Related papers: Lecture notes on Geometric Crystals and their comb…
Crystal basis theory for the queer Lie superalgebra was developed by Grantcharov et al. and it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply…
These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.
This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's…
We shall describe explicitly the decoration functions for certain decorated geometric crystals of classical groups and we shall show that they are represented in terms of monomial realizations of crystal bases.
An expository paper written down after RIMS Model Theory Workshop 2018. To appear in RIMS Kokyuroku.
Crystal skeletons were introduced by Maas-Gari\'epy in 2023 by contracting quasi-crystal components in a crystal graph. On the representation theoretic level, crystal skeletons model the expansion of Schur functions into Gessel's…
Global Categorical Symmetries are a powerful new tool for analyzing quantum field theories. This volume compiles lecture notes from the 2022 and 2023 summer schools on Global Categorical Symmetries, held at the Perimeter Institute for…
Crystal bases are powerful combinatorial tools in the representation theory of quantum groups $U_q(\mathfrak{g})$ for a symmetrizable Kac-Moody algebras $\mathfrak{g}$. The polyhedral realizations are combinatorial descriptions of the…
This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis…
This is a draft of a textbook on differential forms. The primary target audience is sophmore level undergraduates enrolled in what would traditionally be a course in vector calculus. Later chapters will be of interest to advaced…
These lecture notes are based on an introductory course given by the author at the summer school "Noncommutative Algebraic Geometry" at MSRI in June 2012. The emphasis throughout is on examples to illustrate the many different facets of…
Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a…
We construct a geometric crystal for the affine Lie algebra D^{(1)}_n in the sense of Berenstein and Kazhdan. Based on a matrix realization including a spectral parameter, we prove uniqueness and explicit form of the tropical R, the…
Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to…
These lecture notes consist of an introduction to moduli spaces in algebraic geometry, with a strong emphasis placed on examples related to the theory of quiver representations. The goal is to provide the background necessary to understand…
This is a preliminary version of a monograph on homogeneous dynamics and application to some problems of unlikely intersections in Shimura varieties. It consists of four articles, which can be read independently. The first one, by the two…
This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We…
Algebraic and combinatorial properties of a monomial ideal and its radical are compared.
The aim of this introductory lecture is to review the arguments, according to which the symmetry properties of the strong interaction reveal themselves at low energies. I first discuss the symmetries of QCD, then sketch the method used to…
This short note is an extended abstract for my talk at the Nottingham Online Algebraic Geometry Seminar on October 1, 2020. It is based on the joint works with Alexander Kuznetsov arXiv:1802.08097 and arXiv:2001.04148.