Analysis, Geometry and Topology Department | Home |

Institute of Mathematics and Informatics | BG |

Bulgarian Academy of Sciences | EN |

Analysis, Geometry and Topology Department | Home |

Institute of Mathematics and Informatics | BG |

Bulgarian Academy of Sciences | EN |

Institute of Mathematics and Informatics | |

1113 Sofia, Acad. G. Bonchev Str, Bl. 8 | |

Analysis, Geometry and Topology Department |

Bessel's Functions, Mittag-Leffler's Functions, and Generalizations: Properties, Series, Fractional Calculus

Jordanka Paneva-Konovska

Institute of Mathematics and Informatics, BAS

Abstract. Public academic lecture, for "Professor".

17.12.2019 - Antoni Rangachev - Algebraic theory for continuity for meromorphic functions

Algebraic theory for continuity for meromorphic functions

Antoni Rangachev

University of Chicago, IMI-BAS

Abstract. For a reduced complex analytic variety $X$ and a rational number $\alpha$ between $0$ and $1$ I will show that the meromorphic functions on $X$ that are Holder continuous with exponent $\alpha$ form a coherent sheaf that sits between the structure sheaf of $X$ and its normalization. In fact, there are finitely many $\alpha$ that matter. For exponent $\alpha = 1$ the result recovers the Lipschitz saturation considered by Pham-Teissier and Zariski. The proof uses the Riemann extension theorem, which allows us to turn inequalities into integral dependence relations, rational powers of ideals and Rees' valuation theorem.

26.11.2019 - Peter Petrov - Topological Data Analysis (TDA)

Topological Data Analysis (TDA)

Peter Petrov

Institute of Mathematics and Informatics, BAS

Abstract. This talk is an introduction to TDA with an emphasis on persistent homology. It should be understandable for mathematicians as well as biologists, stressing geometric and intuitive explanations rather than formal arguments. Particular examples will be discussed.

10.07.2019 - Ivan Dimovski - Convolutional method in operational calculus

Convolutional method in operational calculus

Ivan Dimovski

Institute of Mathematics and Informatics, BAS

Abstract. This lecture is intended as a kind of replica of the public defense (June 14, 1977) of his thesis presented for Dr. Sci. degree in Mathematics. It is delivered now on occasion of his 85-th anniversary on July 7th.

28.05.2019 - Gueo Grantcharov - Solutions to the Strominger system with torus symmetry

Solutions to the Strominger system with torus symmetry

Gueo Grantcharov

Florida International University, USA

Abstract. In the talk we present a construction of new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over $K3$ surfaces can be generalized to torus bundles over $K3$ orbifolds. In particular, we prove that, for $13 \leq k \leq 22$ and $14 \leq r \leq 22$, the smooth manifolds $S^1 \times \#_k (S^2 \times S^3)$ and $\# r(S^2 \times S^4) \# r+1(S^3 \times S^3)$ have a complex structure with trivial canonical bundle and admit a solution to the Hull-Strominger system. This is a joint work with A. Fino and L. Vezzoni from University of Torino.

Graded algebras, algebraic functions, planar trees, and elliptic integrals

Vesselin Drensky

Institute of Mathematics and Informatics, BAS

Abstract. The starting point of the talk is the problem how to measure how big an infinite dimensional algebra is. It has turned out that this problem concerns not only algebra. It is related with interesting problems in graph theory (enumeration of graphs with prescribed properties), mathematical analysis (e.g. theory of algebraic and transcendent functions). Even elliptic integrals surprisingly appear.

19.02.2019 - Peter Petrov - Cut-and-paste problem for polyhedral and algebraic varieties

Cut-and-paste problem for polyhedral and algebraic varieties

Peter Petrov

Institute of Mathematics and Informatics, BAS

Abstract. The first part of this talk includes the history of Hilbert’s third problem, and the idea of the proof, proposed by Dehn, plus some more recent results. After formulating the problem for varieties and some related to it problems, we will review the main results obtained, and will discuss briefly the case of toric varieties.

30.10.2018 - Peter Petrov - Nash problem in arc spaces

Nash problem in arc spaces

Peter Petrov

Institute of Mathematics and Informatics, BAS

Abstract. After introducing briefly the arc and jet spaces of an algebraic variety with examples I will state the Nash theorem and I will formulate the Nash problem. The main results obtained will be mentioned, together with the counterexamples in dimension 4 (Kollar and Ishii) and 3 (de Fernex). Then will be stated the Nash problem for pairs and will be shown its solution in the case of stable toric varieties, obtaining from it a positive answer of the problem in that case.

Existence and Multiplicity of Solutions for Higher-order $p$-Laplacian Differential and Fractional Equations

Stepan Tersian

Institute of Mathematics and Informatics, BAS

Abstract.
The multiplicity of periodic solutions for $2n$-th order $p$-Laplacian differential equations is studied. Variational method and generalized Clark's theorem are applied. Dirichlet's problem for fractional differential equations is also considered. The results are published in the papers [1], [2] and [3].

[1] P. Drábek, M. Langerová, S. Tersian, Existence and multiplicity of periodic solutions to one-dimensional $p$-Laplacian, *Electronic Journal of Qualitative Theory of Differential Equations* **30** (2016), 1-9.

[2] L. Saavedra, S. Tersian, Existence of solutions for $2n$-th order nonlinear $p$-Laplacian differential equations, *Nonlinear Analysis: Real World Applications* **34** (2017), 507-519.

[3] Lin Li, S. Tersian. Existence and Multiplicity of Periodic Solutions to Fractional $p$-Laplacian Equations, *Differential and Difference Equations with Applications* **230** (2018), pp. 495-507.

Geodesic flow, Laplace spectra, and quantization on Riemannian symmetric spaces of rank one

Gueo Grantcharov

Florida International University, USA

Abstract. In a project, joint with D. Grantcharov, we extend the observation by I. Mladennov and V. Tsanov that the eigenvalues and eigenspaces of the Laplace operator on the sphere correspond via specific quantization scheme to the energy level of the geodesic flow, to Riemannian symmetric spaces of rank one. In the talk I'll discuss also the relations with other constructions like twistor theory and possible extension to other symmetric spaces.

On the geometry of minimal surfaces in four-dimensional Euclidean or Minkowski space

Krasimir Kanchev

Todor Kableshkov University of Transport

Abstract. Preliminary defense of a doctoral thesis.

06.02.2018 - Ivan Minchev - On the Existence of Local Quaternionic Contact Geometries

On the Existence of Local Quaternionic Contact Geometries

Ivan Minchev

Faculty of Mathematics and Informatics, Sofia University

Abstract. In this talk, I will report on some joint work with J. Slovak. We exploit the Cartan-Kaehler theory to prove the local existence of real analytic quaternionic contact structures for any prescribed values of the respective curvature functions and their covariant derivatives at a given point on a manifold. We show that, in a certain sense, the different real analytic quaternionic contact geometries in $4n+3$ dimensions depend, modulo diffeomorphisms, on $2n+2$ real analytic functions of $2n+3$ variables.

On the geometry of minimal surfaces in four-dimensional Euclidean or Minkowski space

Krasimir Kanchev

Todor Kableshkov University of Transport

Abstract. We prove that any minimal surface in n-dimensional Euclidean or Lorentz space locally admits geometrically determined parameters - canonical parameters. For any minimal surface of general type in four-dimensional Euclidean or Minkowski space parametrized by canonical parameters we obtain Weierstrass representations - canonical Weierstrass representations. These formulas allow us to solve explicitly the system of natural partial differential equations and to establish geometric correspondence between minimal surfaces of general type, the solutions to the system of natural partial differential equations and pairs of holomorphic functions in the Gauss plane. On the base of these correspondences we obtain that any minimal surface of general type in Euclidean 4-space determines locally a pair of two minimal surfaces in Euclidean 3-space and vice versa.

Bessel's Functions, Mittag-Leffler's Functions, and Generalizations

Jordanka Paneva-Konovska

Technical University of Sofia

Institute of Mathematics and Informatics, BAS

Abstract. Preliminary defense of a doctor-of-science thesis.

03.11.2017 - Yana Aleksieva - Lorentz Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

Lorentz Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

Yana Aleksieva

Faculty of Mathematics and Informatics, Sofia University

Abstract. Preliminary defense of a doctoral thesis.

Minimal Lorentz Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

Yana Aleksieva

Faculty of Mathematics and Informatics, Sofia University

Abstract.
We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose Gauss curvature $K$ and normal curvature $\kappa$ satisfy the inequality $K^2 - \kappa^2 > 0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that any minimal Lorentz surface of general type is determined (up to a rigid motion) by two invariant functions satisfying a system of two natural partial differential equations. Using a concrete solution to this system we construct an example of a minimal Lorentz surface of general type.

This is a joint work with Velichka Milousheva.

A new framework for numerical analysis of nonlinear systems: the significance of the Stahl's theory and analytic continuation via Pade approximants

Sina Baghsorkhi

University of Michigan, Ann Arbor, USA

Abstract.
An appropriate embedding of polynomial systems of equations into the extended complex plane renders the variables as functions of a single complex variable. The relatively recent developments in the theory of approximation of multi-valued functions in the extended complex plane give rise to a new framework for numerical analysis of these systems that has certain unique features and important industrial applications. In electricity networks the states of the underlying nonlinear AC circuits can be expressed as multi-valued algebraic functions of a single complex variable. The accurate and reliable determination of these states is imperative for control and thus for efficient and stable operation of the electricity networks. The Pade approximation is a powerful tool to solve and analyze this class of problems. This is especially important since conventional numerical methods such as Newton's method that are prevalent in industry may converge to non-physical solutions or fail to converge at all.

The underlying concepts of this new framework, namely the algebraic curves, quadratic differentials and the Stahl's theory are presented along with a critical application of Pade approximants and their zero-pole distribution in the voltage collapse study of the electricity networks.

This is a joint work with Nikolay R. Ikonomov (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria) and Sergey P. Suetin (Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia).

13.06.2017 - Armen Sergeev - Adiabatic Limit in Ginzburg-Landau and Seiberg-Witten Equations

Adiabatic Limit in Ginzburg-Landau and Seiberg-Witten Equations

Armen Sergeev

Steklov Mathematical Institute, Moscow, Russia

Abstract. We study solutions of Ginzburg-Landau equations arising in superconductivity theory. Static solutions, called otherwise vortices, are completely described by the Taubes theorem. However, it is not much known about the structure of dynamical solutions of these equations. The adiabatic limit method allows to describe the slowly moving solutions. In this limit Ginzburg-Landau equations reduce to the adiabatic equation which coincides with the Euler equation for geodesics on the space of vortices with respect to the Riemannian metric determined by the kinetic energy. An analogous adiabatic limit is used for the approximate description of solutions of the Seiberg-Witten equations on 4-dimensional symplectic manifolds. In this limit one obtains instead of geodesics the pseudoholomorphic curves while solutions of Seiberg-Witten equations reduce to the families of vortices defined in the normal planes to the limiting pseudoholomorphic curve. Such families should satisfy a nonlinear $\bar\partial$-equation which can be treated as a complex analogue of the adiabatic equation.

A Calabi-Yau Conjecture for Generalized Kaehler Metrics on a Hyper-Kaehler Manifold

Vestislav Apostolov

Universite du Quebec a Montreal, Canada

Institute of Mathematics and Informatics, BAS

Abstract. In this talk I will introduce and motivate, using the point of view of Geometric Invariant Theory, the conjecture that any generalized Kaehler metric on a compact simple hyper-Kaehler manifold can be obtained from a hyper-Kaehler metric by a deformation due to D. Joyce. I will propose an analytic method for attacking the conjecture, based on the generalized Kaehler-Ricci flow introduced by Tian and Streets, and show its conditional resolution modulo the uniform boundedness of the Aeppli potential along the flow. This is a joint work with Jeff Streets.

Fractional calculus operators of special functions? - The result is well predictable!

Virginia Kiryakova

Institute of Mathematics and Informatics, BAS

Abstract.
Recently many authors are spending lot of time and efforts to evaluate various operators of fractional order integration and differentiation and their generalizations of classes of, or rather particular, special functions. Practically, these are exercises to calculate improper integrals of products of different special functions. The list of such works is rather long and yet growing daily. So, to illustrate our general approach proposed rather earlier but systematized recently (Kiryakova, 1994,...,2017), we limit ourselves to mention here only a few of them as 15 examples. Since there is a great variety of special functions, as well as of operators of fractional calculus, the mentioned job produces a huge flood of publications. Many of them use same formal and standard procedures, and besides, often the results sound not of practical use, with except to increase authors' publication activities.

In this talk, we point out on some few basic classical results, combined with author's ideas and developments, that show how one can do the task at once, in the rather general case: for both operators of generalized fractional calculus and generalized hypergeometric functions. In this way, the greater part of the results in the mentioned publications are well predicted and fall just as rather special cases of the discussed general scheme.

V. Kiryakova, Generalized Fractional Calculus and Applications. Longman & J. Wiley, 1994;

V. Kiryakova, Fractional calculus operators of special functions? - The result is well predictable! Chaos Solitons and Fractals (2017), DOI: 10.1016/j.chaos.2017.03.006, To appear.

Boundary behaviour of invariant distances and metrics in complex analysis

Lyubomir Andreev

Institute of Mathematics and Informatics, BAS

Abstract. Preliminary defense of a doctoral thesis.

07.03.2017 - Gueo Grantcharov - Interactions among some non-Kaehler metrics on complex manifolds

Interactions among some non-Kaehler metrics on complex manifolds

Gueo Grantcharov

Florida International University, USA

Abstract. In the talk will be considered, on particular examples, the existence and non-existence question of 3 types of non-Kaehler metrics on compact complex manifolds - balanced, SKT (or pluriclosed) and astheno-Kaehler.

The qc Yamabe problem on non-spherical quaternionic contact manifolds

Alexander Petkov

Faculty of Mathematics and Informatics, Sofia University

Abstract. The aim of this talk is to demonstrate that the qc Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. Precisely, we establish that the qc Yamabe constant of any such manifold is strictly less than the corresponding constant of the 3-Sasakian sphere, which allows us to give an affirmative answer to the qc Yamabe conjecture.

17.01.2017 - Ivan Minchev - The Yamabe Equation on 3-Sasakian Manifolds

The Yamabe Equation on 3-Sasakian Manifolds

Ivan Minchev

Faculty of Mathematics and Informatics, Sofia University

Abstract. In the talk I will present a solution of the quaternionic contact Yamabe equation on the 3-Sasakian sphere of dimension 4n+3 as well as a related result concerning the uniqueness of solutions of the quaternionic contact Yamabe problem on compact locally 3-Sasakian manifolds.

10.01.2017 - Antoni Rangachev - Excess Intersection and differential equisingularity

Excess Intersection and differential equisingularity

Antoni Rangachev

Northeastern University, Boston, USA

Abstract. We prove an excess intersection formula that determines the change of the top self-intersection numbers of families of Cartier divisors. We apply the formula to settle a long-standing problem of providing numerical control for Whitney (differential) equisingularity for families of arbitrary isolated singularities.

General Rotational Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

Yana Aleksieva

Faculty of Mathematics and Informatics, Sofia University

Abstract. We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classification of minimal general rotational surfaces, general rotational surfaces with parallel normalized mean curvature vector field, flat general rotational surfaces, and general rotational surfaces with flat normal connection.

08.11.2016 - Ralitza Kovacheva - Bernstein's asymptotic formulas for orthogonal polynomials

Bernstein's asymptotic formulas for orthogonal polynomials

Ralitza Kovacheva

Institute of Mathematics and Informatics, BAS

Abstract.
In the present talk, Bernstein's classical asymptotic formulas for polynomials, orthogonal on the interval $[-1,1]$ with respect to weights of the form $\frac{\sigma(x)}{\sqrt{1 - x^2}}$ with $\sigma$ being a Dini-Lipschitz function and nonnegative on $[-1,1]$, will be discussed. Basic methods to derive the formulas will be presented - Bernstein-Szego's method, Nutall's and Stahl-Gonchar-Suetin's ones. A special attention will be given to Nutall's method. A new result, based on Nutall's ideas and extending the validity of Bernstein's formulas to complex-valued Dini-Lipschitz trigonometric weights $\sigma$ will be provided.

The talk is based mainly on the article "Nutall's integral equations and Bernstein's asymptotic formula for a complex weight", N.R Ikonomov, R.K Kovacheva, S.P. Suetin, Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 6, Pages 125-144.

Symplectic intersections and global perturbations of Hamiltonian systems

Dragomir Dragnev

Institute of Mathematics and Informatics, BAS

Abstract. Given a symplectic manifold $M$, we recall that a submanifold $N$ is called coisotropic if the symplectically orthogonal distribution of the tangent bundle of $N$ is a subset of the tangent bundle of $N$. This distribution is integrable in the sense of Frobenius and therefore gives rise to a foliation. We consider the following problem: Given a symplectomorphism $\varphi$ of $M$, under what conditions on $N$ and $\varphi$ we have the following property. There exists a point $x$ on $N$, so that $\varphi(x)$ lies on the leaf through $x$. Further as an application we derive a global perturbation result for Hamiltonian systems.

21.06.2016 - Peter Dalakov - The self-duality equations on a Riemann surface and symplectic geometry

The self-duality equations on a Riemann surface and symplectic geometry

Peter Dalakov

Institute of Mathematics and Informatics, BAS

Abstract. In 1987 N. Hitchin introduced a system of gauge-theoretic equations associated with a compact Riemann surface $X$ and a reductive complex Lie group $G$. These self-duality equations on a Riemann surface have been actively studied since then by differential geometers, algebraic geometers and representation theorists. The moduli space of solutions is a hyperkaehler orbifold, whose twistor family contains two non-isomorphic complex structures, both of which have interpretations as moduli of holomorphic (algebro-geometric) data. One of these corresponds to the moduli space of $G$-Higgs bundles on $X$ and the other to the moduli space of $G$-local systems on $X$. The moduli space of (semi-stable) $G$-Higgs bundles is an algebraic integrable system. In the first part of the talk we shall review the main properties of Hitchin's moduli space. In the second part we shall discuss briefly some recent results (with U. Bruzzo) concerning the holomorphic symplectic and Poisson geometry of the generalised Hitchin system.

03.05.2016 - Oleg Mushkarov - Partial integrability on Thurston manifolds

Partial integrability on Thurston manifolds

Oleg Mushkarov

Institute of Mathematics and Informatics, BAS

Abstract. The lecture is devoted to a generalization of a theorem by H. Kim (Annales Polonici Matematic, 2013) for partial integrability of invariant almost complex structures on some higher dimensional generalizations of the famous Thurston's example (Proc. Amer. Math. Soc. 55 (1976), 467-468) of a compact symplectic 4-manifold which does not admit a Kaehler structure. We will also discuss some open questions on the local existence of holomorphic functions on nilpotent Lie groups and symplectic manifolds.

26.04.2016 - Bogdan Alexandrov - Infinitesimal deformations of nearly parallel $G_2$-structures

Infinitesimal deformations of nearly parallel $G_2$-structures

Bogdan Alexandrov

Faculty of Mathematics and Informatics, Sofia University

Abstract. The subject of this talk are the infinitesimal deformations of nearly parallel $G_2$-structures on compact 7-dimensional manifolds. I will show that they form a subspace of co-closed 3-forms of a certain eigenspace of the Laplace operator. I will also give a similar description of the space of infinitesimal Einstein deformations of such a structure. The results will be illustrated by examples.

22.03.2016 - Norbert Poschadel - About additive dispersion parameters for random variables

About additive dispersion parameters for random variables

Norbert Poschadel

Katholische Universitaet Eichstaett-Ingolstadt, Germany

Abstract.
Variance and standard deviation play a crucial role in probability and statistics. One reason for this might be that the variance of the sum of independent (even of uncorrelated) square integrable random variables is the sum of their variances. If for a dispersion parameter $V$ of the form $V(X) = E(f(E - EX))$, where $f: \mathbb{R} \to \mathbb{R}$ is even, i.e. $f(-x) = f(x), \forall x \in \mathbb{R}$, additivity $V(X + Y) = V(X) + V(Y)$ holds for every two independent real-valued random variables $X$ and $Y$ (such that all involved integrals exist), then necessarily $V$ is a multiple of the variance and thus variance is not only a popular example for a dispersion parameter with additivity for independent random variables, but it can even be characterized by this property.

This result can be generalized to functionals $V$ defined on a class of $\mathbb{R}^n$-valued random vectors: $V$ is additive iff $V(X)$ is a linear combination of the covariances between any two components $X_i$ and $X_j$ of the random vector $X$.

08.03.2016 - Blagovest Sendov - New Proof of the Sector Theorem

New Proof of the Sector Theorem

Blagovest Sendov

Institute of Mathematics and Informatics, BAS

Abstract. The lecture is dedicated to an elementary proof of the Sector Theorem: If a polynomial with real and non-negative coefficients has no zeros in a sector symmetric with respect to the real axis and starts from the origin, then all derivatives of this polynomial have no zeros in the same sector. Some applications of the Sector Theorem will be discussed including a refinement of the Gauss-Lucas theorem.

Existence and multiplicity of solutions to boundary value problems for fourth-order and fractional order differential equations with impulses

Stepan Tersian

"Angel Kanchev" University of Ruse

Abstract.
In the first part of the talk we consider the existence and multiplicity of solutions to impulsive boundary value problem for fourth-order differential equations in phase transitions models. The variational method is applied, based on minimization, mountain-pass and Clark's theorems.

In the second part, the existence and multiplicity of solutions to impulsive boundary value problem for a fractional order differential equation is discussed. Fractional derivatives are involved in both Riemann-Liouville and Caputo sense. The variational method is applied based on minimization theorem and three-critical points theorem due to Bonanno and Candito.

The results are partly published in:

[1] Cabada A., Tersian S., Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations, Boundary Value Problems 2014, 2014: 105, http://www.boundaryvalueproblems.com/content/2014/1/105.

[2] Bonanno, G., Rodriguez-Lopez, R., Tersian, S., Existence of solutions to boundary value problem for impulsive fractional differential equations, Fractional Calculus and Applied Analysis 17 (3), 2014, pp. 717-744, Springer - De Gruyter Open.

Multipoint Pade approximants - growth behavior and distribution of points of interpolation

Ralitza Kovacheva

Institute of Mathematics and Informatics, BAS

Abstract. Given a regular compact set $E$ in the complex plane, a unit Borel measure $\mu$ supported by $E$ and a function $f$ holomorphic on $E$, we consider the sequence of $\mu$-multipoint Pade approximants $\{\pi^m_{n,m_n}\}$, $n \to \infty$, $m_n = o(n)$ of $f$. The growth behavior of $\{\pi^m_{n,m_n}\}$ is considered under some restrictions on the points of interpolations and on the function $f$. Furthermore, an inverse problem will be discussed - what is the influence of the behavior of the sequence $\{\pi^m_{n,m_n}\}$ on the asymptotic distribution of the points of interpolation?.

19.01.2016 - Elitza Hristova - Momentum images of representations

Momentum images of representations

Elitza Hristova

Institute of Mathematics and Informatics, BAS

Abstract. Let $K$ be a compact semisimple Lie group, $V$ an irreducible unitary representation of $K$, and $P(V)$ the corresponding projective space. In this talk we consider the momentum map for the action of $K$ on $P(V)$ and discuss the question of describing the image of the momentum map for this action. By a theorem of Kirwan, the intersection of the momentum image with a positive Weyl chamber is a convex polytope, called the momentum polytope. In this talk, we are interested in describing the momentum polytope in terms of the highest weight of the representation. We start with a short introduction to the subject and a description of some known results and then we describe some of our results. The talk is based on a joint work with Valdemar Tsanov and Tomasz Maciazek.

15.12.2015 - Betul Bulca - Semi-parallel Surfaces in Euclidean Spaces

Semi-parallel Surfaces in Euclidean Spaces

Betul Bulca

Uludag University, Bursa, Turkey

Abstract. In the present study we consider semi-parallel surfaces in Euclidean spaces. We study Wintgen ideal surfaces which are satisfying the semi-parallelitiy condition. We classified semi-parallel meridian surface and semi-parallel tensor product surfaces in 4-dimensional Euclidean space. Recently, we have given some examples of extended semi-parallel surfaces in Euclidean 4-space.

01.12.2015 - Yana Aleksieva - Lorentz Surfaces in Four-dimensional Pseudo-Euclidean Spaces

Lorentz Surfaces in Four-dimensional Pseudo-Euclidean Spaces

Yana Aleksieva

Faculty of Mathematics and Informatics, Sofia University

Abstract.
We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce canonical parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations.

The talk is for applying to self-study doctorate in section "Analysis, Geometry and Topology".

13.10.2015 - Lyubomir Andreev - Boundary behavior of invariant metrics on planar domains

Boundary behavior of invariant metrics on planar domains

Lyubomir Andreev

Institute of Mathematics and Informatics, BAS

Abstract.
There will be presented several results concerning the boundary behavior of the Caratheodory, Kobayashi and Bergman metrics near boundary points of planar domains under different assumptions of smoothness for the boundary of the domain.

The talk is based on a joint work with Nikolai Nikolov.

20.10.2015 - Nikolai Nikolov - Estimates of the Kobayashi and quasi-hyperbolic distances

Estimates of the Kobayashi and quasi-hyperbolic distances

Nikolai Nikolov

Institute of Mathematics and Informatics, BAS

Abstract.
Upper estimates of the Kobayashi and quasi-hyperbolic distances near Dini-smooth boundary points of domains in $\mathbb{C}^n$ and $\mathbb{R}^n$, respectively, are obtained. The best universal lower bound for the quasi-hyperbolic distance is found.

The talk is based on a joint paper with Lyubomir Andreev.

13.10.2015 - Nikolay Ikonomov - On the Limit Zero Distribution of Type I Hermite-Pade Polynomials

On the Limit Zero Distribution of Type I Hermite-Pade Polynomials

Nikolay Ikonomov

Institute of Mathematics and Informatics, BAS

Abstract.
We present the results of new numerical experiments on zero distribution of type I Hermite-Pade polynomials of order $n = 200$ for three different collections of three functions $[1, f_1, f_2]$. We obtained these results numerically, and they not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points of second order, and finally, in the third case they both have branch points of third order.

The talk is based on "On the limit zero distribution of type I Hermite-Pade polynomials", N.R. Ikonomov, R.K. Kovacheva, S.P. Suetin, arXiv:1506.08031.

06.10.2015 - Andreas Kirsch - The Factorization Method for Inverse Scattering Problems

The Factorization Method for Inverse Scattering Problems

Andreas Kirsch

Karlsruhe Institute of Technology, Germany

Abstract. We consider the inverse scattering problem to determine the shape of a sound-soft obstacle from the knowledge of the scattered fields corresponding to time-harmonic acoustic plane waves. We first formulate the direct and inverse scattering problems rigorously and report on classical approaches for solving the inverse problem numerically. Then we present a non-iterative approach which belongs to a class of sampling methods. We construct a binary criterion to determine whether or not a given point belongs to the scatterer. This method is mathematically rigorous and elegant and provides an explicit formula for the characteristic function of the unknown scatterer. One of the advantages of this Factorization Method is that it does not need to know the kind of boundary condition in advance or the number of components of the scatterer.

29.09.2015 - Gueo Grantcharov - On Quaternionic Kahler / Hyper-Kahler Correspondence

On Quaternionic Kahler / Hyper-Kahler Correspondence

Gueo Grantcharov

Florida International University, USA

Abstract. The QK/HK correspondence is a differential-geometric representation of some ideas in string theory related to moduli spaces of solutions of self-duality equations. To a hyper-kahler manifold with an isometry preserving one of the structures and rotating the others corresponds a quaternionic kahler one. Apart from the theoretical physics it has deep connections to algebraic geometry and is under active development. In the talk a basic introduction for people with different backgrounds and some directions of future research will be given.

Some Classifications of Submanifolds in Semi-Euclidean Spaces Considering Their Position Vector

Nurettin Cenk Turgay

Istanbul Technical University, Turkey

Abstract.
Position vector is one of the most basic objects studied to understand geometrical properties of submanifolds of (semi-)Euclidean spaces. In this direction, the notion of generalized constant ratio (GCR) submanifolds has been introduced very recently. Let $M$ be a hypersurface of a semi-Euclidean space $E^m_s$ and $x$ its position vector. $M$ is said to be a generalized constant ratio hypersurface if the tangential component $x^T$ of $x$ is a principal direction of $M$.

On the other hand, biharmonic submanifolds have caught interest of many geometers so far. A submanifold $M$ is said to be biharmonic if $\Delta^2 x = 0$ and biconservative if a weaker condition is satisfied.

In this talk, we will give a summary of results very recently obtained on hypersurfaces in semi-Euclidean spaces considering their position vector. We will also present some open problems that we are currently studying.

Acknowledgements: This work is an announcement of results obtained during a project of scientific and Technological Research Council of Turkey, Project Number: 114F199 (TUBITAK).

09.06.2015 - Nikolay Ikonomov - Generalized Pade approximants and extremal distribution of points

Generalized Pade approximants and extremal distribution of points

Nikolay Ikonomov

Institute of Mathematics and Informatics, BAS

Abstract. Given a regular plane condenser $(E, F)$, let $\alpha$ and $\beta$ be triangular point tables; $\alpha \in E$, $\beta \in F$. Let a function $f$ be analytic on $E$. We prove that, if a sequence of generalized Pade approximants, associated with the point tables $\alpha$ and $\beta$, converge maximally to $f$, then the points $(\alpha, \beta)$ are extremally distributed with respect to the condenser $(E, F)$.

19.05.2015 - Petar Rusev - Bessel's Functions

Bessel's Functions

Petar Rusev

Institute of Mathematics and Informatics, BAS

Abstract. Presentation of a book.

On the Trisection of an Angle - seriously speaking!

Georgi Dimkov and Dessislava Dimkova

Institute of Mathematics and Informatics, BAS

Abstract. More than 20 centuries ago ancient mathematicians ascertained that the following three problems: doubling the cube, trisecting the angle and squaring the circle cannot be solved using compass and straightedge. More precisely speaking: the compass can be opened arbitrarily wide, but there are no markings on it, the straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. The ancient Greek mathematicians tried to find different tools to solve the problems. So appeared the ideas for improving of usual instruments, going out of the constructions of straight lines and circle, construction of new instruments. The talk aims to present the development of the ideas as well as their realizations.

28.04.2015 - Radoslav Rashkov - An invitation to String Theory

An invitation to String Theory

Radoslav Rashkov

Faculty of Mathematics and Informatics, Sofia University

Abstract. Superstring theory is supposed to serve as the most successful theory unifying all the fundamental interactions in Nature. In this talk I'll present basic concepts of (super) String theory and I'll review some of the recent developments as well. First I'll give a very brief overview of the Physics origin of String theory and its development over the years. Next I'll introduce the main ingredients of the theory and their properties. In many aspects String theory can be thought of as a mixture of Physics and Mathematics. The last part of the talk is devoted to mathematical aspects of String theory and mathematical tools used there. I'll conclude with some applications and open issues.

21.04.2015 - Doychin Tolev - On Sums of Values of the Legendre Symbol

On Sums of Values of the Legendre Symbol

Doychin Tolev

Faculty of Mathematics and Informatics, Sofia University

Abstract. In this lecture a short presentation of some classical results (due to I. Vinogradov, A. Weil and others) concerning sums of values of the Legendre symbol will be given. We will consider more precisely certain special sums of such type (the Jacobstal sums) as well as their connection to the problem of representation of primes as sums of two squares.

Families of Mittag-Leffler Type Functions and Convergent Series in Them

Jordanka Paneva-Konovska

Technical University of Sofia

Abstract. Series defined by means of the Mittag-Leffler type functions are considered, namely, Mittag-Leffler functions and their Prabhakar and multi-index generalizations. Their domains of convergence are found and the behaviour of such series on the boundaries of these domains is studed. Analogues of the classical theorems for the power series like Cauchy-Hadamard, Abel, Tauber and Littlewood as well as Fatou type theorems are proposed.

Completely monotone functions in the study of a class of fractional evolution equations

Emilia Bazhlekova

Institute of Mathematics and Informatics, BAS

Abstract. After a short introduction to the theory of completely monotone functions and Bernstein functions, a class of fractional evolution equations is considered, containing different models of the so-called slow diffusion. Each problem of this class can be rewritten as an abstract Volterra integral equation, which kernel satisfies certain complete monotonicity properties. Based on these properties of the kernel, existence of a unique solution is proven for the considered class of equations and some properties of the solution are derived.

17.03.2015 - Peter Popivanov - On the Local Solvability of Linear Partial Differential Equations

On the Local Solvability of Linear Partial Differential Equations

Peter Popivanov

Institute of Mathematics and Informatics, BAS

Abstract. This talk is a short survey on several results concerning the local solvability of linearv PDE. We consider different classes of non-solvable PDE, including H. Lewy and Mizohata operators, and describe the set of $C_0^\infty$ right-hand sides for which a distribution solution exists. Methods of complex analysis are used to do this.

Generalized fractional derivatives of Riemann-Liouville and Caputo type

Virginia Kiryakova

Institute of Mathematics and Informatics, BAS

Abstract.
In Fractional Calculus (FC), as in the (classical) Calculus, the notions of derivatives and integrals (of first, second, etc. or arbitrary, incl. non-integer order) are co-related. One of the most frequent approach in FC is to define first the Riemann-Liouville (R-L) integral of fractional order, and then by means of suitable integer-order differentiation operation applied over it (or under its sign) a fractional derivative is defined - in the R-L sense (or in Caputo sense). The first mentioned (R-L type) is closer to the theoretical entertainments in analysis, but has some shortages - from the point of view of interpretation of the initial conditions for Cauchy problems (stated also by means of fractional order derivatives/ integrals), and also for the analysts' confusion that such a derivative of a constant is not zero in general. The Caputo (C-) derivative, arising in geophysical studies, helps to overcome these problems and to describe models of applied problems with physically consistent initial conditions. Let us mention however that recently some authors dispute the advantages of the C-derivative against the R-L one, with examples from control theory.

The operators of the generalized fractional calculus (integrals and derivatives) represent commuting m-tuple ($m=1,2,3,\ldots$) compositions of operators of the classical FC with power weights (the so-called Erdelyi-Kober operators), represented in compact and explicit form by means of integral, integro-differential (R-L type) or differential-integral (C- type) operators, where the kernels are special functions of most general hypergeometric kind.

In this survey we present the genesis of the definition of the generalized fractional derivatives (of fractional multi-order) of R-L type, introduce the new ones of Caputo type, and analyze their properties and cases of coincidence of the definitions (for example for the hyper-Bessel differential operators or order m = multi-order ($1,1,\ldots,1$), and for the Gelfond-Leontiev generalized differentiation operators). We consider some more particular examples of the derivatives of both types and of Cauchy problems for fractional order differential equations with R-L or C-derivatives and initial conditions of the corresponding type. Note that quite natural, the solutions of such problems are related to the Mittag-Leffler function or its multi-index analogues.

10.02.2015 - Blagovest Sendov - Proof of the Sector Theorem

Proof of the Sector Theorem

Blagovest Sendov

Institute of Mathematics and Informatics, BAS

Abstract. Let $S(\phi) = \{z: |\arg(z)| \geq \phi\}$ be a sector on the complex plane. If $\phi \geq \frac{\pi}{2}$, then $S(\phi)$ is a convex set and, according to Gauss-Lucas theorem, if all the zeros of the polynomial $p(z)$ are on the sector $S(\phi)$, then the same is true for the zeros of all its derivatives. In the lecture it is proved that, if the polynomial $p(z)$ is with real and non negative coefficients, then the same is true also for $\phi < \frac{\pi}{2}$, when the sector is not a convex set.

11.11.2014 - Blagovest Sendov - The Notion of Apolarity

The Notion of Apolarity

Blagovest Sendov

Institute of Mathematics and Informatics, BAS

Abstract. In the first part of the talk the Grace theorem and its applications will be discussed. In the second part the notion of locus of a complex polynomial will be introduced and its main properties will be considered. In conclusion, some problems concerning loci will be given.

04.11.2014 - Stoyu Barov - On closed sets with convex projections in $\mathbb{R}^n$

On closed sets with convex projections in $\mathbb{R}^n$

Stoyu Barov

Institute of Mathematics and Informatics, BAS

Abstract. Let $n \geq 2$, $B$ be a closed and nonconvex subset of $\mathbb{R}^n$ such that the projections of $B$ onto all hyperplanes are convex and line-free. What can we say about such a $B$? We show that in this case the dimension of $B$ is at least $n-2$. In addition, we present some interesting examples.

28.10.2014 - Lyubomir Andreev - Two equivalent descriptions of the Poincare sphere

Two equivalent descriptions of the Poincare sphere

Lyubomir Andreev

Institute of Mathematics and Informatics, BAS

Abstract. The Poincare sphere is known as the first example of 3 dimensional compact manifold for which the first homology group is trivial but it is not homeomorphic to $S^3$. There are many different ways to construct this manifold due to its various applications in topology. In the present talk will be discussed the equivalence between two descriptions of this manifold. On one hand it is the coset $S^3/ I^*$ where $I^*$ is the preimage under the two fold covering $SU(2) \to SO(3)$ of the group $I$ of isometries with center $(0,0,0)$ of the regular icosahedron, and on the other hand it is the Brieskorn manifold \[ M(2,3,5) = \{(z_1,z_2,z_3) \in C^3 | z_1^2 + z_2^3 + z_3^5 = 0 , |z_1|^2 + |z_2|^2 + |z_3|^2 = 1\} . \]

21.10.2014 - Maria Trybula - The Kobayashi balls of (C-)convex domains

The Kobayashi balls of (C-)convex domains

Maria Trybula

Jagiellonian University in Krakow, Poland

Abstract. A pure geometric description of the Kobayashi balls of (C-)convex domains in terms of the so-called minimal basis will be shown.

Surfaces with Lightlike Mean Curvature Vector Field in Four-dimensional Pseudo-Euclidean Spaces

Velichka Milousheva

Institute of Mathematics and Informatics, BAS

Abstract. We consider surfaces in four-dimensional pseudo-Euclidean spaces with the property that the mean curvature vector is lightlike at each point. We introduce a geometrically determined moving frame field at each point of such a surface and using the derivative formulas for this frame field we obtain a system of invariant functions satisfying some integrability conditions. Our main theorem states that these invariants determine the surface up to a motion. We find examples of surfaces with lightlike mean curvature vector field in the class of the meridian surfaces of elliptic, hyperbolic or parabolic type.

07.10.2014 - Georgi Ganchev - Special surfaces of codimension one or two and their natural PDE's

Special surfaces of codimension one or two and their natural PDE's

Georgi Ganchev

Institute of Mathematics and Informatics, BAS

Abstract. We consider Weingarten surfaces in the three-dimensional Euclidean or Minkowski space. These surfaces admit canonical (natural) parameters, which allows us to solve the Lund-Regge problem for this class of surfaces. The background PDE's, describing the surfaces from the subclass of linear-fractional Weingarten surfaces, are found. We discuss basic ideas from the theory of surfaces of codimension one which can be applied to the theory of surfaces of codimension two.

On the Geometry of Minimal Surfaces in the Four-dimensional Euclidean or Minkowski Space

Krassimir Kanchev

Todor Kableshkov University of Transport

Abstract. The talk is for applying to self-study doctorate in section "Analysis, Geometry and Topology".

23.09.2014 - Julian Lawrynowicz - On an Extension of Harmonicity and Holomorphy

On an Extension of Harmonicity and Holomorphy

Julian Lawrynowicz

University of Lodz, Poland

Abstract. The concept of harmonicity and holomorphy related with the Laplace equation $\Delta s \equiv (\partial^2 / \partial x^2) + (\partial^2 / \partial y^2) = 0$, $(x, y) \in \mathbb{R}^2$, is extended with the use of equation \[ (\partial / \partial t) s = -\Gamma s_x + \Lambda(\Delta + \Delta_\tau) s \] with \[ \Delta + \Delta_1 = (\partial^2 / \partial x^2) - a^2 (\partial^2 / \partial \theta^2), \Delta_2 = - a^2 (\partial^2 / \partial \theta^2), \] \[ \Delta_3 = (\partial^2 / \partial z^2) - a^2 (\partial^2 / \partial \theta^2), \Delta_4 = (\partial^2 / \partial z^2) - (\partial^2 / \partial \xi^2) - a^2 (\partial^2 / \partial \theta^2), \] \[ \Delta_5 = (\partial^2 / \partial z^2) - (\partial^2 / \partial \xi^2) - (\partial^2 / \partial \eta^2) - a^2 (\partial^2 / \partial \theta^2), \] where $\Gamma$ and $\Lambda$ are $C^1$-scalar functions of $(x, \theta) \in \mathbb{R}^2, \ldots, (x, y, z, \xi, \eta, \theta) \in \mathbb{R}^6$ for $\tau = 1, \ldots, 5$, respectively, $t \in \mathbb{R}$, $\theta \in \mathbb{R}$ and $x*$ is an arbitrary admissible function. We discuss the fundamental solutions for the equations in question (more precisely, of the corresponding linearized problem) which is a parabolic equation of the second kind. For effective solutions and $\tau \equiv 1, 2, 3, 4 (\mod 8)$ it is convenient to involve the quaternionic structure, for $\tau \equiv 5, 6, 7, 0 (\mod 8)$ - the paraquaternionic structure. Physically, it is natural to describe, with help of the equation involved, relaxation processes attaching $(x, y, z)$ to the first chosen particle, $(\xi, \eta, \zeta)$ - to the second one, $\theta$ to temperature, entropy or order parameter, and $t$ - to time.

09.09.2014 - Kadri Arslan - Surfaces in Euclidean Space $E^{n+2}$

Surfaces in Euclidean Space $E^{n+2}$

Kadri Arslan

Uludag University, Bursa, Turkey

Abstract.
In the present study we consider basic important concepts of differential geometry of surfaces in $E^{n+2}$, particularly including tangential and normal spaces along with the concept of orthonormal normal frames and orthogonal transformations between them. Furthermore, we define the three fundamental forms, Gauss equations and Weingarten equations. Finally we concentrate on curvature properties of the surfaces in $E^{n+2}$. In the second part we derive the integrability conditions of Codazzi-Mainardi, Ricci and the theorema egregium from the equations of Gauss and Weingarten. In particular, we introduce the Riemannian curvature tensor and the curvature tensor of the normal bundle. Further, we give some examples on 4-dimensional Euclidean space $E^4$.

[1] B. Y. Chen, Geometry of Submanifols, Dekker, New York (1973).

[2] Yu. Aminov, The Geometry of Submanifolds. Gordon and Breach Science Publishers, Singapore, 2001.

[3] B. Bulca and K. Arslan, Surfaces Given with the Monge Patch in E4: J. Math. Physics, Analysis, Geometry 05/2013.

01.07.2014 - Yulian Tsankov - Operational calculi for boundary value problems

Operational calculi for boundary value problems

Yulian Tsankov

Faculty of Mathematics and Informatics, Sofia University

Abstract. Open session of the scientific jury for the defense of a doctoral thesis.

24.06.2014 - Johann Davidov - Generalized complex structures (a la Hitchin)

Generalized complex structures (a la Hitchin)

Johann Davidov

Institute of Mathematics and Informatics, BAS

Abstract.
The theory of generalized complex structures was initiated by N. Hitchin about 10 years ago and further developed by his students M. Gualtieri and G. Cavalcanti. The notion of a generalized complex structures contains that of a complex and a symplectic structure as special cases, and can be considered as a complex analog of the notion of a Dirac structure introduced by T. Courant and A. Weinstein to unify the Poisson and presymplectic geometries. This and the fact that the generalized complex structures play an important role in the physical string theory motivate the increasing interest to the generalized complex geometry.

The main purpose of this talk is to present some basic ideas, concepts and results of the theory of generalized complex structures.

Maximally Convergent Sequences of Rational Functions: Growth Behavior and Zero Distribution

Ralitza Kovacheva

Institute of Mathematics and Informatics, BAS

Abstract. We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree $\leq n$ and denominator degree $\leq m_n$ for meromorphic functions $f$ on a compact set $E$ of $\mathbb{C}$ where $m_n=o(n)$ as $n \to \infty$. We obtain a Jentzsch-Szego type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain $E_{\rho(f)}$ of meromorphy of $f$ if $f$ has a singularity of multivalued character on the boundary of $E_{\rho(f)}$. The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Pade approximation and real rational best approximants are considered.

03.06.2014 - Lyubomir Andreev - Local properties of complex convex domains

Local properties of complex convex domains

Lyubomir Andreev

Institute of Mathematics and Informatics, BAS

Abstract. In this talk there will be considered several connected to each other notions of convexity of domains which generalize the classical notion of convexity, namely complex convexity, weak local linear convexity, weak local convexity and pseudoconvexity. Under concrete assumptions about the type of convexity of one domain and the smoothness of its boundary we establish some local properties connected with the existence of subdomains with the same type of convexity and smoothness of the boundary, which contain sections of the domain with open balls.

27.05.2014 - Petar Rusev - The Mittag-Leffler function

The Mittag-Leffler function

Petar Rusev

Institute of Mathematics and Informatics, BAS

Abstract. The Mittag-Leffler method for summation of divergent series gives, in fact, a method for analytical continuation of holomorphic functions defined by convergent Maclorain's series.

20.05.2014 - Blagovest Sendov - Problems connected with the Gauss-Lucas theorem

Problems connected with the Gauss-Lucas theorem

Blagovest Sendov

Institute of Mathematics and Informatics, BAS

Abstract.
The talk is based on the ideas of the resent publication [1]. Several ideas and conjectures for straightening of the Gauss-Lucas theorem will be stated.

[1] Rudinger, A.: Straightening the Gauss-Lucas theorem for polynomials with zeros in the interior of the convex hull. arXiv: 1405.0689v1 [math.CV] 4 May 2014.

13.05.2014 - Ivan Dimovski - Operational calculus of Mikusinski and its generalizations

Operational calculus of Mikusinski and its generalizations

Ivan Dimovski

Institute of Mathematics and Informatics, BAS

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