This article is only in Czech.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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Přednášky v jarním semestru 2016
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
Přednášky v podzimním semestru 2014
- 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-manifoldsÂÂÂÂÂÂÂÂÂ of a compact semisimple groupÂÂÂÂÂÂÂÂÂ $G$ 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
TBA
- 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
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