### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

#### Program v akademickém roce 2019/2020

4. 6. 2020

doc. RNDr. Artur Sergyeyev, Ph.D., Nelokální zákony zachování pro Przanowského rovnici

Abstrakt:

Uvádíme nekonečnou hierarchii nelokálních zákonů zachování pro Przanowského rovnici, integrabilní PDR druhého řádu lokálně ekvivalentní s antisamoduálními Einsteinovými rovnicemi ve vakuu s nenulovou kosmologickou konstantou. Zmíněná hierarchie byla získána s využitím neisospektrální laxovské representace pro zkoumanou rovnici. Jako vedlejší důsledek jsme získali nekonečněrozměrné diferenciální nakrytí nad Przanowského rovnicí. Jedná se o společnou práci s prof. Krasil'shchikem, podrobnosti viz I. Krasil’shchik, A. Sergyeyev, Ann. Henri Poincaré 20 (2019), 2699-2715, https://doi.org/10.1007/S00023-019-00816-0

12. 12. 2019

doc. Igor Khavkine, PhD. (Matematický ústav AV ČR, Praha), Initial data for closed conformal Killing-Yano 2-forms

Abstrakt:

The Kerr-NUT-(A)dS family of exactly integrable higher dimensional black hole solutions of Einstein's equations is characterized by the existence of a non-degenerate closed conformal Killing-Yano (cCYK) 2-form. Using an exhaustive search, we identify a family of 2nd order propagation identities for the cCYK equation on 2-forms in n>4 dimensions. These identities allow us to project the cCYK equations onto a spacelike surface and thus characterize the initial data for Einstein's equations whose development admits a cCYK.

5. 12. 2019

RNDr. Hynek Baran, Ph.D., Infinitely Many Commuting Nonlocal Symmetries for Modified Martínez Alonso–Shabat Equation

Abstrakt:

We present a recursion operator and an infinite hierarchy of nonlocal commuting symmetries for the modified Martínez Alonso–Shabat equation uy uxz+α ux uty−(uz+α ut)uxy=0.

14. 11. 2019

doc. Roman Popovych, D.Sc. (University of Vienna, Rakousko), Conditional symmetries of linear partial differential equations with two independent variables

Abstrakt:

We discuss reduction operators, i.e., operators of nonclassical (conditional) symmetry, of (1+1)-dimensional linear partial differential equations.

In particular, the reduction operators of (1+1)-dimensional second-order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. This “no-go” result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints.

The consideration of the above equations and of the (1+1)-dimensional linear rod equation is used to illustrate a new theorem on linear reduction operators of general linear partial differential equations.

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

#### Program v akademickém roce 2018/2019

27. 6. 2019

dr hab. Andrzej Frydryszak, prof. UWr (Uniwersytet Wrocławski, Polsko) Functions of Nilpotent Variables and Entanglement

Abstrakt:

I will give an introduction to the description of a new quantum mechanical resource called entanglement. In its basic version it is related to the question whether a tensor which is used to represent a physical state of a compound system is a simple tensor or not. Then, a natural desire arises to define a 'measure' of its departure from being simple. This yields to the study of sets of invariants of products of the SU(2) groups. I will present elements of the formalism of functions of nilpotent variables and I will explain how entanglement questions concerning the pure states find natural solutions in this approach, producing relevant invariants.

20. 6. 2019

RNDr. Petr Vojčák, Ph.D. Nelokální symetrie čtyřdimenzionální Martínez Alonsovy - Shabatovy rovnice

Abstrakt:

V přednášce budou prezentovány průběžné výsledky výzkumu nakrytí a hierarchií nelokálních symetrií čtyřdimenzionální Martínez Alonsovy - Shabatovy rovnice.

25. 4. 2019

RNDr. Petr Vojčák, Ph.D., Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation

18. 4. 2019

doc. RNDr. Artur Sergyeyev, Ph.D., The Gardner method for symmetries (after the paper by A. Rasin and J. Schiff)

15. 2. 2019

Roman Popovych, D.Sc. (University of Vienna, Rakousko), Extended symmetry analysis of isothermal no-slip drift flux model II

Jedná se o volné pokračování předchozí přednášky.

Seminář proběhne v mimořádném termínu v pátek 15. 2. od 14.00.

7. 2. 2019

Roman Popovych, D.Sc. (University of Vienna, Rakousko), Extended symmetry analysis of isothermal no-slip drift flux model

Abstrakt:

We perform extended group analysis for a system of differential equations modeling an isothermal no-slip drift flux. The maximal Lie invariance algebra of this system is proved to be infinite-dimensional. We also find the complete point symmetry group of this system, including discrete symmetries, using the megaideal-based version of the algebraic method. Optimal lists of one- and two-dimensional subalgebras of the maximal Lie invariance algebra in question are constructed and employed for obtaining reductions of the system under study. Since this system contains a subsystem of two equations that involves only two of three dependent variables, we also perform group analysis of this subsystem. The latter can be linearized by a composition of a fiber-preserving point transformation with a two-dimensional hodograph transformation to the Klein-Gordon equation. We also employ both the linearization and the generalized hodograph method for constructing the general solution of the entire system under study. We find inter alia genuinely generalized symmetries for this system and present the connection between them and the Lie symmetries of the subsystem we mentioned earlier. Hydrodynamic conservation laws and their generalizations are also constructed.

This is joint work with S. Opanasenko, A. Bihlo and A. Sergyeyev.

6. 12. 2018

doc. RNDr. Artur Sergyeyev, Ph.D. The Gardner method for symmetries (after Rasin and Schiff)

Přednáška začne v 14.50.

22. 11. 2018

Priscila Leal da Silva, Ph.D. (Universidade Federal de São Carlos, Brazílie), Classification of bounded travelling wave solutions of Dullin-Gotwald-Holm equations

Abstrakt:

In this talk we will discuss the problem of existence of bounded traveling wave solutions of Dullin-Gottwald-Holm equations. The classification presents both smooth and weak solutions of the equations under consideration.

15. 11. 2018

Roman Popovych, D.Sc. (University of Vienna, Rakousko), Effective generalized equivalence groups for classes of differential equations

Seminář prof. Popovyche začne ve 14:30.

1. 11. 2018

Igor Leite Freire, D.Sc. (UFABC, Brazílie), Some results on the rotation Camassa-Holm equation

Abstrakt:

Recently, a mathematical model of the equatorial water waves with Coriollis effect was derived. This model is a generalization of the Dullin-Gottwald-Holm equation, which itself is a generalization of the Camassa-Holm equation. In this talk we shall discuss some results on the model in question, known as the rotation Camassa-Holm equation.

18. 10. 2018

Roman Popovych, D.Sc. (University of Vienna, Rakousko), Generalized symmetries and conservation laws of (1+1)-dimensional Klein--Gordon equation

Abstrakt:

We explicitly find the algebra of generalized symmetries of the (1+1)-dimensional Klein--Gordon equation, which allows us to describe this algebra in terms of the universal enveloping algebra of the essential Lie invariance algebra of the Klein--Gordon equation. Then we single out variational symmetries of this equation and compute the space of its local conservation laws.

11. 10. 2018

doc. RNDr. Artur Sergyeyev, Ph.D., Integrable systems in 4D from contact geometry

Abstrakt:

In this talk we present a survey of results from the paper A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376.

4. 10. 2018

Igor Leite Freire, D.Sc. (UFABC, Brazílie), A look on some results about Camassa-Holm type equations

Abstrakt:

We discuss some results on families of equations including the Camassa-Holm and Novikov equations, and the generalizations thereof. The talk is based on joint works with S.C. Anco, P. L. da Silva, and D.C. Ferraioli.

26. 9. 2018

Seminář je spojen se slavnostním seminářem prof. Smítala a doc. Štefánkové - zahájení mimořádně od 12:00 v aule:

Martin Černohorský, Úspěch historie a lingvistiky při rehabilitaci Newtona a jeho prvního axiomu pohybu vykládaného po tři staletí neoprávněně jako jen pouhý zvláštní případ druhého axiomu (kolokviální přednáška).

Semináře se obvykle konají ve čtvrtek od 14.00 hod. v budově Matematického ústavu, Na Rybníčku 1 v Opavě, v místnosti R1. Všichni zájemci jsou srdečně zváni.

#### Program v akademickém roce 2017/2018

30. května 2018

Jakub Vašíček, Symmetries and conservation laws for a generalization of Kawahara equation
Seminář se koná v mimořádném termínu 30. 5. 2018 v 15:35.

3. května 2018

doc. RNDr. Artur Sergyeyev, Ph.D., Integrabilní systémy v dimenzi (3+1) s algebraickými Laxovskými páry

5. dubna 2018

Mgr. Jíři Jahn, Ph.D., O jedné zajímavé vlastnosti kanonických transformací a jejím využití

11. prosince 2017

prof. RNDr. Josef Mikeš, DrSc. (Univerzita Palackého v Olomouci), Geodetická zobrazení a jejich zobecnění
Seminář se mimořádně koná v pondělí od 14:45 v místnosti RZ

7. prosince 2017

Mgr. Pavel Novák, Weylův tensor a úvod do tetrádního formalismu

Mgr. Adam Hlaváč, Exact solutions of the constant astigmatism equation revisited

Seminář se mimořádně koná ve středu od 15:35

mgr. Aleksandra Lelito (AGH, Kraków, Polsko), Nonlocal conservation laws for some three-dimensional partial differential equations

prof. dr hab. Maciej Błaszak (Adam Mickiewicz University, Poznań, Polsko), Non-homogeneous dispersionless systems and quasi-Stäckel Hamiltonians

Prof. Willard Miller Jr. (University of Minnesota, USA), Superintegrability and exactly solvable problems in classical and quantum mechanics

26. října 2017

RNDr. Petr Blaschke, Ph.D., Proč by se měly hypergeometrické funkce vyučovat na střední škole?

14. září 2017

Dr. Maxim Pavlov (Moskva), Four-dimensional Linearly Degenerate Equations of Second Order and their Three-dimensional Hydrodynamic Reductions

Abstrakt:

We consider one of four-dimensional equations belonging to Hyper-Kahler hierarchy.

We show that infinitely many three-dimensional equations can be extracted as reductions from this four-dimensional equation.

7. září 2017

Dr. Maxim Pavlov (Moskva), Integrability of Exceptional Hydrodynamic Type Systems

Abstrakt:

We consider integrable non-diagonalisable hydrodynamic type systems with a single multiple root of a characteristic polynomial of a velocity matrix.

We apply the Extended Hodograph Method to construct a general solution.

For the Mikhalev three-dimensional linearly degenerate equation of second order we present simplest solution selected by three-component integrable non-diagonalisable hydrodynamic type system with a triple root.

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

#### Program v akademickém roce 2016/2017

15. května 2017

Mgr. Pavel Novák, Ekvivalence čtyřdimenzionálních metrik

10. dubna 2017

Mgr. Adam Hlaváč, Plochy konstantního astigmatismu -- nejnovější výsledky

13. března 2017

Roman Popovych, D.Sc., Structures formed by point transformations in classes of differential equations III

6. března 2017

Roman Popovych, D.Sc., Structures formed by point transformations in classes of differential equations II

27. února 2017

Roman Popovych, D.Sc., Structures formed by point transformations in classes of differential equations

12. prosince 2016

Roman Popovych, D.Sc., Contractions of Lie algebras

5. prosince 2016

prof. I.S. Krasil'shchik, DrSc. (Moskva, Rusko), Reductions, coverings and nonlocal symmetries of 3D Lax integrable equation

Roman Popovych, D.Sc., Direct and inverse problems for conservation laws II

Roman Popovych, D.Sc., Direct and inverse problems for conservation laws

Diego Catalano Ferraioli, Ph.D. (UFBA, Brazílie), Local isometric immersions of pseudo-spherical surfaces described by equations of the form zt=A(x,t,z)zxx+B(x,t,z,zx)

23. října 2016

Mgr. Adam Hlaváč, More exact solutions of the constant astigmatism equation

10. října 2016

RNDr. Hynek Baran, Ph.D., Databáze exaktních řešení Einsteinových rovnic v Maple

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

#### Program v akademickém roce 2015/2016

26.05.2016

Oleg Morozov, Dr.Sc. (AGH University of Science and Technology, Krakov, Polsko), Deformed cohomologies of symmetry pseudogroups and zero-curvature representations of differential equations

19.05.2016

P. Holba, Bc. J. Sedláková, J. Vašíček, Prezentace bakalářských a diplomových prácí

12.05.2016

prof. dr hab. Andrzej Borowiec (Wroclaw University, Polsko), Mathematical Aspects of Cosmology

14.04.2016

RNDr. Petr Blaschke, Ph.D., Pedální rovnice křivek a temný Keplerův problem

07.04.2016

doc. RNDr. Artur Sergyeyev, Ph.D., Integrabilní systémy s čtyřmi nezávisle proměnnými: nové příklady

Abstrakt:

Společná práce s M. Blaszakem, podrobnosti viz arXiv:1605.07592.

17.03.2016

Nonlocal conservation laws of the constant astigmatism equation

Abstrakt:

For the constant astigmatism equation, we construct a system of nonlocal conservation laws (an abelian covering) closed under the reciprocal transformations. We give functionally independent potentials modulo a Wronskian type relation.

This is joint work with Michal Marvan; please see arXiv:1602.06861 for details.

03.03.2016

doc. Pasha Zusmanovich, Ph.D., M.Sc. (Ostravská univerzita)

Abstrakt:

The Ado theorem is a basic fact in the theory of Lie algebras saying that any finite-dimensional Lie algebra admits a faithful finite-dimensional representation. Somewhat surprisingly, the standard proof of such a basic fact utilizes non-trivial facts about universal enveloping algebras and is quite involved. We will present an entirely different proof intrinsic to the category of finite-dimensional Lie algebras. If time will permit, we will also discuss a variant of this theorem for "commutative analogs" of Lie algebras, i.e. commutative algebras satisfying the Jacobi identity.

17.12.2015

On the constant astigmatism equation and surfaces of constant astigmatism

03.12.2015

doc. RNDr. Artur Sergyeyev, Ph.D., Zákony zachování ABC rovnice

Abstrakt:

Společná práce s I.S. Krasil'shchikem a O.I. Morozovem; viz arXiv:1511.09430

26.11.2015

RNDr. Petr Vojčák, Ph.D., Nakrytí a nelokální symetrie (3+1)-dimenzionálního zobecnění bezdisperzní Kadomtsevovy--Petviashviliho rovnice

Abstrakt:

V přednášce budeme uvažovat následující systém PDR:

qz = 2uz+wx+2wwz,
vz = 2qx-3ux-2wy+2wuz-2wwx+2uwz,
ut = vuz+qux-uvz-wvx+vy,
wt = qy-2vx+4wux-wqx+qwx+vwz-uqz.

Tento systém, který byl poprvé prezentován A. Sergyeyevem (viz. arXiv:1401.2122v2), poskytuje (3+1)-dimenzionální integrabilní zobecnění bezdisperzní Kadomtsevovy--Petviashviliho (dKP) rovnice. Hlavním cílem přednášky je prezentovat nalezená nakrytí uvedeného systému a rovněž i výsledky hledání nelokálních symetrií, které se objevují v jednotlivých nakrytích.

05.11.2015

prof. nadzw. AGH dr. hab. Vsevolod A. Vladimirov (AGH, Krakow, Polsko), On continual models connected with the chains of pre-stressed elastic bodies II

29.10.2015

prof. nadzw. AGH dr. hab. Vsevolod A. Vladimirov (AGH, Krakow, Polsko), On continual models connected with the chains of pre-stressed elastic bodies

(mimořádně v 15.25)

Abstrakt:

We consider a one-dimensional chain of strongly pre-stressed bodies interacting with each other by means of nonlinear forces. Passing to continual analog of such a system, we can obtain different nonlinear PDS, depending on a type of elastic force. Thus, in the case when the interaction has the form F(z)=A zn+B z, |A|=O(|B|), n>1, we get a Boussinesq equation, while in the case when B=0 Nesterenko's equation is obtained. However, if we assume that |A|=O(1), while |B|<<1 then, using a formal multi-scaled decomposition, we get a nonlinear evolutionary PDE, describing compactons (both bright and dark ones). Next, we will show that the compacton solutions possess many interesting features. In particular, they evolve in a self-similar mode and restore their shapes after mutual collisions.

15.10.2015

doc. RNDr. Michal Marvan, CSc., On an integrable class of Chebyshev nets

Abstrakt:

We study surfaces equipped with a Chebyshev net such that the Gauss curvature K and a curvature G of the net satisfy a linear condition αK + βG + γ = 0, where α,β,γ are constants. These surfaces form an integrable class. We point out some of its noteworthy peculiarities.

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

#### Program v akademickém roce 2014/2015

22.05.2015

Roman Popovych, D.Sc. (Wolfgang Pauli Institute, Rakousko),
Noether theorems

Abstrakt:

After recalling classical Noether's first and second theorems on conservation laws
of differential equations, we enhance and generalize Noether's second theorem.

More specifically, we prove that a system of differential equations is abnormal,
i.e., it has an identically vanishing differential consequence, if and only if
it possesses a trivial conserved vector corresponding to a nontrivial
characteristic.

Moreover, the above properties are also equivalent to that the system admits a family
of characteristics that is parameterized by an arbitrary function of all independent
variables in a local fashion and whose Frechet derivative with respect to the parameter-
function does not vanish on solutions of the system for a value of the parameter-function.
The theorem is illustrated by physically relevant examples.

14.05.2015

doc. RNDr. Artur Sergyeyev, Ph.D.
Noncommutative integrability of non-Hamiltonian systems

07.05.2015

Mgr. Jiří Jahn
Úvod do teorie komplexních variet

Abstrakt:

Cílem přednášky je seznámit posluchače se základními pojmy a výsledky z
oblasti komplexních variet. V průběhu přednášky se pokusíme vysvětlit, v čem
odpovídající protějšky elementárních konstrukci dobře známých z kategorie
hladkých variet (vektorová pole, diferenciální formy), a pokud čas dovolí,
skončíme komplexní obdobou Riemannovy variety (tou je tzv. Hermiteovska
varieta), resp. jejím asi nejdůležitějším speciálním případem (tzv. Kählerova varieta).

30.04.2015

doc. RNDr. Artur Sergyeyev, Ph.D.
Integrabilita S-deformovatelných ploch

Abstrakt:

viz arXiv:1501.07171 (společná práce s prof. I.S. Krasil'shchikem)

23.04.2015

Mgr. Diana Barseghyan, Ph.D. (Ostravská univerzita a Ústav jaderné fyziky AV ČR),
Schrödinger operators exhibiting spectral transition

Abstrakt:

In the first part of the talk we analyse two-dimensional Schrödinger operators
with the potential unbounded from below which demonstrate spectral tran-
sition. More precisely, we consider two-dimensional Schrödinger operators
with the potential |xy| p − λ(x 2 + y 2) p/(p+2) where p ≥ 1 and λ ≥ 0. We
show that there is a critical value of λ such that the spectrum for λ < λ crit
is bounded below and purely discrete, while for λ > λ crit it is unbounded
from below. In the subcritical case we prove upper and lower bounds for
the eigenvalue sums. As the second example we discuss a modification of
Smilansky model in which a singular potential ‘channel’ is replaced by a reg-
ular, below unbounded potential which shrinks as it becomes deeper. We
demonstrate that, similarly to the original model, such a system exhibits a
spectral transition with respect to the coupling constant, and determine the
critical value above which a new spectral branch opens.

The goal of the second part of the talk is to derive estimates of eigenvalue
moments for Dirichlet Laplacians and Schrödinger operators in regions having
infinite cusps which are geometrically nontrivial being curved; we are going
to show how those geometric properties enter the eigenvalue bounds.

In the third part of the talk we investigate Dirichlet Laplacian in a straight
twisted tube of a non-circular cross section, in particular, its discrete spec-
trum coming from a local slowdown of the twist. We prove a Lieb-Thirring-
type estimate for the spectral moments and present two examples illustrating
how the bound depends on the tube cross section.

16.04.2015

RNDr. Petr Vojčák, Ph.D.,
Jets: efektivní cesta od kosymetrie k zákonu zachování.

Abstrakt:

Hledání zákonů zachování systému diferenciálních rovnic v n nezávislých
proměnných lze rozdělit do dvou víceméně samostatných kroků.
Nejdříve hledáme řešení adjungovaného linearizovaného systému, kterými
jsou tzv. charakteristiky zákonů zachování (kosymetrie).

V druhém kroku pomocí nalezených kosymetrií sestavíme lagrangiány, pro
které hledáme jejich rozklad na totální divergenci nějaké n-tice
diferenciálních funkcí.

V této přednášce představíme proceduru naprogramovanou v softwaru Maple s
užitím balíčku Jets, která za jistých předpokladů poskytuje velmi
efektivní nástroj pro nalezení zákonů zachování příslušných k daným
kosymetriím. Představíme ovšem také několik příkladů, ve kterých uvedená
procedura selhává a prodiskutujeme příčiny, proč tomu tak je.

Jedná se o společnou práci s Hynkem Baranem.

09.04.2015

On multisoliton solutions of the constant astigmatism equation

Abstrakt:

26.03.2015

RNDr. Jiřina Jahnová, Ph.D.,
On the proof of Yoshida's conjecture

Abstrakt:

19.03.2015

doc. RNDr. Artur Sergyeyev, Ph.D.
Bezdispersní integrabilní systémy: od dimenze (2+1) k (3+1)

15.01.2015

prof. Maciej Blaszak (Uniwersytet im. Adama Mickiewicza, Poznan, Polsko)
Hamiltonian systems: from classical to quantum models IV: Quantum separability

08.01.2015

prof. Maciej Blaszak (Uniwersytet im. Adama Mickiewicza, Poznan, Polsko)
Hamiltonian systems: from classical to quantum models II: Riemannian
representation of quantum Hamiltonian dynamics

18.12.2014

prof. I.S. Krasil'shchik, DrSc.,
Symmetry reduction of Lax-integrable 3D systems and their properties

Abstrakt:

We present a complete description of 2-dimensional equations that arise as symmetry reductions of four 3-dimensional Lax-integrable equations:

(1) the universal hierarchy equation uyy=uzuxy-uyuxz;
(2) the 3D rdDym equation uty=uxuxy-uyuxx;
(3) the equation uty=utuxy-uyutx, which we call the modified Veronese web equation;
(4) Pavlov's equation uyy=utx+uyuxx-uxuxy.

For the reductions possessing finite-dimensional symmetry algebras, we describe nonlinear coverings and infinite hierarchies of nonlocal conservation laws.

This is joint work with H. Baran, O. Morozov, and P. Vojčák.

11.12.2014

Andrey Mironov, D.Sc. (Sobolev Institute of Mathematics,  Novosibirsk, Rusko) Algebro-geometric solutions of the equations of minimal Lagrangian surfaces in CP2

Abstrakt:

We suggest a method for constructing minimal Lagrangian immersions of R2 in CP2 with induced diagonal metric in terms of Baker–Akhiezer functions of algebraic curves.

04.12.2014

Andrey Mironov, D.Sc. (Sobolev Institute of Mathematics,  Novosibirsk, Rusko)
Integrable geodesic flows on 2-torus

Abstrakt:

We discuss classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on the coefficients of the polynomial integral we study the region (the so-called elliptic region) where there are complex-conjugate eigenvalues. We show that for quartic integrals the other two eigenvalues are real and necessarily genuinely nonlinear. This observation together with the property of the system to be rich (semi-Hamiltonian) enables us to classify elliptic regions completely. The case of complex-conjugate eigenvalues for the system corresponding to the integral of degree 3 is done similarly. These results show that if new integrable examples exist they could be found only within the region of hyperbolicity of the quasi-linear system.

01.12.2014 (mimořádně v pondělí v zasedací místnosti rektorátu SU v 14.45)

prof. Raffaele Vitolo (Universita del Salento, Itálie)
On the geometry of homogeneous third-order Hamiltonian operators

Abstrakt:

We will present old and new results on the affine and projective geometry of homogeneous third order Hamiltonian operators. We give a classification of such operators for a number of components less than or equal to 3. If such operators are part of a bi-Hamiltonian formulation of a given system of PDEs, then the structure of the operator can be expressed in terms of a sequence of homogeneous conservation law densities in involution coming from the Magri scheme. This fact can be used in opposite direction to reconstruct the third-order homogeneous operator and find new bi-Hamiltonian systems. The WDVV hydrodynamic type systems are considered as old and new examples of PDEs that can be presented as Hamiltonian systems using third-order homogeneous Hamiltonian operators.

27.11.2014

prof. Mirjana Djoric (University of Belgrade, Srbsko)
Submanifolds of complex projective space

Abstrakt:

If for a real submanifold M of a complex manifold (M',J), the holomorphic tangent space Hx(M)=JTx(M)Tx(M) has constant dimension with respect to x ε M,  the submanifold M is called the CR submanifold and the constant complex dimension is called the CR dimension of M.

In  M. Djoric, M. Okumura, CR submanifolds of complex projective space, Dev. in Math. 19, Springer, 2009 we collected  the elementary facts about complex manifolds and their submanifolds and introduced the reader to the study of CR submanifolds of complex manifolds, especially complex projective space.

Since in the case of maximal CR dimension there are two geometric structures: an almost contact structure F, induced from the complex structure J of the ambient space M', and a submanifold structure, represented by the second fundamental tensor h of M in M', it is interesting to study certain conditions on F and h, and we present some of these results.

In this lecture we also discuss submanifolds of real codimension two of a complex manifold, especially of a complex space form. An n-dimensional complex hypersurface, which is a CR submanifold of CR dimension (n-2)/2, is a submanifold of real codimension two, but there also exist submanifolds of real  codimension two which are not CR submanifolds. We  present some new classification theorems for  submanifolds of real codimension two of a complex space form under the algebraic condition on the second fundamental form of the submanifold and the endomorphism induced from the almost complex structure on the tangent bundle of the submanifold.

This talk is based on joint research with M. Okumura.

20.11.2014

Pasha Zusmanovich, PhD. (Ostravská univerzita)
Cohomology and deformations of current Lie algebras

Abstrakt:

I will discuss cohomology and deformations of current Lie algebras, i.e.
Lie algebras formed as a tensor product of a Lie algebra and an
associative commutative algebra, as well as a more general class of Lie
algebras arising from any pair of algebras over Koszul dual operads. I
will try to explain why these algebras are interesting, and touch upon
their numerous applications in mathematics and physics.

10.11.2014 (mimořádně v pondělí v zasedací místnosti MÚ v 15.35)

O.I. Morozov, D.Sc. (AGH, Krakow, Polsko)
Integrable dispersionless PDEs in 4D, their symmetry pseudogroups and deformations

Abstrakt:

We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry pseudogroups have a distinguished graded structure, uniquely determining the equations. This is used to deform the equations into new integrable equations of Monge-Ampere type having large symmetry pseudogroups. We classify these integrable symmetric deformations, and we discuss their geometry and integrability.

06.11.2014

prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Invariants or observables in relativity theory II

Abstrakt:

See the abstract of the lecture of 03.11.2014.

04.11.2014 (mimořádně v úterý v místnosti R2 v 14.45)

Mgr. Vojtěch Pravda, Ph.D. (Matematický ústav AV ČR, Praha)
Universal spacetimes

Abstrakt:

Typical vacuum solutions of the Einstein equations do not solve the
vacuum equations of modified gravities (such as Lovelock gravity, quadratic
gravity etc). However, there is one exceptional class of Einstein spacetimes that
are immune to all corrections to Einstein equations following from the modified
gravities. These, so called universal spacetimes, solve vacuum equations of all
gravitational theories with the Lagrangian being a polynomial curvature invariant
constructed from the metric, the Riemann tensor and its derivatives of arbitrary
order. It is known since 1990s that pp-waves are universal. In this talk, we will
present our recent results on various more general classes of universal
spacetimes.

03.11.2014 (mimořádně v pondělí v zasedací místnosti rektorátu SU v 14.45)

prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Invariants or observables in relativity theory I

Abstrakt:

In these lectures, based on joint research with Alexei Kotov and Valerii Yumaguzhin, I will discuss differenatial invariants of the diffeomorphism groups in the following three cases:
1. Riemann manifolds
2. Einstein manifolds, i.e. solutions of the vacuum Einstein equations.
3. Einstein-Maxwell manifolds, i.e. solutions of the Einstein-Maxwell equations.
In all three cases it will be shown how to compute numbers of independent differential invariants and find Hilbert and Poincare functions of the corresponding differential invariant algebras.
It is also will be discussed methods of finding basic differential invariants and the corresponding factor-equations.

30.10.2014

RNDr. Petr Vojčák, Ph.D.,
Coverings and nonlocal symmetries of some 3-dimensional PDEs

Abstrakt:

In this talk we will consider four 3-dimensional Lax-integrable partial
differential equations, namely
(1) the 3D rdDym equation uty=uxyux-uxxuy,
(2) the universal hierarchy equation uyy=uxyuz-uxzuy,
(3) the modified Veronese web equation uty=uxyut-utxuy,
(4) Pavlov's equation uyy=utx+uxxuy-uxyux.
For the equations in question, we will construct some coverings and also
This is a joint work with H. Baran, I.S. Krasil'shchik and O.I. Morozov.

23.10.2014

RNDr. Jiřina Jahnová, Ph.D.,
On symmetries and conservation laws of the Majda–Biello system

Abstrakt:

In 2003, A.J. Majda and J.A. Biello derived and studied the so-called
reduced equations for equatorial baroclinic–barotropic waves, to which we
refer as to the Majda–Biello system. The equations in question describe
the nonlinear interaction of long-wavelength equatorial Rossby waves and
barotropic Rossby waves with a significant midlatitude projection in the
presence of suitable horizontally and vertically sheared zonal mean flows.

In this talk we present a Hamiltonian structure for the Majda–Biello
system and describe all generalized symmetries and conservation laws for
the latter. It turns out that there are only three symmetries
corresponding to x-translations, t-translations and to a scaling of t, x,
u and v, and four conservation laws, one of which is associated with the
conservation of energy, the second conserved quantity is just the
Hamiltonian functional and the other two are Casimir functionals of the
Hamiltonian operator admitted by our system. Our result provides inter
alia a rigorous proof of the fact that the Majda–Biello system has just
the conservation laws mentioned in the paper by Majda and Biello.

16.10.2014

Roman Popovych, D.Sc. (Wolfgang Pauli Institute, Rakousko),
Linear potential framework of linear evolution equations

Abstrakt:

We study the linear potential framework for (1+1)-dimensional linear evolution equations of order not less than two. Related objects include linear potential systems and associated high-level potential equations as well as their conservations laws, cosymmetries, Lie symmetries and generalized symmetries. The main tool of the study is the use of multiple dual Darboux transformations, which map modified potential equations to the corresponding initial equations and establish the inverse maps between the associated adjoint equations. In particular, every potential conservation law of any (1+1)-dimensional linear evolution equation of even order proves to be linear and induced by a local conservation law of the same equation.

09.10.2014

Roman Popovych, D.Sc. (Wolfgang Pauli Institute, Rakousko),
Algebraic methods of group analysis of differential equations

Abstrakt:

Direct group classification is a tool for selection of modeling differential equations from classes of such equations parameterized by arbitrary constants of functions. The following criterion is used for the selection: Modeling differential equations have to admit the most extensive symmetry groups from the possible ones. The study of group classification problems is interesting not only from the purely mathematical point of view, but is also important for physical applications. The complexity of group classification led to the development of a great variety of specialized techniques, which are conventionally partitioned into two approaches. The first approach is based on the direct compatibility analysis and integration of the corresponding determining equations up to a relevant equivalence relation. We discuss the other approach, which is of algebraic nature. Any version of the algebraic method of group classification involves, in some way, the classification of algebras of vector fields up to certain equivalence induced by point transformations. The key question is what set of vector fields should be classified and what kind of equivalence should be used. Depending on this and completeness of solution, one can talk about partial preliminary group classification, complete preliminary group classification and complete group classification.

Within the framework of group classification, an important role is played by the notion of normalized classes of differential equations. Thus, for a weakly normalized class, complete preliminary group classification and complete group classification coincide. If the class is semi-normalized, the group classification up to equivalence generated by the associated equivalence group coincides with the group classification up to general point equivalence. As normalized classes are both semi-normalized and weakly normalized, it is especially convenient to carry out group classification in such classes by the algebraic method. This is why the normalization property can be used as a criterion for selecting classes of differential equations to be classified or for splitting of such classes into subclasses which are appropriate for group classification.

To illustrate the approach developed, we present, in particular, the complete solution of the group classification problem for the class of nonlinear wave equations arising in the theory of elasticity. The symmetry analysis of such equations had been initiated in twenty years ago, but only partial preliminary group classification of this class was carried out. We also briefly overview group classifications of various classes of linear and nonlinear Schrödinger equations.

#### Program v akademickém roce 2013/2014

22.05.2014

dr hab. Katarzyna Grabowska (Uniwersytet Warszawski, Polsko),
Variational calculus with constraints on general algebroids II

Abstrakt:

After a necessary introduction in the first part of the seminar we can use the Tulczyjew triple for general algebroids to describe mechanical systems with constraints. In the second part of the seminar constraints of different kinds (holonomic, nonholonomic and vaconomic) will be discussed.
The algebroid formulation makes the distinction between different types of constraints more clear than it is for systems defined on a tangent bundle. We shall illustrate our theory with a number of important examples.

The second part of the seminar will be based on the paper: K. Grabowska, J. Grabowski: Variational calculus with constraints on general algebroids, J. Phys. A 41 (2008), 175204, 25pp.

24.04.2014

RNDr. Jiřina Jahnová, Ph.D.,
The Painlevé property of differential equations III

Abstrakt:

In the third part of the talk we continue the description of the so-called Painlevé test for ODEs. We also briefly discuss the relationship between the Painlevé property and different approaches to integrability of dynamical systems. Finally, we present fragments of Painlevé analysis for partial differential equations.

17.04.2014

prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Applications: higher symmetries and differential constraints

Abstrakt:

Different applications, especially higher symmetries and differential constrains, will be discussed.

10.04. 2014

prof. D.V. Alekseevsky (MU Brno),
Cohomogeneity one Kaehler and Kaehler-Einstein Manifolds

Abstrakt:

A Riemannian manifold M is called to be a cohomogeneity one if there is an isometry group G with a codimension one orbit. We give a description of cohomogeneity one Kaehler manifolds of a compact semisimple Lie group G in terms of painted Dynkin diagrams and parametrized intervals in a T-Weyl chamber. We describe when such manifolds admit an invariant Kaehler-Einstein metric. In this case the Einstein equation is reduced to a second order ODE for function f(t) which describe the parametrization of the interval.

08.04.2014 (mimořádně v úterý v místnosti R2 v 13.05)

prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Obstructions for integrability: Spencer cohomology and Weyl tensors

Abstrakt:

The classical Spencer-Goldshmidt approach to Integrability (Cartan-Kahler theorem) together with an alternative approach based on Weyl tensors will be discussed.

03.04.2014

RNDr. Jiřina Jahnová, Ph.D.,
The Painlevé property of differential equations II

Abstrakt:

In the second part of the talk we describe the collection of necessary conditions of the Painlevé property for ODEs - the so-called Painlevé test. We also briefly discuss the methods that can be used to prove their sufficiency, and the relationship between the Painlevé property and different approaches to integrability. Finally we present fragments of Painlevé analysis for partial differential equations.

27.03.2014

RNDr. Jiřina Jahnová, Ph.D.,
The Painlevé property of differential equations I

Abstrakt:

In the first part of the talk we define the Painlevé property of ordinary differential equations and consider linear ODEs as an example of equations that have the Painlevé property. Finally we briefly present the main results that were obtained by Painlevé, Fuchs, Gambier and others within the so-called Painlevé program which was established at the beginning of the 20th century by Paul Painlevé.

20.03.2013

doc. RNDr. Artur Sergyeyev, Ph.D.,
Formal symmetries and integrability

Abstrakt:

We give a brief survey of the formal symmetry approach to classification of integrable (1+1)-dimensional evolution equations.

13. 03. 2014

On invariant solutions of the constant astigmatism equation

Abstrakt:

In the talk some new results concerning the constant astigmatism equation (CAE) will be given.

1. We find solutions of the CAE that correspond to surfaces obtained by Lipschitz in 1887. They are invariant with respect to a suitable combination of the known Lie symmetries of the CAE.

2. We obtain a pair of reciprocal transformations for the CAE. Each of them coincides with a certain nonlocal symmetry of the CAE. A corresponding invariant solution is also computed.

06. 03. 2014

doc. RNDr. Artur Sergyeyev, Ph.D.,
Nonlocal symmetries for the four-dimensional Martinez Alonso--Shabat equation

Abstrakt:

We construct an infinite hierarchy of commuting nonlocal symmetries (and ot just the shadows, as it is usually the case in the literature) for the Martinez Alonso--Shabat equation.

This is joint work with Oleg Morozov, see arXiv:1401.7942 for details.

27. 02. 2014

doc. RNDr. Artur Sergyeyev, Ph.D.,
A new class of integrable (3+1)-dimensional dispersionless systems

Abstrakt:

We introduce a new class of (3+1)-dimensional dispersionless integrable systems having Lax pairs written in terms of contact vector fields. Our results show inter alia that (3+1)-dimensional integrable systems are considerably less exceptional than it was believed.

In particular, we present a new (3+1)-dimensional dispersionless integrable system with an arbitrarily large finite number of components. In the simplest special case this system yields a (3+1)-dimensional integrable generalization of the dispersionless Kadomtsev--Petviashvili equation.

For further details please see arXiv:1401.2122.

12. 12. 2013

prof. Iosif Krasil'shchik, DrSc.,
Generalized jet spaces

Abstrakt:

I shall define generalized jet spaces and shall try to give a geometric explanation of Lax pairs that incorporate differentiations with respect to "spectral parameter". If time allows, I shall discuss other applications.

05. 12. 2013

Giovanni Moreno, Ph.D.,
Meta-symplectic geometry of 3rd order Monge-Ampère equations

Abstrakt:

In this talk I will present the geometry of a special type of scalar PDEs in one unknown function and two independent variables: those whose characteristics correspond to a 3D sub-distribution of the Cartan distribution on the Lagrangian Grassmannian bundle $M^{(1)}$ of a 5D contact manifold $M$. The main result is that any PDE of this type can be written as a linear combination of the minors of the Hankel matrix on $M^{(2)}$, with coefficients on $M^{(1)}$, i.e., they generalize the classical notion of Monge-Ampère equations, accordingly to some authors (Boillat, Ferapontov).

This is joint work with G. Manno.

28. 11. 2013

Dr. Diego Catalano Ferraioli (Federal  University of Bahia, Brazil),
Fourth order evolution equations of pseudo-spherical type

Abstrakt:

It is given a classification of pseudo-spherical evolution equations of the form u_t = u_xxxx + G(u, u_x , u_xx , u_xxx), under some suitable conditions on the associated 1-forms ω_i = f_i1 dx + f_i2 dt, 1 ≤ i ≤ 3. These equations can be equivalently described as compatibility condition (or zero-curvature representation) of an associated overdetermined linear system. The classiﬁcation provides 4  huge classes of such equations which are explicitly described.

7. 11. 2013

Dr. Jonathan Kress (University of New South  Wales, Australia),
Invariant classification of conformally superintegrable systems

Abstrakt:

A Hamiltonian system with a 2n-dimensional phase space is said to be Liouville integrable if it possesses n integrals of the motion that are mutually in involution.  The system is said to be superintegrable if it possesses more than n integrals up to a maximum of 2n-1.  Many well known integrable systems, such as the inverse square central force and harmonic oscillator systems, are in fact superintegrable.

Superintegrable systems with second order constants of the motion have been extensively studied because of their close connection with special functions and separation of variables.  This talk will present a classification of second order non-degenerate superintegrable systems on three-dimensional conformally flat spaces.  Each such system is associated with a configuration of 6 points in the extended-complex plane and invariants of these configurations under the action of the conformal group can be used to classify the systems.  This is a joint work with Joshua Capel.

5.11. 2013 (výjimečně v útery v 16.25 v R1)

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras V.

31. 10. 2013

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras IV.

29. 10. 2013 (výjimečně v útery v 16.25 v R1)

Giovanni Manno, Ph.D. (University of Padova, Italy),
Contact Geometry of Monge-Ampère equations.

Abstrakt:

Monge-Ampère equations (MAEs) with an arbitrary number of independent variables can be interpreted as particular hypersurfaces of a Lagrangian Grassmann bundle. After investigating the relation between MAEs and their characteristics, we discuss how to obtain normal forms for parabolic MAEs with two independent variables.

24. 10. 2013

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras II.

Abstrakt:

I plan to give a series of lectures on some geometric and algebraic methods in the theory of integrable nonlinear PDEs. It is well known that Backlund transformations (BTs) and zero-curvature representations (ZCRs) help to construct interesting explicit solutions for a wide class of nonlinear PDEs. I will begin my lectures with some classical examples, showing how soliton solutions of the Korteweg--de Vries (KdV) equation can be obtained by means of BTs and ZCRs of KdV.

After that, I will describe a general geometric theory for BTs and ZCRs of PDEs. This theory is based on the use of infinite-dimensional Lie algebras, infinite jet bundles, and Krasilshchik--Vinogradov coverings of PDEs.

For any PDE satisfying some non-degeneracy conditions, I will define a family of Lie algebras, which are called the fundamental Lie algebras of this PDE. Fundamental Lie algebras are defined in a coordinate-independent way and are new geometric invariants for PDEs.

Recall that, for every topological space $X$ and every point $a\in X$, one has the fundamental group $\pi_1(X,a)$. The above-mentioned Lie algebras are called fundamental, because their role for PDEs is somewhat similar to the role of fundamental groups for topological spaces.

Fundamental Lie algebras are closely related to ZCRs and BTs. In these lectures, we will concentrate on the case of (1+1)-dimensional PDEs. For such PDEs, it will be shown how fundamental Lie algebras classify all ZCRs up to local gauge equivalence. Also, I will show how to describe fundamental Lie algebras in terms of generators and relations. Using these algebras, one obtains necessary conditions for integrability and necessary conditions for existence of BTs for the considered class of PDEs.

In this construction, jets of arbitrary order are allowed. In the case of low-order jets, fundamental Lie algebras generalize Wahlquist--Estabrook prolongation algebras.

In the structure of fundamental Lie algebras for KdV, Krichever--Novikov, nonlinear Schrodinger, (multicomponent) Landau--Lifshitz type equations, we encounter infinite-dimensional subalgebras of Kac--Moody algebras and infinite-dimensional Lie algebras of certain matrix-valued functions on some algebraic curves. Applications to classification of some PDEs with respect to BTs and methods to construct BTs will be discussed as well.

10. 10. 2013

Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko),
Benney hydrodynamic chain and DKP hierarchy. Their N-component hydrodynamic reductions.
Löwner equations and the Gibbons--Tsarev system. Particular solutions

Abstrakt:

We consider the Gibbons--Tsarev system and discuss possibilities
to construct their particular solutions parameterized by arbitrary constants.

03. 10. 2013

doc. RNDr. Artur Sergyeyev, Ph.D.,
Coupling constant metamorphosis and integrability

Semináře se obvykle konají ve čtvrtek od 14.45 hod. v budově Matematického ústavu, Na Rybníčku 1 v Opavě, v místnosti R1. Všichni zájemci jsou srdečně zváni.

Schroedinger operators exhibiting spectral transition

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev, prof. Krasil'shchik

#### Program v akademickém roce 2013/2014

22.05.2014

dr hab. Katarzyna Grabowska (Uniwersytet Warszawski, Polsko), Variational calculus with constraints on general algebroids II

Abstrakt:

After a necessary introduction in the first part of the seminar we can use the Tulczyjew triple for general algebroids to describe mechanical systems with constraints. In the second part of the seminar constraints of different kinds (holonomic, nonholonomic and vaconomic) will be discussed.
The algebroid formulation makes the distinction between different types of constraints more clear than it is for systems defined on a tangent bundle. We shall illustrate our theory with a number of important examples.

The second part of the seminar will be based on the paper: K. Grabowska, J. Grabowski: Variational calculus with constraints on general algebroids, J. Phys. A 41 (2008), 175204, 25pp.

24.04.2014

RNDr. Jiřina Jahnová, Ph.D., The Painlevé property of differential equations III

Abstrakt:

In the third part of the talk we continue the description of the so-called Painlevé test for ODEs. We also briefly discuss the relationship between the Painlevé property and different approaches to integrability of dynamical systems. Finally, we present fragments of Painlevé analysis for partial differential equations.

17.04.2014

prof. V.V. Lychagin (Universitetet i Tromsø, Norsko) Applications: higher symmetries and differential constraints

Abstrakt:

Different applications, especially higher symmetries and differential constrains, will be discussed.

10.04. 2014

prof. D.V. Alekseevsky (MU Brno), Cohomogeneity one Kaehler and Kaehler-Einstein Manifolds

Abstrakt:

A Riemannian manifold M is called to be a cohomogeneity one if there is an isometry group G with a codimension one orbit. We give a description of cohomogeneity one Kaehler manifolds of a compact semisimple Lie group G in terms of painted Dynkin diagrams and parametrized intervals in a T-Weyl chamber. We describe when such manifolds admit an invariant Kaehler-Einstein metric. In this case the Einstein equation is reduced to a second order ODE for function f(t) which describe the parametrization of the interval.

08.04.2014 (mimořádně v úterý v místnosti R2 v 13.05)

prof. V.V. Lychagin (Universitetet i Tromsø, Norsko) Obstructions for integrability: Spencer cohomology and Weyl tensors

Abstrakt:

The classical Spencer-Goldshmidt approach to Integrability (Cartan-Kahler theorem) together with an alternative approach based on Weyl tensors will be discussed.

03.04.2014

RNDr. Jiřina Jahnová, Ph.D., The Painlevé property of differential equations II

Abstrakt:

In the second part of the talk we describe the collection of necessary conditions of the Painlevé property for ODEs - the so-called Painlevé test. We also briefly discuss the methods that can be used to prove their sufficiency, and the relationship between the Painlevé property and different approaches to integrability. Finally we present fragments of Painlevé analysis for partial differential equations.

27.03.2014

RNDr. Jiřina Jahnová, Ph.D., The Painlevé property of differential equations I

Abstrakt:

In the first part of the talk we define the Painlevé property of ordinary differential equations and consider linear ODEs as an example of equations that have the Painlevé property. Finally we briefly present the main results that were obtained by Painlevé, Fuchs, Gambier and others within the so-called Painlevé program which was established at the beginning of the 20th century by Paul Painlevé.

20.03.2013

doc. RNDr. Artur Sergyeyev, Ph.D., Formal symmetries and integrability

Abstrakt:

We give a brief survey of the formal symmetry approach to classification of integrable (1+1)-dimensional evolution equations.

13. 03. 2014

Mgr. Adam Hlaváč, On invariant solutions of the constant astigmatism equation

Abstrakt:

In the talk some new results concerning the constant astigmatism equation (CAE) will be given.

1. We find solutions of the CAE that correspond to surfaces obtained by Lipschitz in 1887. They are invariant with respect to a suitable combination of the known Lie symmetries of the CAE.

2. We obtain a pair of reciprocal transformations for the CAE. Each of them coincides with a certain nonlocal symmetry of the CAE. A corresponding invariant solution is also computed.

06. 03. 2014

doc. RNDr. Artur Sergyeyev, Ph.D., Nonlocal symmetries for the four-dimensional Martinez Alonso--Shabat equation

Abstrakt:

We construct an infinite hierarchy of commuting nonlocal symmetries (and ot just the shadows, as it is usually the case in the literature) for the Martinez Alonso--Shabat equation.

This is joint work with Oleg Morozov, see arXiv:1401.7942 for details.

27. 02. 2014

doc. RNDr. Artur Sergyeyev, Ph.D., A new class of integrable (3+1)-dimensional dispersionless systems

Abstrakt:

We introduce a new class of (3+1)-dimensional dispersionless integrable systems having Lax pairs written in terms of contact vector fields. Our results show inter alia that (3+1)-dimensional integrable systems are considerably less exceptional than it was believed.

In particular, we present a new (3+1)-dimensional dispersionless integrable system with an arbitrarily large finite number of components. In the simplest special case this system yields a (3+1)-dimensional integrable generalization of the dispersionless Kadomtsev--Petviashvili equation.

For further details please see arXiv:1401.2122.

12. 12. 2013

prof. Iosif Krasil'shchik, DrSc., Generalized jet spaces

Abstrakt:

I shall define generalized jet spaces and shall try to give a geometric explanation of Lax pairs that incorporate differentiations with respect to "spectral parameter". If time allows, I shall discuss other applications.

05. 12. 2013

Giovanni Moreno, Ph.D., Meta-symplectic geometry of 3rd order Monge-Ampère equations

Abstrakt:

In this talk I will present the geometry of a special type of scalar PDEs in one unknown function and two independent variables: those whose characteristics correspond to a 3D sub-distribution of the Cartan distribution on the Lagrangian Grassmannian bundle $M^{(1)}$ of a 5D contact manifold $M$. The main result is that any PDE of this type can be written as a linear combination of the minors of the Hankel matrix on $M^{(2)}$, with coefficients on $M^{(1)}$, i.e., they generalize the classical notion of Monge-Ampère equations, accordingly to some authors (Boillat, Ferapontov).

This is joint work with G. Manno.

28. 11. 2013

Dr. Diego Catalano Ferraioli (Federal  University of Bahia, Brazil), Fourth order evolution equations of pseudo-spherical type

Abstrakt:

It is given a classification of pseudo-spherical evolution equations of the form u_t = u_xxxx + G(u, u_x , u_xx , u_xxx), under some suitable conditions on the associated 1-forms ω_i = f_i1 dx + f_i2 dt, 1 ≤ i ≤ 3. These equations can be equivalently described as compatibility condition (or zero-curvature representation) of an associated overdetermined linear system. The classiﬁcation provides 4  huge classes of such equations which are explicitly described.

7. 11. 2013

Dr. Jonathan Kress (University of New South  Wales, Australia), Invariant classification of conformally superintegrable systems

Abstrakt:

A Hamiltonian system with a 2n-dimensional phase space is said to be Liouville integrable if it possesses n integrals of the motion that are mutually in involution.  The system is said to be superintegrable if it possesses more than n integrals up to a maximum of 2n-1.  Many well known integrable systems, such as the inverse square central force and harmonic oscillator systems, are in fact superintegrable.

Superintegrable systems with second order constants of the motion have been extensively studied because of their close connection with special functions and separation of variables.  This talk will present a classification of second order non-degenerate superintegrable systems on three-dimensional conformally flat spaces.  Each such system is associated with a configuration of 6 points in the extended-complex plane and invariants of these configurations under the action of the conformal group can be used to classify the systems.  This is a joint work with Joshua Capel.

5.11. 2013 (výjimečně v útery v 16.25 v R1)

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras V.

31. 10. 2013

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras IV.

29. 10. 2013 (výjimečně v útery v 16.25 v R1)

Giovanni Manno, Ph.D. (University of Padova, Italy), Contact Geometry of Monge-Ampère equations.

Abstrakt:

Monge-Ampère equations (MAEs) with an arbitrary number of independent variables can be interpreted as particular hypersurfaces of a Lagrangian Grassmann bundle. After investigating the relation between MAEs and their characteristics, we discuss how to obtain normal forms for parabolic MAEs with two independent variables.

24. 10. 2013

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras II.

Abstrakt:

I plan to give a series of lectures on some geometric and algebraic methods in the theory of integrable nonlinear PDEs. It is well known that Backlund transformations (BTs) and zero-curvature representations (ZCRs) help to construct interesting explicit solutions for a wide class of nonlinear PDEs. I will begin my lectures with some classical examples, showing how soliton solutions of the Korteweg--de Vries (KdV) equation can be obtained by means of BTs and ZCRs of KdV.

After that, I will describe a general geometric theory for BTs and ZCRs of PDEs. This theory is based on the use of infinite-dimensional Lie algebras, infinite jet bundles, and Krasilshchik--Vinogradov coverings of PDEs.

For any PDE satisfying some non-degeneracy conditions, I will define a family of Lie algebras, which are called the fundamental Lie algebras of this PDE. Fundamental Lie algebras are defined in a coordinate-independent way and are new geometric invariants for PDEs.

Recall that, for every topological space $X$ and every point $a\in X$, one has the fundamental group $\pi_1(X,a)$. The above-mentioned Lie algebras are called fundamental, because their role for PDEs is somewhat similar to the role of fundamental groups for topological spaces.

Fundamental Lie algebras are closely related to ZCRs and BTs. In these lectures, we will concentrate on the case of (1+1)-dimensional PDEs. For such PDEs, it will be shown how fundamental Lie algebras classify all ZCRs up to local gauge equivalence. Also, I will show how to describe fundamental Lie algebras in terms of generators and relations. Using these algebras, one obtains necessary conditions for integrability and necessary conditions for existence of BTs for the considered class of PDEs.

In this construction, jets of arbitrary order are allowed. In the case of low-order jets, fundamental Lie algebras generalize Wahlquist--Estabrook prolongation algebras.

In the structure of fundamental Lie algebras for KdV, Krichever--Novikov, nonlinear Schrodinger, (multicomponent) Landau--Lifshitz type equations, we encounter infinite-dimensional subalgebras of Kac--Moody algebras and infinite-dimensional Lie algebras of certain matrix-valued functions on some algebraic curves. Applications to classification of some PDEs with respect to BTs and methods to construct BTs will be discussed as well.

10. 10. 2013

Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko),
Benney hydrodynamic chain and DKP hierarchy. Their N-component hydrodynamic reductions.
Löwner equations and the Gibbons--Tsarev system. Particular solutions

Abstrakt:

We consider the Gibbons--Tsarev system and discuss possibilities
to construct their particular solutions parameterized by arbitrary constants.

03. 10. 2013

doc. RNDr. Artur Sergyeyev, Ph.D., Coupling constant metamorphosis and integrability

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev, prof. Krasil'shchik

#### Program v akademickém roce 2012/2013

25.7.2013

Michael Bächtold, Dr. Dipl. Math. ETH (Hochschule Luzern, Switzerland), Flags and polarization in geometric singularities of jets

16. 5. 2013

Prof. Iosif Krasil'shchik, DrSc. On geometry of flat connections equation

Abstrakt:

Let χ: P → M be a locally trivial fiber bundle and ♥ be a connection in this bundle. The flatness condition is a first order equation F1 in the bundle χ1,0: J¹χ → P. The infinite prolongation of this equation is endowed with two geometries: one is the usual Cartan geometry while the second arises due to existence of "universal connection" on (F1). The equation at hand plays an important role in the theory of coverings.
I shall discuss the above matters in reasonable detail.

30. 4. 2013 (výjimečně v 16:25 v R2)

prof. Evgeny Ferapontov (Loughborough University, UK), Dispersive deformations of dispersionless integrable systems

Abstrakt:

In this last installment of the lecture series, we shall describe the procedure enabling one to reconstruct the higher-order (dispersive) terms based on the requirement that the dispersive system inherits' all hydrodynamic reductions of its dispersionless limit and give examples of the classification results in this direction.

25. 4. 2013 (výjimečně v 16:00 v R1; bude to druhý seminář tentýž den)

Prof. Evgeny Ferapontov (Loughborough University, UK), Classification of 2+1D dispersionless integrable systems

Abstrakt:

We shall continue here the introduction to the method of hydrodynamic reductions, give a survey of classification results obtained using this method and, if time permits, to begin the discussion of the problem of reconstruction of dispersive terms in multidimensional integrable systems.

25. 4. 2013

Prof. Evgeny Ferapontov (Loughborough University, UK) Geometric aspects and the integrability of multidimensional quasilinear PDEs

Abstrakt:

In these series (this and two subsequent lectures) it will be demonstrated that the problem of classification of integrable systems in higher dimensions (i.e., having more than two independent variables) can be efficiently approached based on the following two-step procedure:

- firsly, one classifies dispersionless integrable systems based on the method of hydrodynamic reductions. A review of the results in this direction will be given.

- secondly, one reconstructs higher order (dispersive) terms based on the requirement that the dispersive system inherits' all hydrodynamic reductions of its dispersionless limit. Examples of classification results will be discussed.

In the first talk of the series we will provide an introduction to multidimensional integrable systems of hydrodynamic type, introduce the method of hydrodynamic reductions and discuss its role as an integrability test for dispersionless systems in higher dimensions.

24. 4. 2013 (výjimečně v 15:35 v R1 místo druhého semináře z matematické analýzy)

prof. Vladimir Rubtsov (Université d'Angers, France), Symmetries, conservation laws and differential invariants of the Monge-Ampère equations

Abstrakt:

This is a second (and last) lecture in the series. We plan to cover here

• Symmetry and conservation laws of the Monge-Ampère equations, an analogue of the Noether Theorem;
• The S. Lie problem and invariants of the Monge-Ampère operators in dimensions 2 and 3 (dim M = 2, 3).

23. 4. 2013 (výjimečně v 16:25 v R2)

prof. Vladimir Rubtsov (Université d'Angers, France), Contact and Symplectic Geometry of Monge-Ampère equations: Introduction

Abstrakt:

In these two-lecture series we discuss a geometric approach to the Monge-Ampère equations on a smooth manifold M.  This approach goes back to the original ideas of E. Cartan and his Ph.D. student T. Lepage. Its modern version was developed in 1978-79 by V. Lychagin and it is based on the rich geometric structures of cotangent and 1-jet bundles. The ideas and methods of this Lychagin approach were approved by many interesting results obtained both in theoretic (like a solution of the Sophus Lie classification and linearization problem) and in various applied areas, including the numerical problems of the weather forecast. The Lychagin approach uses the canonical (symplectic) structure on the phase space T*M or/and the contact structure on the 1-jet space J¹M.

The plan of my first lecture is as follows:

• the 1-jet bundle J¹M; the contact structure on J¹M; Legendre submanifolds in J¹M, contact diffeomorphisms and contact vector fields;
• Monge-Ampère equations and their generalized solutions;
• Differential forms on J¹M and Hodge-Lepage theory.

The main prerequisite for my lectures is a basic knowledge of differential calculus on manifolds (vector fields, differential forms, the de Rham complex).

18. 4. 2013

prof. Iosif Krasil'shchik, DrSc. On interrelation of integrability properties in finite-dimensional coverings

Abstrakt:

Let τ: E' → E be a finite-dimensional covering. A natural question (asked to me by Artur Sergyeyev some time ago) arises: how integrability properties of equations E and E' are related to each other?

11. 4. 2013

doc. RNDr. Artur Sergyeyev, Ph.D. Master symmetries: a very short introduction

Abstrakt:

In this talk we will describe the notion of master symmetry and its applications. Our two main examples will be the Landau-Lifshitz and the Kadomtsev-Petviashvili equations.

4. 4. 2013

Natascha Neumärker, Ph.D. Integrability in discrete dynamical systems -- A survey of selected aspects II

Abstrakt:

This second part of the talk (the first one took place on March 7, 2013) will be devoted to integrability detection in maps by means of singularity confinement, algebraic entropy and reductions over finite fields. It will be essentially self-contained.

21. 3. 2013

Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko), Multi-phase solutions of the KdV type equations.

Abstrakt:

We present a method of extracting polynomial solutions with respect to spectral parameter for KdV, Kaup—Boussinesq equation and all other so called Coupled KdV system embedded in 2x2 zero curvature representation.

Also we discuss general 2x2 zero curvature representation.
Video

7. 3. 2013

Natascha Neumärker, Ph.D. Integrability in discrete dynamical systems -- A survey of selected aspects

Abstrakt:

This talk will be a survey of selected aspects of integrability in discrete dynamical systems on an introductory level. I will present notions of integrability in the settings of discrete analogues of Lagrange systems, polynomial automorphisms of and birational maps on the plane. The second part of the talk (to take place on April 4, 2013) will be devoted to integrability detection in maps by means of singularity confinement, algebraic entropy and reductions over finite fields.

21. 2. 2013

Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko), Integrability of the Skyrme—Faddeev Model.

Abstrakt:

We consider the Skyrme—Faddeev Lagrangian, which yields a very Euler—Lagrange equations. This is a NON-integrable four-dimensional system in partial derivatives of the second order.

In this talk we:

1. find a periodic solution;
2. construct the corresponding Whitham averaged system;
3. find its dispersionless limit, which we call "dispersionless SF system";
4. extract the so called "integrable sector".
5. found a general solution of this "Reduced SF system".

13. 12. 2012

doc. RNDr. Tomáš Kopf, Ph.D. Statistics on jet spaces

Abstrakt:

Statistic generative models may assume a dynamics given by differential equations.
Possible approximate inversions of such models are discussed.

28. 11. 2012

doc. Oleg Morozov, DrSc. (University of Tromsø, Norway) Recursion Operators for the Universal Hierarchy Equation via Cartan's Method of Equivalence

Seminář se bude výjimečně konat ve středu 28. listopadu od 15.35 hod. v místnosti R1.

Abstrakt:

Cartan's method of equivalence is applied to find a Backlund autotransformation for the tangent and cotangent coverings of the universal hierarchy equation. The transformations provide recursion operators for symmetries and cosymmetries of this equation.

22. 11. 2012

prof. Iosif Krasil'shchik, DrSc. On symmetries and conservation laws of the Gibbons--Tsarev equation (joint work with M. Marvan and H. Baran)

Abstrakt:

We study the Gibbons--Tsarev equation z_yy + z_x z_xy - z_y z_xx + 1= 0 and prove the existence of infinite number of nonlocal symmetries and conservation laws for it.

22. 11. 2012

doc. RNDr. Artur Sergyeyev, Ph.D. A few glimpses from ECM'2012

15. 11. 2012

prof. Jerzy Kijowski (CFT PAN, Warsaw, Poland) Fundamental symplectic structures arising from the calculus of variations

Abstrakt:

In this talk I will show that every variational problem leads to a natural jet bundle P over the space of independent variables M. Fibers of this bundle are equipped with a canonical symplectic structure. For every point m of the base manifold M, the set of all possible jets of solutions of the variational problem is proved to be a Lagrangian (i.e. maximal, isotropic) submanifold of the symplectic space P_m. This way calculus of variations and its consequences (e.g. Hamiltonian formulation, Hamilton-Jacobi theory etc.) acquire powerful tools provided by symplectic geometry.

08. 11. 2012

prof. Gaetano Vilasi (Universita degli Studi di Salerno, Italy) Einstein metrics with two-dimensional Killing leaves: geometrical and physical aspects

Abstrakt:

Local and global aspects of the solutions of Einstein's field equations, for the class of Riemannian metrics admitting a non-Abelian Lie algebra of Killing fields generating a 2-dimensional distribution, will be explicitly described. Physical properties of the obtained metrics will be described naturally as well.

25. 10. 2012

Mgr. Jakub Hruška, Ph.D. (MFF UK, Praha) Conformal null Einstein-Maxwell fields

Abstrakt:

The use of conformal transformation as a method for generating solutions of
Einstein's equations has been mainly studied in the cases where the original
spacetime is vacuum. The generated spacetimes then frequently belong to the
class of pp-waves. In the present work, the electrovacuum spacetimes are
studied, i.e the solutions of coupled Einstein's and Maxwell's equations.
By using the conformal transformation, it is possible to circumvent
solving the later equations. This method is concretely studied for null
Einstein-Maxwell fields and it turns out that the admissible spacetimes
are pp-waves again. However, if the method is generalized, it is possible
to enlarge the class of conformal null Einstein-Maxwell fields to a wider
family of Kundt spacetimes.

11. 10. 2012

prof. Iosif Krasil'shchik, DrSc. Yet another look on recursion operators

Abstrakt:

In 1995, Michal Marvan [M. Marvan, Another look on recursion
operators, in: Differential Geometry and Applications, Proc. Conf.
Brno, 1995 (Masaryk University, Brno, 1996) 393–402.] used the
vertical tangent space VE of a PDE E as a natural environment for
geometrical approach to recursion operators. He treated the latter as
auto-Backlund transformations of VE. I shall discuss another way to
use VE for construction of ROs. A generalization of this scheme will
also be considered.

4. 10. 2012

RNDr. Hynek Baran, Ph.D. Cluster computing on diffieties

27. 9. 2012

prof. Raffaele Vitolo (Universita del Salento, Italy) On the symbolic computation of integrability related operators. Part II: examples of computations.

Abstrakt:

In the second part I will describe examples of computations with CDIFF, from basic local and nonlocal Hamiltonian operators for KdV and Boussinesq equation to the newly found nonlocal Hamiltonian-recursion operator for the Plebanski equation. I will also discuss a work in progress (joint with M. Pavlov) about the WDVV equation in the first-order three-component presentation. For a potential form of this equation we found a Lagrangian representation that could be possibly used in the classification of integrable systems having two homogeneous local Hamiltonian structures.

26. 9. 2012

prof. Raffaele Vitolo (Universita del Salento, Italy) On the symbolic computation of integrability related operators. Part I: symbolic software.

Seminář se bude výjimečně konat od 15.35 hod. v místnosti R1.

Abstrakt:

In the first part I will describe CDIFF, a REDUCE package for computing integrability related structures. The software provides an environment for computations on differential equations. Basically it is aimed at computing total derivatives on any super-differential equation. The main use of super-differential equation is the so-called tangent and cotangent covering, by Kersten-Krasil'shchik-Verbovetsky. CDIFF is able to deal with anticommuting variables and equations defined with them. Total derivatives on tangent and cotangent covering can be used to define the linearization operator, the Euler operator, and the bracket between two bivectors to prove Hamiltonianity and/or compatibility. In the second part I will describe examples of computations with CDIFF, from basic local and nonlocal Hamiltonian operators for KdV and Boussinesq equation to the newly found nonlocal Hamiltonian-recursion operator for the Plebanski equation. I will also discuss a work in progress (joint with M. Pavlov) about the WDVV equation in the first-order three-component presentation. For a potential form of this equation we found a Lagrangian representation that could be possibly used in the classification of integrable systems having two homogeneous local Hamiltonian structures.

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev, prof. Krasil'shchik

#### Program v akademickém roce 2011/2012

31. 5. 2012

prof. Iosif Krasil'shchik, DrSc., Open problems in integrable systems II

Seminář se koná výjimečně v místnosti R2

24. 5. 2012

Mgr. Petr Vojčák, Recursion operators, nonlocal symmetries  and related
structures for integrable systems  (presentation of the Ph.D. thesis)

17. 5. 2012

Giovanni Moreno, Ph.D., Characteristics and singularities of solutions of nonlinear PDEs.

Abstrakt:

Prolongation procedure allows to frame nonlinear PDEs in the context of infinite jet bundles, where powerful homological methods can be exploited. However, by its definition, prolongation discards important information about singular solutions, which may have physical significance. In this talk I will show how to recover such information from the sequence of vertical bundles of an infinitely prolonged PDE.

prof. Iosif Krasil'shchik, DrSc., Open problems in integrable systems I

10. 5. 2012

Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko), Third local Lagrangian representation for the WDVV associativity equations with N=3.

Abstrakt:

For the first nonlocal (in negative direction) Hamiltonian operator of the system under study, we have found the associated Lagrangian representation of the latter.

3. 5. 2012

prof. Iosif Krasil'shchik, DrSc., First integrals. Lie-Bianchi Theorem. Linearization and
computation of symmetries.

26. 4. 2012

prof. Iosif Krasil'shchik, DrSc., Geometrization of ODEs. Geometry of distributions. Symmetries.

19. 4. 2012

Giovanni Moreno, Ph.D., Natural structures in iterated jet spaces.
Video

Abstrakt:
Mechanics is a well-known context where iterated (co)tangent bundles and their tautological structures play a prominent role. A similar role, in the geometrical theory of nonlinear PDEs, is played by iterated jet bundles. The very definition of higher-order jets can be seen as a natural equation in an iterated jet bundle. Moreover, it seems that such important notions as the equation of singularities of solutions and the space of initial data of a nonlinear PDEs, or the natural boundary conditions in variational calculus, can be given a natural geometric interpretation in the framework of iterated jet spaces.

10. 4. 2012

Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko), Finite-component reductions of non-integrable kinetic equations.

Seminář se koná výjimečně v úterý od 14.45 hod. v místnosti LVT1. Všichni zájemci jsou srdečně zváni.

5. 4. 2012

Oleg Morozov, DrSc. (University of Tromsø, Norsko) Coverings, recursion operators, and nonlocal symmetries of integrable differential equations.
Video

Abstrakt:
The talk is devoted to recent results about recursion operators and nonlocal symmetries for nonlinear partial differential equations with more that two independent variables.

28. 3. 2012

doc. RNDr. Zdeněk Dušek, Ph.D. (Univerzita Palackého v Olomouci) Existence of lightlike homogeneous geodesics.

Seminář se koná výjimečně ve středu od 14.45 hod. v místnosti R1. Všichni zájemci jsou srdečně zváni.

Abstrakt:
In previous projects, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point.
In the present project this affine method is refined and adapted to the pseudo-Riemannian and Lorentzian case. Results about existence of light-like homogeneous geodesics will be presented and illustrated with an example.

22. 3. 2012

RNDr. Jiřina Vodová, Low-order Hamiltonian operators having momentum.

15. 3. 2012

Giovanni Moreno, Ph.D., An introduction to jet spaces and their natural structures.

8. 3. 2012

prof. Iosif Krasil'shchik, DrSc., Geometry of differential equations and integrable systems.
Video

Abstrakt:
I shall try to explain on elementary level how geometrical approach to PDEs helps to compute main structures responsible for integrability: recursion, Hamiltonian, and symplectic operators.

References:
Joseph Krasilshchik, Alexander Verbovetsky: Geometry of jet spaces and integrable systems, Geometry and Physics (2011),
doi:10.1016/j.geomphys.2010.10.012, http://arxiv.org/abs/1002.0077

1. 3. 2012

doc. RNDr. Artur Sergyeyev, Ph.D., První seznámení se supersymetrií.

23. 2. 2012

doc. RNDr. Michal Marvan, CSc., Operátory rekurze jako Bäcklundovy autotransformace linearizovaných rovnic.

1. 12. 2011

doc. RNDr. Zdeněk Kočan, Ph.D., Diskrétní dynamické systémy a chaos.

24. 11. 2011

Dr. Blazej M. Szablikowski, (Univerzita Adama Mickiewicze, Poznan, Polsko), Classical r-matrix approach to Frobenius manifolds: taking advantage of Rota—Baxter and classical Yang—Baxter relations.

10. 11. 2011

Mgr. Jiří Jahn, RNDr. Jiřina Vodová, Singularity diferenciálních rovnic v komplexním oboru I.

3. 11. 2011

doc. RNDr. Michal Marvan, CSc., Likvidace negramotnosti v Lobačevského geometrii.

20. 10. 2011

doc. RNDr. Michal Marvan, CSc., Bourův problém II.

13. 10. 2011

doc. RNDr. Michal Marvan, CSc., Bourův problém I.

6. 10. 2011

doc. RNDr. Artur Sergyeyev, Ph.D., Několik poznámek o řešení nelineárních parciálních diferenciálních rovnic prvního řádu s jednou závisle proměnnou.

29. 9. 2011

doc. RNDr. Artur Sergyeyev, Ph.D., Integrabilita Hamiltonovských dynamických systémů: od Hamiltona k Jacobimu přes Liouville'a.

22. 9. 2011

Mgr. Adam Hlaváč, Bäcklundova transformace pro rovnici konstantního astigmatismu.

15. 9. 2011

Dr. Sergey Zykov, (Universita di Salento, Italy), Classification of integrable hydrodynamic chains of triangular type.

Abstrakt:
Integrable quasilinear PDEs can be produced from hydrodynamic chains. In earlier work only a handful of examples of such chains was found.

In this talk we present the most general hydrodynamic chain which has the Riemann surface and therefore possesses an infinite number of conservation laws. The coefficients of the chain under study are parametrized using hypergeometric functions. The sequence of conservation laws is produced by a generating function.

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

#### Program v akademickém roce 2010/2011

19. 5. 2011
RNDr. Jiřina Vodová, Darbouxovy souřadnice pro systém Hamiltonovských operátorů Kace, de Soleho a Wakimota.

12. 5. 2011
doc. RNDr. Josef Klusoň, Ph.D. (MU, Brno), Modifikované teorie gravitace.

5. 5. 2011
Bc. Veronika Bernhauerová, Stabilita v Darwinovské dynamice.

28. 4. 2011
RNDr. Oldřich Stolín, Ph.D., Problém ekvivalence metrik se dvěma konutujícími Killingovými vektorovými poli - negenerické případy.

21. 4. 2011
Seminář se nekoná.

14. 4. 2011
prof. RNDr. Josef Mikeš, DrSc. (UP, Olomouc), Geodetická zobrazení a jejich fundamentální rovnice.

7. 4. 2011
Mgr. Adam Hlaváč, Stručný přehled diferenciální geometrie vnořených ploch.

31. 3. 2011
doc. RNDr. Artur Sergyeyev, Ph.D., Formální symetrie a integrabilita.

24. 3. 2011
doc. RNDr. Michal Marvan, CSc., O výpočtu fluxí zákonů zachování.

17. 3. 2011
doc. RNDr. Artur Sergyeyev, Ph.D., Integrabilní systémy hydrodynamického typu: stručný úvod.

10. 3. 2011
doc. RNDr. Stanislav Krajči, PhD. (UPJŠ, Košice, SK), Čo je pravda?

16. 12. 2010
RNDr. Jan Kotůlek, Ph.D., Klikatá cesta ke slávě: Problémy Diracovy teorie elektronu v letech 1928–1934

Abstrakt:
Diracova teorie z roku 1928 má punc jednoho z největších úspěchů meziválečné matematické fyziky. Dobové prameny však jasně ukazují, že Dirac musel o její přijetí tvrdě bojovat. Související problém negativních energií se mu nedařilo vyřešit téměř dva roky a i poté navrženou teorii elektronových děr (hole theory) musel vzhledem k ostré kritice v létě 1931 přepracovat. Nejpalčivější problémy se však během roku 1933 vyjasnily a Diracovi byla v prosinci 1933 udělena Nobelova cena.

Již roku 1934 byla však interpretace Diracovy rovnice pomocí teorie elektronových děr překonána prací W. Pauliho a V. Weisskopfa a postupně přestala být používána. Jelikož některé nejasnosti v interpretaci přetrvaly, Dirac se k otázce formulace relativistické pohybové rovnice několikrát vrátil, ale zdánlivě dosažitelné upřesnění teorie již nepřinesl. Uzavřenou a bezespornou interpretaci se ovšem nepodařilo nalézt dodnes.

Přesto se po druhé světové válce na tyto rozpory zapomnělo a příběh Diracovy rovnice byl postupně přikrášlován pozdějšími vzpomínkami. Ukazujeme, že Dirac sám přispěl k růstu svého mýtu a že jeho kolegové a také někteří historici dodnes odmítají vidět Diracovu teorii v dobových souvislostech.

9. 12. 2010
Mgr. Petr Vojčák, Operátor rekurze, hamiltonovská a symplektická struktura systému Mikhailova-Novikova-Wangové

Abstrakt:
V přednášce bude představen nový integrabilní systém pátého řádu s dvěma nezávislými a dvěma závislými proměnnými, který byl nedávno objeven Mikhailovem, Novikovem a Wangovou. Poznamenejme, že tento systém připouští mj. dobře známou Kaup-Kupershmidtovu rovnici jako redukci. Užitím tzv. symbolické metody Mikhailov se spolupracovníky ukázali, že tento systém má nekonečně mnoho zobecněných symetrií řádu 1, 5 mod 6. Nicméně operátor rekurze, symplektická a hamiltonovská struktura nebyly dosud známy a společně se svými vlastnostmi budou prezentovány v této přednášce.

2. 12. 2010
RNDr. Hynek Baran, Ph.D., Jets, a way to empower calculations on differential equations in total derivatives on diffieties II.

25. 11. 2010
RNDr. Peter Sebestyén, Ph.D., Normální tvary ireducibilních reprezentací nulové křivosti (ZCR) s hodnotami v algebře sl(n).

Abstract:
V roce 1997 M. Marvan publikoval klasifikaci normálních tvarů ZCR v algebře sl(2). Následně v roce 2005 byla publikována klasifikace ZCR v algebře sl(3) (Sebestyén). Přirozenou snahou bylo vytvořit klasifikaci v algebře sl(n) pro libovolné n>=2. Povedlo se to v případě, kdy charakteristický element ZCR má Jordanův normální tvar sestávající z jediné Jordanovy buňky (Sebestyén 2007). Na tomto semináři stručně shrnu publikované výsledky, uvedu dosud nepublikované výsledky a nastíním možnosti budoucího výzkumu.

18. 11. 2010
doc. RNDr. Artur Sergyeyev, Ph.D., Může lineární evoluční rovnice mít nelineární zákony zachování?

Abstract:
V této přednášce se soustředíme na případ lineární evoluční rovnice řádu n>1 s dvěma nezávisle proměnnými. Ukazuje se, že taková rovnice může mít (až na přičtení triviálních zákonů zachování) pouze zákony zachování, které jsou nanejvýš kvadratické v závisle proměnné a jejich derivacích. V případě sudého n všechny zákony zachování jsou lineární v závisle proměnné a jejich derivacích. Uvedene výsledky pocházejí z článku R.O. Popovych and A. Sergyeyev, Conservation laws and normal forms of evolution equations, Phys. Lett. A 374 (2010) 2210–2217, arXiv:1003.1648.

11. 11. 2010
doc. RNDr. Tomáš Kopf, Ph.D., Informační geometrie

Abstract:
Parametrizovaná rozdělení pravděpodobnosti byla základními objekty již v počátcích statistické teorie. Takto zadanou množinu lze pochopitelně chápat jako varietu a zkoumat geometrickými metodami. Pozoruhodnou okolností však je, že statistická varieta je automaticky vybavena zajimavými strukturami, jako naprříklad Kullbackovou-Leiblerovou divergencí, odpovidajícími dualistickými konexemi a Fisherovou informační metrikou. Zvláštní pozornost bude věnována výsledkům Shun-ichi Amariho.

4. 11. 2010
doc. RNDr. Michal Marvan, CSc., Modifikovaná Rustova klasifikace přípustných uspořádání derivací

Abstract:
Úplnou klasifikaci přípustných uspořádání derivací podal C.J. Rust v disertaci Rankings of derivatives for elimination algorithms and formal solvability of analytic partial differential equations. Každé takové uspořádání je nejednoznačně popsáno jistou soustavou matic se společnými řádky, jejichž struktura je určena stromovým grafem. V přednášce bude prezentován modifikovaný popis, který je v podstatě jednoznačný, mnohem stručnější, a proto vhodnější pro implementaci.

21. 10. 2010
doc. RNDr. Tomáš Kopf, Ph.D.,Variační volná energie a Bayesovská statistika

Abstract:
V řadě situací mohou prosté skutečnosti mít složité důsledky, jak lze popsat generativními modely. Jednou zmožností, jak z těchto složitých důsledků usoudit zpět na jejích jednoduché příčiny, je přibližná inverze těchto generativních modelů, založená na principu nejmenší volné energie. Seminář je přehledem literatury k tomuto tematu.

7. 10. 2010
RNDr. Hynek Baran, Ph.D., Jets, a way to empower calculations on differential equations in total derivatives on diffieties.

30. 9. 2010
doc. RNDr. Michal Marvan, CSc., Some classification results for integrable surfaces.

Abstract:
Recognizing integrability is among the important unsolved problems in soliton theory. Obtaining a reasonably complete classification of integrable PDE still remains a challenge. Nevertheless, if a PDE comes with a known non-parametric zero curvature representation, then the problem becomes linear and is much easier.

This is the case, e.g., with the Gauss–Mainardi–Codazzi equations of immersed surfaces. Classification results obtained so far include previously unknown integrable classes of Weingarten surfaces. Isolated subcases, such as surfaces of constant curvature, have been known since the nineteenth century. Some of them, such as "surfaces of constant astigmatism" and "a new class of Weingarten surfaces" have fallen into oblivion.

### Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

#### Program v akademickém roce 2009/2010

18. 3. 2010

Dr. Maxim Pavlov (The Lebedev Physical Institute of Russian Academy of Sciences, Moscow, Russia), Integrability of hydrodynamic chains. Collisionless kinetic equations. Three dimensional quasilinear systems.

Abstract:
Theory of integrable hydrodynamic chains in fact was established by B. Kupershmidt in 1983. However, just in recent years (due to significant progress in symbolic computations and hardware) a full classification of such hydrodynamic chains in some simple classes was given.

In this talk we concentrate on the Benney hydrodynamic chain, its local Hamiltonian structures, particular solutions, and relationships with the Khokhlov--Zabolotskaya equation (also well known as a dispersionless limit of the Kadomtsev--Petviashvili equation) and with Vlasov (collisionless Boltzmann equation) kinetic equation..

15. 3. 2010
Prof. RNDr. Josef Mikeš, DrSc. (Univerzita Palackého v Olomouci), PDR Cauchyova typu a některé geometrické úlohy
Seminář se výjimečně koná již od 14.15 hod. v zasedací místnosti rektorátu v budově Na Rybníčku 1, Opava.

17. 12. 2009
RNDr. Oldřich Stolín, Ph.D., Petrovova klasifikace metrik se dvěmi komutujicími Killingovými vektorovými poli.

Abstract:
Considered is the problem of local equivalence and the Petrov classification of generic four-dimensional metrics possessing two commuting and orthogonally transitive Killing vector fields. Some results by classifying the Kerr—NUT—(anti)de Sitter space-time will be presented.

10. 12. 2009
RNDr. Zdeněk Dušek, Ph.D. (PřF UP Olomouc), Homogenní geodetiky na homogenních afinních varietách.

Abstract:
For studying homogeneous geodesics in Riemannian and pseudo- Riemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which involves the reductive decomposition g = h+ m of the Lie algebra g of the isometry group G and the scalar product on m induced by the metric. In the affine differential geometry, there is not such a universal formula.

In the talk, the new method for studying affine homogeneous geodesics will be presented. As an application, homogeneous geodesics for homoge- neous affine connections in dimension 2 will be described and families of affine g.o. spaces in dimension 2 will be found. Some results about homogeneous geodesics in dimension 3 and the main result about the existence of homoge- neous geodesic in arbitrary odd dimension will be mentioned.

3. 12. 2009
Dr. Le Hong Van (MÚ AVČR), Classification of 3-forms and 4-forms on R^8.

Abstract:
Two k-forms on $R ^ n$ are called equivalent, if there exists a transformation in $Gl (R ^n)$ which send one k-form to the other. For $k = 1, 2$ the classification of equivalent k-forms is well known. In my talk I shall explain a method to classify equivalent 3-forms in $\R ^ 8$ due to Djokovic and a recently developed method to classify equivalent 4-forms on $R ^ 8$.

12. 11. 2009
RNDr. Zdeněk Dušek, Ph.D. (PřF UP Olomouc), Homogenní geodetiky na riemannovských a pseudo-riemannovských homogenních varietách.

5. 11. 2009
Steven J. Verpoort, M.Sc., Ph.D. (PřF MU Brno), Some Aspects of The Geometry of the Second Fundamental Form.

### Seminář z diferenciální geometrie a jejich aplikací - vedoucí doc. Marvan

#### Program v akademickém roce 2008/2009

22. 5. 2009
Petr Vojčák, Hyers-Langova a kompaktní diferencovatelnost z pohledu tečných kuželů.

15. 5. 2009
Alžběta Haková, Variační integrační faktory pro parciální diferenciální rovnice prvního řádu.

3. 4. 2009
Diego Catalano Ferraioli, An application of nonlocal symmetries in the reduction of ODEs.

18. 12. 2008
Diego Catalano Ferraioli (Universita delgi Studi di Milano, Italy), Differential invariants of generic parabolic Monge—Ampere equations.

12. 12. 2008
Jan Kotůlek, Spinové struktury na nekomutativním toru.

5. 12. 2008
Alžběta Haková, název bude upřesněn.

17. 10. 2008
Tomáš Kopf, Konec úplně kladných zobrazení.

### Seminář z diferenciální geometrie a jejich aplikací - vedoucí doc. Marvan

#### Program v akademickém roce 2007/2008

4. 10. 2007
dr. Blazej Szablikowski (A. Mickiewicz University, Poznań), Geometric aspects of integrable systems. I. Introduction to evolution systems.

11. 10. 2007
dr. Blazej Szablikowski (A. Mickiewicz University, Poznań), Geometric aspects of integrable systems. II. Construction of symmetries and conserved quantities for KdV and introduction to infinite-dimensional vector fields.

24. 10. 2007
účastníkům semináře se doporučuje vyslechnout ve 14,00 v posluchárně R1 kolokviální přednášku Dr. Le Hong Van (Matematický ústav AV ČR, Praha), Introduction to Gromov-Witten invariants.

25. 10. 2007
Michal Marvan, Kompatibilní uspořádání derivací.

1. 11. 2007
dr. Blazej Szablikowski (A. Mickiewicz University, Poznań), Geometric aspects of integrable systems III. Calculus of infinite-dimensional vector fields.

8. 11. 2007
dr. Blazej Szablikowski (A. Mickiewicz University, Poznań), Geometric aspects of integrable systems IV. Infinite-dimensional Hamiltonian theory.

15. 11. 2007
dr. Blazej Szablikowski (A. Mickiewicz University, Poznań), Geometric aspects of integrable systems V. Bihamiltonian theory.

22. 11. 2007
dr. Blazej Szablikowski (A. Mickiewicz University, Poznań), Geometric aspects of integrable systems VI. Classical R-matrix formalism.

29. 11. 2007
dr. Blazej Szablikowski (A. Mickiewicz University, Poznań), Geometric aspects of integrable systems VII. Lie-Poisson structures.

13. 12. 2007
Klaus Bering Larsen, PhD (MÚ Brno), Non-commutative Batalin-Vilkovsky algebras, homotopy Lie algebras and the Courant bracket.

28. 2. 2008
Peter Sebestyén (Univerzita Palackého, Olomouc), O klasifikaci ireducibilních reprezentací nulové křivosti.

6. 3. 2008
Vojtěch Pravda (MÚ AVČR, Praha), Všechny prostoročasy s nulovými invarianty křivosti ve čtyřech i ve vyšších dimenzích.

27. 3. 2008
Michal Marvan, Horizontální kalibrační kohomologie.

10. 4. 2008
Michal Marvan, Horizontální kalibrační kohomologie II.

15. 5. 2008
Michal Marvan, Riemannova geometrie vnořených ploch II.

### Seminář z diferenciální geometrie a jejich aplikací - vedoucí doc. Marvan

#### Program v akademickém roce 2006/2007

19. 10. 2006
Tomáš Kopf, Kvantová teorie her.

29. 3. 2007
Denis Kochan, Ph.D. (Department of Theoretical Physics FMFI, Comenius University), Kvantovanie disipatívnych systémov.

### Seminář z diferenciální geometrie a jejich aplikací - vedoucí doc. Marvan

#### Program v akademickém roce 2005/2006

2. 3. 2006
Michal Marvan, O minimálním souboru podmínek integrability v Riquierově teorii.

15. 12. 2005
Alžběta Haková, Úvod do homologické algebry I.

8. 12. 2005
prof. Valeriy Yumaguzhin (Pereslavl State University, Pereslavl'-Zalesskij, Russia), Problem of equivalence of Monge-Ampere equations.

1. 12. 2005
Lubomír Klapka, Sincovova rovnice.

3. 11. 2005
Artur Sergyeyev, Maximal superintegrability of Benenti systems.

6. 10. 2005
Michal Marvan, Riquier-Janetova teorie.