Higher order estimates for Monge-Ampere type equations motivated by quaternionic geometry

Marcin Sroka, Jagiellonian University, Kraków, Poland

Abstract. In the talk I will discuss obstacles for obtaining higher order a priori estimates for equation introduced by Alesker and Verbitsky in 2010 on HKT manifolds in relation to Calabi type conjecture on those manifolds (and similar type PDEs studied since on hypercomplex manifolds). After presenting the state of art, I will focus on the key second order estimate. For the real Monge-Ampere equation on Riemannian manifolds it follows essentially by Pogorelov calculation, for the complex Monge-Ampere equation, in relation to Calabi conjecture, it was proven by Aubin and Yau. I will use those two as comparison to underline new difficulties related to concavity properties of quaternionic Monge-Ampere operator.

Unimodality Preservation by Ratios of Functional Series and Integral Transforms

Dmitrii Karp, Holon Institute of Technology, Holon, Israel

Abstract. Elementary, but very useful lemma due to Biernacki and Krzyz (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their corresponding coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios and integral transforms as well as for unimodality in place of monotonicity. In the talk, we discuss conditions on the functional sequence and the kernel of an integral transform ensuring such property. Numerous series and integral transforms appearing in applications satisfy our sufficient conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. The key role in our considerations is played by the notion of sign regularity.

Presentation

Complex Dirac structures on flag manifolds

Carlos Augusto Bassani Varea, Universidade Tecnológica Federal do Paraná, Brazil

Abstract. In this talk we present a description of the invariant complex Dirac structures with constant real index on a maximal flag manifold in terms of the roots of the Lie algebra which defines the flag manifold. As a particular case, when the real index is zero, we have the description of all invariant generalized complex structures on a maximal flag manifold. We also present a classification of the complex Dirac structures under the action of B-transformation. This is a joint work with Cristian Ortiz (IME-USP).

New and little known properties of the Fox and Fox-Wright functions

Dmitrii Karp, Holon Institute of Technology, Holon, Israel

Abstract. In the talk, I will introduce the functions H of Fox and W of Wright (its general case known as Fox-Wright function) and some motivation behind them. Then, I will discuss the extension of Gauss’ expansion and summation formulas for the hypergeometric function to the case of general Fox-Wright function and Norlund’s expansion for the Meijer’s G function to the case of Fox’s H function. I will further present positivity conditions for the so-called delta-neutral case of Fox’s H function making it a probability distribution (widely used in statistics), which rely on complete monotonicity of certain product ratios of Gamma functions. Moreover, I will exhibit a new integral equation for Fox’s H function and a conjecture regarding its zeros. Finally, some open problems will be presented.

Burau Representation and Application to Reducibility and Exchangeability of Braids

Alexander Stоimenov, Dongguk University, Republic of Korea

Abstract. I will give an introduction to the braid groups, closure operation, Markov theorem, and exchange move. Then I will introduce the Burau representation, and discuss its application to reducibiliy and exchangeability of braids.

Independence Structures in Random Matrix Theory and Random Partitions

Yacine Barhoumi-Andréani, University of Bochum, Germany

Abstract. We consider two models of random objects: eigenvalues of random matrices and random integer partitions for two classical measures: the GUE and the Schur measure. We express the largest element of these sets as maxima of independent random variables. The method uses orthogonal polynomials on the real line in the random matrix case and on the unit circle in the second case. The distribution involved in the random matrix case uses the sigma-form of Painlevé IV functions whose theory will be recalled.

Functions holomorphic over finite-dimensional complex commutative algebras

Marin Genov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract. Every morphism 𝜑:𝐴 →𝐵 in the category of finite-dimensional complex commutative associative unital (Banach) algebras gives rise to a structure sheaf of functions of a single 𝐴-variable and taking values in 𝐵 and whose differential respects the 𝐴-module structure of 𝐵. Even though these are holomorphic mappings of several complex variables, it turns out they also share many features of the theory of holomorphic functions of a single complex variable. At the same time this also enables the construction of relative complex analysis and relative (on the level of coefficients and in the sense of Grothendieck) complex-analytic geometry. In this talk I will discuss some basic aspects and questions of the local analytic theory.

Riemannian four-manifolds and twistor spaces: some rigidity results

Davide Dameno, Department of Mathematics “Federigo Enriques”, University of Milan

Abstract. It is well-known that four-dimensional Riemannian manifolds carry many peculiar properties, which give rise to the existence of unique canonical metrics. In order to find conditions for the existence of such metrics, in 1978 Atiyah, Hitchin and Singer adapted Penrose’s construction of twistor spaces to the Riemannian context, paving the way for the study of many other characterizations of curvature properties for Riemannian four-manifolds. After giving an overview of the Riemannian and Hermitian structures of twistor spaces in the four-dimensional case, we present some new rigidity results for Riemannian four-manifolds whose twistor spaces satisfy specific vanishing conditions (such as Bochner-flatness). This is based on joint work with Professors Giovanni Catino and Paolo Mastrolia.

Introduction to spectral graph theory, Part 3

Peter Petrov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Introduction to spectral graph theory, Part 2

Peter Petrov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract. In the second lecture we will ntroduce the incidence matrix and the line graph of given graph with their properties. Spectral graph drawing and partition into clusters will be discussed also. Unlike the first talk when the stress was on the motivation, in this talk it will be on results accompanied by typical examples.

Introduction to spectral graph theory

Peter Petrov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences.

Abstract. This is the first lecture of a series of 2-3 lectures on algebraic graph theory. A very basic knowledge in linear algebra should be enough as a prerequisite, so that the talk could be understood by people from informatics or biology, also by students. Simple examples will accompany any notion. All knowledge needed from graph theory will be given. The Laplacian of a simple graph will be defined, showing in particular how it could be calculated and what for it could be used.

Isotropic Killing vector fields and structures on complex surfaces

Gueo Grantcharov, Florida International University, USA

Abstract. In a 4-dimensional vector space with scalar product of signature (2,2), two independent vectors spanning a maximal isotropic (null) plane determine a canonical action of the para-quaternioins. We noticed that on an oriented 4-manifold with such pseudo-Riemannian metric, existence of two isotropic (null) Killing vector fields leads to integrability of the induced structure – called para-hypercomplex, and the metric is anti-selfdual. Using the Kodaira classification one can describe the topology of the underlying 4-manifold in the compact case. In this talk, examples of such structures on several of the 4-manifolds will be provided and some restrictions for a compact complex surface to admit split signature Hermitian metric with one non-vanishing null Killing vector field will be established. The talk is based on a joint project with J. Davidov and O. Mushkarov.

On Squeezing Function for Planar Domains

Ahmed Yekta Ökten, Institut de Mathématiques de Toulouse, France

Abstract. Let Ω be a domain in ℂ^𝑛 such that the set 𝐸(Ω, 𝐵^𝑛) of injective holomorphic maps from Ω into the unit ball 𝐵^𝑛 ⊂ ℂ^𝑛 is non-empty. The squeezing function of Ω, denoted by 𝑆_Ω is defined as

𝑆_Ω(𝑧) = sup{𝑟 ∈ (0, 1): 𝑟𝐵^𝑛 ⊂ 𝑓(Ω),      𝑓 ∈ 𝐸(Ω, 𝐵^𝑛),      𝑓 (𝑧) = 0}.

It follows from the definition that the squeezing function is biholomorphically invariant and roughly speaking, it measures how much a domain looks like the unit ball looking at a fixed point. As expected, the study of the squeezing function leads to nice results about the properties of the invariant metrics on complex domains. The behaviour of the squeezing function is well studied however very few non-trivial explicit formulas of squeezing functions have been found.

In this talk we will establish the explicit formulas of squeezing functions on doubly connected planar domains in an elementary way. With the same method we will also provide bounds to squeezing functions of higher connected domains. Finally, we will conclude by mentioning other results and further questions about explicit formulas of squeezing functions on planar domains.

Kähler Manifolds of Quasi-constant Holomorphic Sectional Curvature and Generalized Sasakian Space Forms

Cornelia-Livia Bejan, “Gheorghe Asachi” Technical University of Iasi, Romania

Abstract. Two geometric notions, namely Kähler manifolds of quasi-constant holomorphic sectional curvature and generalized Sasakian space forms, are related to each other, for the first time. Some conditions under which each of these structures induces the other one, are provided here. Several results are obtained on direct products (which are special cases of Naveira’s classification), warped products or hypersurfaces of manifolds and relevant examples are included. A result of Niebergall and Ryan is generalized here. Some necessary and sufficient conditions for Einsteinian hypersurfaces are given at the end. The talk is based on a joint work with S. Guler.

Towards Homological Mirror Symmetry for weighted projective planes over the complex numbers

Marin Genov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Index counting theories and linear stability of periodic waves

Sevdzhan Hakkaev, Shumen University

Abstract. In this talk, we will present some results on the stability of periodic waves. In the models that we will consider, stability problem leads to the study of the spectral problems of the form and computations of the quantities involved in the stability index.

The Dimension Dind оf Finite Topological T0-Spaces

Dimitrios Georgiou, University of Patras, Greece

Abstract. A.V. Arhangelskii introduced the dimension Dind [2] and some properties of this dimension have been studied in [1, 3]. In this talk, we present the study of this dimension for finite T0-spaces. Especially, we present that in the realm of finite T0-spaces, Dind is less than or equal to the small inductive dimension ind, the large inductive dimension Ind and the covering dimension dim. We also give the “gaps” between Dind and the dimensions ind, Ind and dim, presenting various examples which shows these “gaps”. Moreover, in this field of spaces, we present characterizations of Dind, inserting the meaning of the maximal family of pairwise disjoint open sets, and give properties of this dimension.
References:[1] Chatyrko V.A., Pasynkov B.A., On sum and product theorems for dimension Dind, Houston J. Math. 28 (2002), no. 1, 119-131.[2] Egorov V., Podstavkin Ju., A definition of dimension, (Russian) Dokl. Akad. Nauk SSSR 178 1968, 774-777.[3] Kulpa W., A note on the dimension Dind, Colloq. Math. 24 (1971/72), 181-183.
The results of this talk are the research work of the paper “THE DIMENSION Dind OF FINITE TOPOLOGICAL T0-SPACES”, the authors of which are D. Georgiou, Y. Hattori, A. Megaritis and F. Sereti. This paper has been accepted for publication to the journal Mathematica Slovaca.

General Rotational Surfaces in Pseudo-Euclidean 4-Spaces

Victoria Bencheva, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract. Rotational surfaces are rich source of examples both in Euclidean and pseudo-Euclidean spaces. We consider the so-called general rotational surfaces of elliptic and hyperbolic type in the Minkowski 4-space and the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces introduced by C. Moore in the Euclidean space R^4. We describe analytically some basic subclasses of general rotational surfaces, namely: minimal, flat, with flat normal connection, with parallel normalized mean curvature vector field.

Formal neighbourhoods in the space of arcs

Peter Petrov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract: After introducing briefly jet and arc spaces I will discuss the notion of formal neighbourhood of a point. Then the theorem of Drinfeld-Grinberg-Kazhdan representing the formal neighbourhood of a non-degenerate arc will be formulated. Finally, I will discuss some possible generalizations.

Viskovatov algorithm for Hermite-Padé polynomials

Nikolay Ikonomov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract. We will discuss the problem of characterizing geometric objects in the space by their invariants. For minimal surfaces the problem was solved in the works of Ganchev. On the other hand, Ganchev and Mihova considered the question in some more general cases. We propose a solution in the general case, and we characterize a surface by two invariant functions, subject of a differential equation. We remind about the classical algorithm of Viskovatov, which can be applied to one power series, and is used to compute classical Padé approximant. We need alternative computation of classical Hermite-Padé approximant (two power series), and we have already computed it by matrix method. We propose this new extension of the Viskovatov algorithm for much faster computation of Hermite-Padé approximant, and we compare with the matrix method.

The talk is based on https://arxiv.org/abs/2007.03370.
This is a joint work with Sergey Suetin, MIAN-RAS.

Algebraic Non-hyperbolicity of Hyperkähler Manifolds

Ljudmila Kamenova, Stony Brook University, USA

Abstract. A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperkahler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperkahler manifold is infinite, then it is algebraically non-hyperbolic. These results are joint with Misha Verbitsky.