2020-2021 Colloquia

In this talk, a subordination principle for the solution operators to a family of the linear multi-dimensional space-time-fractional diffusion-wave equations is addressed. These equations are obtained from the diffusion equation by replacing the first order time-derivative by the Caputo fractional derivative of order β, 0 < β ≤ 2 and the Laplace operator by the fractional Laplacian −(−∆) α 2 with 0 < α ≤ 2. First, a special representation of the fundamental solution to these equations is obtained in form of a Mellin-Barnes type integral. This representation is then employed for derivation of a subordination formula that connects the solutions to the space-time-fractional diffusion-wave equations with different orders α and β of the fractional derivatives. The talk is mainly based on the results published in [1].

[1] Yu. Luchko, Subordination principles for the multi-dimensional space-time-fractional diffusionwave equation. Theory of Probability and Mathematical Statistics 98, 1, 2018, 121-141.

 

A number of methods were developed to numerically solve fractional differential equations. This work contains two methods for numerically solving non-linear fractional differential equations, necessity of multiple approaches is demonstrated to assure their validity. The new substitution method is analyzed and used in this work as a base for all calculations. The convergence theorem assures precision of the method, discretization schemes present implementation. Sufficient condition for a well-condition problem for the method is developed and proved. Solutions for Bessel equation in fractional derivatives were attempted before and the topic interests many scientists. This work presents the theory for solving the generalized fractional Bessel equation, conditions for existence of a solution and its uniqueness for equations with Caputo derivatives. A step further expands the Bessel equation into a more general quasi-linear fractional Bessel equation, where the matching of powers and the order of the derivatives is not required for all terms but one. The discovered methodology for quasi-linear Bessel equation is further developed in its application to the homogeneous equations with constant coefficients and fractional equations with power functions as coefficients. Simple equations in this domain were considered before and analytical solutions were identified. We expand the class of equations and cross check the results with simpler equations. Some cases of fractional Cauchy Euler equations were addressed in the past. Generalized fractional Cauchy Euler equations for both Riemann-Liouville and Caputo fractional derivatives are analyzed next in this work. The solutions similar to the classical Cauchy Euler equation are identified. Independence of solutions is proved.

 

The first half of the talk will be an overview of relations between poylogarithms, Riemann zeta values and multiple zeta values, and their relations to algebraic geometry. Some people who have worked on this topic are Deligne, Zagier, Kontsevich, Manin, Goncharov and many others. The next half of the course will be on a generalization of multiple zeta values to number fields, which I call multiple Dedkind zeta values. At the end I will mention a current work with Pavel Sokolov on analogues of polylogarithms to number fields, which we call Dedekind polylogarithms and their applications to values L-functions.

 

An almost complex structure on a manifold is a way to define an operation similar to multiplication by i (the imaginary unit) on the tangent space of the manifold; a manifold with an almost complex structure is called an almost complex manifold. 

On the other hand, a complex manifold is a manifold modeled over complex vector spaces. 

I will explain these concepts carefully and compare them, showing several examples, especially those based on algebra structures like the quaternions and the octonions. Then I will show examples of almost complex manifolds that cannot be complex manifolds, but can be as close as we want to complex manifolds in a certain sense. This work is in collaboration with Scott Wilson, from Queen's College and the GC.

If there is enough time, I will also talk about almost complex curves on the 6-sphere.

Roughly speaking, a Lie bialgebra is a Lie algebra g such that its dual space g* is equipped with its own Lie algebra structure. The Lie algebras g and g* are compatible in the sense that they define a Lie algebra on g⨁g* which contain g and g* as subalgebras and for which the natural scalar product on g⨁g* is invariant. This Lie algebra structure on g⨁g* is denoted as D(g) and is called a double Lie algebra. A Lie group whose Lie algebra is D(g) is called a double Lie group. In this talk, I will discuss the problem of constructing left invariant complex structures on double Lie groups (which will in effect turn these objects into complex manifolds). The only prerequisites for this talk are basic differential geometry and familiarity with Lie groups and Lie algebras. No familiarity with complex manifolds is assumed. We will review the relevant background on complex manifolds as part of the talk. This project is joint work with Fei Ye.