lowest possible dimension will be called minimal. We show that when \(G\) is semi-simple and non-compact, there exists a compact subgroup \(K\) \(\subset\) \(G\) such that the minimal orbits of \(G\) are in bijection with the minimal \(K\)-orbits on a \(K\)-invariant proper subspace \( \mathbb{W}\subset \mathbb{V} \). In the case that \(G\) is split-real, \(K\) is the trivial subgroup and the minimal projective orbit is unique.

Přednášky v podzimním semestru 2022

  • 7. 11. 2022 (start at 10:00, lecture room M5 and MSTeams)
    Jan Slovák
    Bundles of Weyl structures and invariant calculus for parabolic geometries

    For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian and projective geometries. More recently, several general tools were presented for the entire class of parabolic geometries, i.e., the Cartan geometries modelled on homogeneous spaces G/P with P a parabolic subgroup in a  semi-simple Lie group G. Similarly to conformal Riemannian and projective structures, all these geometries determine a class of distinguished affine connections, which carry an affine structure modelled on differential 1-forms Υ. They correspond to reductions of P to its reductive Levi factor, and they are called the Weyl structures similarly to the conformal case. The standard definition of differential invariants in this setting is as affine invariants of these connections, which do not depend on the choice within the class. In this article, we describe a universal calculus which provides an important first step to determine such invariants. We present a natural procedure how to construct all affine invariants of Weyl connections, which depend only tensorially on the deformations Υ.

  • 3. 10. 2022 (start at 10:00, lecture room M5 and MSTeams)
    Sam Blitz
    Introduction to Conformal Hypersurface Geometry

    Hypersurfaces embedded in conformal manifolds appear naturally in string theory, and their geometry is an invaluable tool for studying interesting problems in differential geometry. While Weyl's classical invariant theory makes easy work of diffeomorphism invariants in Riemannian geometry, it is less obvious how to restrict these invariants to only those that respect an underlying conformal structure. This remains the case when studying embedded hypersurfaces. We are thus faced with the problem of how one might generate such invariants; in doing so, we are led to tractors, an invaluable tool for studying conformal geometry, amongst other things. By analogizing to the Riemannian setting, we can develop a hypersurface tractor calculus, enabling us to generate a (perhaps incomplete) set of conformal hypersurface invariants.

Přednášky v jarním semestru 2022

  • 9. 5. 2022 (start at 10:00, lecture room M5 and MSTeams)
    Josef Šilhan
    Modified Patterson-Walker construction

    The classical Patterson-Walker (spit signature) metric is defined on the cotangent bundle of an affine manifold. A slight generalisation leads to a projective-to conformal version of this construction. There are several ways how to modify this construction and we shall discuss certain class of these modifications in details. In particular, we shall identify conformal symmetries 'upstairs' in terms of suitable projective data 'downstairs'. We shall also discuss almost Einstein scales in a similar way. (This is a joint work with M. Hammerl, K. Sagerschnig a V. Žádník.)

  • 3. 5. 2022 (start at 10:00, lecture room M3 and MSTeams)
    Gerd Schmalz
    CR manifolds with symmetries and the embeddabilty problem

    It is a classical result by Jacobowitz that a hypersurface type CR manifold $M$ with complex structure $V$ can be locally realised as a hypersurface in complex space if and only if there exists a complex vector field $Z$ transversal to $V$ and such that $[Z,V]\subseteq V$. Hill and Nacinovich proved the following generalisation: If $(M,V)$ is a  CR manifold of type $(n,k)$ and there exists a solvable Lie algebra of complex vector fields of dimension $\ell\le k$ transversal to $V$ then $(M,V)$ can be embedded into a CR manifold $(M'V')$ of type $(n+\ell, k-\ell)$. In particular, if $k=\ell$ this is an embedding into complex space. I will present a generalisation for Hill and Nacinovich's theorem without the assumption of solvability of the Lie algebra. This is joint work with M. Cowling and A.Ottazzi (UNSW) and Masoud Ganji (UNE).
  • 14. 2. 2022
    Ondřej Hulík
    Generalized and Exceptional geometry and its relation to supergravity and M theory

Přednášky v podzimním semestru 2021

  • 29. 11. 2021
    Mauro Mantegazza
    Towards a non-commutative generalization of the jet functor

  • 22. 11. 2021
    Giovanni Russo
    Nearly Kahler six-manifolds with two-torus symmetry

  • 25. 10. 2021
    David Sykes
    On geometry of 2-nondegenrate, hypersurface-type Cauchy-Rieman structures encoded by dynamical Legendrian contact structures

  • 11. 10. 2021
    Henrik Winther
    Differential geometry of $SO*(2n)$-structures and $SO*(2n)Sp(1)$-structures

Přednášky v podzimním semestru 2017

  • 13. 11. 2017
    Radoslaw Kycia
    Topological modelling of nuclear pasta phases

  • 6. 11. 2017
    Jan Slovák
    Cartan-Kähler theory -- application to quaternionic contact structures

  • 30. 10. 2017
    Josef Šilhan
    Interesting connections in the projective class, and obstructions

Přednášky v jarním semestru 2017

  • 22. 5. 2017
    Henrik Winther
    Non-degenerate Almost- and Para-Complex structures, and their Symmetries

  • 24. 4. 2017
    Sumit Kaushik

    Curve Evolution under Extrinsic metrics for DTI processing

  • 10. 4. 2017
    Jan Gregorovič
    On standard models of CR and para--CR manifolds

  • 20. 3. 2017
    Vojtěch Žádník
    The curves, or from Frenet apparatus to Thomas tractors and back again

  • 6. 3. 2017
    Jan Slovák:
    Constant curvature models in sub-Riemannian geometry

Přednášky v podzimním semestru 2016

  • 12. 12. 2016
    Josef Šilhan
    Curves in conformal and projective geometries

  • 21. 11., 2016, the lecture room M5:
    Miroslav Kureš

  • 14. 11. 2016, the lecture room M5:
    Phan Thanh Nam

    Recent progress on the ionization problem

  • 31. 10. 2016
    10.00 - 10.50
    Ivan Minchev
    On the generality of quaternionic contact structures

    11.00 - 11.50
    Gerd Schmalz
    Chern-Moser theory for para-CR-manifolds and degenerate multi-contact structures

  • 24. 10. 2016
    Ilya Kossovskiy

    The Associated Differential Equations Method in CR-geometry

  • 10. 10. 2016
    Arman Taghavi-Chabert

Přednášky v jarním semestru 2016

  • 13. 6. 2016
    Stefan Ivanov
    The quaternionic contact Yamabe problem

  • 9. 5. 2016
    Josef Šilhan
    Higher supersymmetries

  • 25. 4. 2016
    Diana Barseghyan
    (University of Ostrava)
    Spectral analysis of a class of Schroedinger operators exhibiting a
    parameter-dependent spectral transition

  • 11. 4. 2016
    Anton Galaev
    Special Kähler-Lorentz metrics

  • 14. 3. 2016
    Jan Slovák

  • 7. 3. 2016
    Vladimir Ezhov (Flinders University/ MPIM Bonn)
    New familiy of unbounded homogeneous tube domains
    in Cn

  • 29. 2. 2016
    Ioannis Chrysikos
    Spin and metaplectic structures on homogeneous spaces

  • 22. 2. 2016
    Matthias Hammerl (University of Greifswald):
    Holography of BGG-Solutions

Přednášky v podzimním semestru 2015

  • 7. 12. 2015
    Katja Sagerschnik
    Nurowski's conformal structures with almost Einstein scales

  • 30. 11. 2015
    Matthew Randall

  • 23. 11. 2015
    Vojtěch Žádník
    Report on Lie contact structures

  • 2.11.2015
    Gueo Grantcharov (the Florida International University, USA):
    On HKT geometry

  • 26.10.2015
    Matthew Burke
    Synthetic Lie Theory Part I: An Introduction to Synthetic Differential Geometry

    Parts II and III will be presented on the seminar on algebra on October 29 and Novenber 5 (1pm, seminar room)

  • 12.5.2015
    Anton Galaev
    Holonomy algebras of Einstein pseudo-Riemannian manifolds

Přednášky v jarním semestru 2015

  • 18.5.2015
    Yaroslav Bazaykin

    Numerical analysis of topological characteristics of three-dimensional geological models of oil and gas fields

    Zoran Škoda
    Hopf algebroids of differential operators

  • 11.5.2015
    Ioannis Chrysikos
    Killing spinors with torsion and applications

  • 27. 4. 2015
    Zdeněk Dušek
    How many are affine connections of special types

  • 20. 4. 2015
    Michal Marvan

    Integrable surfaces II

  • 13. 4. 2015
    Arman Taghavi-Chabert:
    Twistorial description of null foliations

  • 30.3.2015
    Dmitri Alekseevsky

    Conformally homogeneous manifolds

  • 2. 3. 2015
    Anton Galaev
    Classification of third-order symmetric Lorentzian manifolds

  • 9. 3. 2015
    Jan Slovák
    Linearized metrizability problem for Parabolic Geometries
    -- general procedure and examples

Přednášky v podzimním semestru 2014

  • 15.12. 2014
    Christian Gustad
    Regular and singular contact actions of Lie algebras

  • 8. 12. 2014
    Ioannis Chrysikos
    Killing and twistor spinors with torsion on compact naturally reductive spaces

  • 1. 12. 2014
    Jean-Philippe Michel
    Dirac operators and their symbols

  • 10. 11. 2014
    Georgy Sharygin
    Local formulas for characteristic classes

  • 3. 11. 2014
    Dmitri Alekseevski
    Homogeneous pseudo-Riemannian Einstein metrics associated with
    graded semisimple Lie algebras

  • 13. 10. 2014
    Ivan Kolář
    On the vertical Weil bundles

  • 6. 10. 2014
    Anton Galaev
    How to compute the holonomy algebra of a Lorentzian manifold

  • 24. 9. 2014
    Stefan Ivanov
    The Lichnerowicz-Obata sphere theorems on a quaternionic contact
    manifold of dimension bigger than seven"

  • 22. 9. 2014
    Dimiter Vassilev
    Lichnerowicz-Obata sub-laplacian eigenvalue theorem
    in CR geometry under a positive "Ricci" bound

Přednášky v jarním semestru 2014

  • 12. 5. 2014
    Dimiter Vassilev
    Quaternionic contact Liouville theorem and applications

  • 5. 5. 2014
    Ivan Kolář
    On general connections

  • 14. 4 .2013
    Dmitri Alekseevski

  • 7. 4. 2014
    Yaroslav Bazaykin
    Stability of integral persistence diagrams

  • 24. 3. 2014 ,12:00--13:50, lecture room M4
    25. 3. 2014, 10:00--11:50, lecture room MS1
    27. 3. 2014, 10:00--11:50, lecture room M4
    31. 3. 2014, 12:00--13:50, lecture room M4
    Evgeny Malkovich
    Construction of metrics with special holonomies via geometrical flows
    (series of lectures)

  • 10. 3. 2014
    Katja Sagerschnig
    Conformal structures in dimension four and generic rank two distributions in
    dimension five

  • 3. 3. 2014
    Anton Galaev
    Irreducible holonomy algebras of Riemannian supermanifolds

Přednášky v podzimním semestru 2013

  • 25. 11. 2013
    Jonathan Kress
    Recent progress in superintegrable systems

  • 18.11. 2013
    Ivan Minchev
    Quaternionic contact hypersurfaces in hyper-Kaehler geometry

  • 11. 11. 2013
    Anton Galaev
    Holonomy groups of superconnections on supermanifolds

  • 4. 11. 2013
    Giovanni Moreno
    Meta-symplectic geometry of 3rd order Monge-Ampére equations

  • 21. 10. 2013
    Jan Gregorovič
    Generalized symmetries of homogeneous parabolic geometries

  • 7. 10. 2013
    Ivan Kolář
    General connections and the Froehlicher-Nijenhuis bracket

  • 30. 9. 2013
    Dmitri Alekseevsky
    Classification of cohomogeneikty one Kaehler G-manifolds  of a compact semisimple group  $G$ in terms of painted Dynkin diagrams

Přednášky v jarním semestru 2013

  • 13. 5. 2013
    John Ryan
    Dirac type operators and conformal groups

  • 15. 4. 2013
    Christian Gustad

  • 8. 4. 2013
    Anton Galaev
    Pseudo-Riemannian manifolds with recurrent spinor fields

  • 25. 3. 2013
    Josef Šilhan
    On symmetries of the Laplacian

  • 18. 3. 2013
    Ivan Kolář
    On natural transformations of Weil bundles

  • 4. 3. 2013
    Rikard von Unge
    Generalized Kahler reduction

  • 25. 2. 2013
    Joseph Krasiľshchik
    On geometry of integrable systems and its cohomological background

Přednášky v podzimním semestru 2012

  • 10. 12. 2012
    Jan Slovák
    Conformal Fedosov manifolds

  • 26. 11. 2012
    Dmitri Alekseevsky

  • 19. 11. 2012
    Ivan Kolář
    On the functorial prolongations of fiber bundles

  • 12. 11. 2012
    Ioannis Chrysikos
    The Dirac operator on non-naturally reductive spaces

  • 29. 10. 2012
    Andrea Santi
    On the automorphism group of extended Poincaré structures

  • 15. 10. 2012
    Anton Galaev
    Algorithm for deRham-Wu decomposition for Riemannian and
    Lorentzian manifolds

  • 1. 10. 2012
    Jan Gregorovič
    Geometric structures invariant to symmetries
Aktualizováno Úterý, 19 Září 2023 16:19