Joint International Workshop in Algebra and Analysis

Speakers

Schedule

Abstracts

Xiaosong Sun (Jilin University)

Title: Derivations and Their Connections with Affine Geometry

Abstract: Some notable problems in affine geometry are closely related to the algebraic theory of derivations. The aim of the talk is to introduce the connections between the two fields. We will also introduce some of our works on locally nilpotent derivations and the Makar-Limanov invariant, which are related to the Jacobian conjecture and the cancellation problem in affine geometry.

Xiaosong Sun is a member of the School of Mathematics at Jilin University, Changchun. His work concerns locally nilpotent derivations, Mathieu–Zhao subspaces, the LND conjecture and the Jacobian conjecture.

Changyu Ren (Jilin University)

Title: A General Form of Newton–Maclaurin Type Inequalities

Abstract: In this talk, I will introduce a general form of the Newton–Maclaurin type inequality. We have extended the Newton–Maclaurin inequality for elementary symmetric functions to the case of linear combinations of multiple elementary symmetric functions.

Changyu Ren is a professor at Jilin University working in partial differential equations and geometric analysis. His recent work includes curvature estimates for Hessian equations and prescribed curvature problems for hypersurfaces in Minkowski space, published in Communications on Pure and Applied Mathematics, the American Journal of Mathematics, the Journal of Functional Analysis, Analysis & PDE and IMRN.

Rustam Turdibaev (New Uzbekistan University)

Title: Necklace Lie Algebra and the Liezation of Leibniz Necklace Algebra

Abstract: The necklace Lie algebra of a quiver is classically defined on the commutator quotient $A/\mathrm{Com}(A)$ of the path algebra $A$ of its doubled quiver. Through Van den Bergh's double Poisson formalism, this structure lifts to a left Leibniz bracket $\{-,-\}$ on $A$ itself. In turn, the theory of Leibniz algebras attaches to $(A,\{-,-\})$ another canonical Lie quotient, the liezation $A/\mathrm{Leib}(A)$, where the Leibniz kernel $\mathrm{Leib}(A)$ is spanned by the self-brackets $\{a,a\}$. In this talk we investigate the precise conditions under which these two quotients coincide. Over a field of characteristic zero, we show that $\mathrm{Leib}(A)=\mathrm{Com}(A)$ if and only if every arrow in the quiver with distinct endpoints possesses a loop at either its source or its target. As a consequence, for the free algebra $k\langle x_1,\dots,x_g,y_1,\dots,y_g\rangle$ equipped with Kontsevich's bracket, the necklace Lie algebra is precisely the liezation of the necklace Leibniz algebra, meaning that the relations imposed by taking traces are exactly those forced by antisymmetry.

Reza Seyyedali (New Uzbekistan University)

Title: Complex Monge–Ampère on Kähler Toric Surfaces

Abstract: In his celebrated work, Yau proved the existence and regularity of solutions of the complex Monge–Ampère equation with smooth right hand side on compact Kähler manifolds. In work in progress, we study the complex Monge–Ampère equations on a toric surface with a less regular right hand side.

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