Related papers: Problems for problem session
This is a survey written for the Proceedings of the AMS Summer Institute in Algebraic Geometry held in Seattle in 2005. Topics discussed in the survey include the ample and the effective cone of the moduli space of curves, Kodaira…
This is an expanded version of my Shaw Prize Lecture delivered at the Chinese University of Hong Kong.
In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my…
This is the manuscript for Proceedings of International Conference and Workshop on Valuation Theory held at University of Saskachewan, Canada in 1999. I have succeeded in showing that any two-dimensional hypersurface singularities of germs…
We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing a certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some…
This survey article on Hilbert's first and second problems is adapted from a one-hour colloquium lecture given at the University of Auckland in May, 2000, just three months before the 100th anniversary of Hilbert's lecture. It includes an…
This is a series of three lectures I gave at the Korea Institute of Advanced Study in June 2019 at a workshop about "Algebraic and Symplectic Aspects of Degenerations of Complex Surfaces". I focus on the symplectic aspects, in particular on…
It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational…
The original version of the paper was published in Contemporary Mathematics 378 ``Groups, Languages, Algorithms''; 2005, pp. 319-348. This is a modified version with Appendix that holds a corrected formulation of Proposition 4.1.
These notes are based on the lecture courses given at the Ruhr-Universit{\"a}t-Bochum (03--08.02.1997) and at the Universit{\'e} Paul Sabatier (Toulouse, 08-12.01.1996).
The paper surveys various Waring type problems in groups, Lie algebras, and associative algebras.
This is a brief overview of a few selected chapters on automorphism groups of affine varieties. It includes some open questions.
Notes used for a course held in 2016 in the School of Advances in Group Theory and Applications, for some lectures given in 2018 for the students of the Master in Mathematics of the Vrije Universiteit Brussels, a course for master and Ph.D.…
Second-order equations of motion on a group manifold that appear in a large class of so-called chiral theories are presented. These equations are presented and explicitely solved for cases of semi-simple, finite-dimensional Lie groups. With…
Traditionally, scholars in physics education research pay attention to students solving well-structured learning activities, which provide restricted room for collaboration and idea-generation due to their close-ended nature. In order to…
This manuscript provides a more detailed treatment of the material from my lecture series at the 2022 Arizona Winter School on Automorphic Forms Beyond $GL_2$. The main focus of this manuscript is automorphic forms on unitary groups, with a…
Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and…
We present a general algorithm for constructing a free resolution for unit groups of orders in semisimple rational algebras. The approach is based on computing a contractible $G$-complex employing the theory of minimal classes of quadratic…
These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected…