**Laboratory for Complex Analysis and Differential Equations**

## About the Laboratory

This laboratory was established as part of a scientific research project supported with a monetary grant awarded by the Government of the Russian Federation under a grant competition designed to provide governmental support to scientific research projects implemented under the supervision of the world's leading scientists at Russian institutions of higher learning (Resolution of the RF Government No.220 of April 9, 2010).

Link to the official website of the Laboratory

**Grant Agreement No.:**

14.Y26.31.0006

**Name of the institution of higher learning:**

Siberian Federal University

**Fields of scientific research:**

Mathematics

**Project goal:**

To conduct world-class research in the field of multidimensional complex analysis and differential equations and to solve a number of fundamental problems using new integrated methods.

**Project progress to date:**

The project staff are performing a detailed investigation of hypergeometrical D-modules. The solutions for such modules contain general algebraic functions, as well as a significant number of special functions of modern mathematical physics. The aforementioned transmission of information will make it possible to form a new language for implementation of analytical extensions of general algebraic functions, as well as hypergeometrical functions with multiple variables. To implement this program, the project team will develop a theory of multidimensional deductions, as well as a theory of distribution of algebraic sets based on the notions of amoeba and co-amoeba – materialized and implied parts of the image of an algebraic set on the logarithmic scale.

Incorrect boundary problems for elliptical equations are usually investigated using the method of integral expressions and the method of analytical extensions. Alongside the spectral theory of self-adjoined operators, these methods make it possible to put together constructive Carleman formulae for elliptical systems and complexes. The project intends to investigate incorrect problems for parabolic equations.

Mutar transformation widely known in classical differential geometry was successfully used to create new examples of integrated two-dimensional Schrödinger equations. The project staff plan to continue investigating the spectral properties of the potentials thus obtained, as well as their connection with direct and reverse problems of the dissipation theory.

According to the de Franchis theorem (1913), the number of holomorphic reflections between Riemannian surfaces of different genera of large units is finite. The exact upper boundary of that value is still unknown and its determination represents a known problem. The project plans to estimate the number of holomorphic reflections when both surfaces are hyper-elliptical. In different discretizations of Riemannian surfaces, their role is performed by graphs, while holomorphic reflections are replaced with harmonic ones. The dimensionality of a group of homologues of a graph is known as graph genus. In these conditions, it is possible to prove analogues of the Riemann-Hurwitz and Riemann-Roch theorems. The project group plans to identify discrete analogues of classical Hurwitz theorems, Wiemann theorems, etc. to estimate the size of the final group operating on the surface of a given genus. The scientists plan to establish a connection between the theory of uniformization of Riemann surfaces and orbifolds developed by Kline, Poincaré and Kebe, and the graph theory by the Bassa-Serra group, and to examine the spectral properties of the Laplace operator on graphs.

## Leading scientist

Full Name: Laptev Ari Link to the scientist's profile |

**Academic degree and title:**

PhD in Physics and Mathematics, Professor

**Job Title:**

Director of Intitut Mittag-Leffler (Sweden),

Professor of Pure Mathematics, Imperial College London (UK),

Professor of Mathematics, KTH, Stockholm (Sweden)

**Field of scientific interests:**

Spectral theory of partial differential equations. Spectral estimates for the Schrödinger equation. Lieb-Thirring inequalities.

**Scientific Recognition:**

Member of the Swedish Royal Academy of Sciences (since 2011).

President of the European Mathematical Society (2007–2010).

Member of the editorial boards of seven international scientific journals, including "Acta Mathematica".

**Awards:**

Wolfson Research Merit Award from the London Royal Society for outstanding achievements in researching Lieb-Thirring inequalities (2007).