권재훈 (Seoul National University)
Title
Characters for Lie superalgebra and a canonical basis for quantum group
Abstract
In this talk, we introduce the representation theory of a general linear Lie superalgebra. Our review will focus on Brundan's seminal work, where he solved the irreducible character problem for general linear Lie superalgebras using the theory of a (dual) canonical basis for the quantum group.
서의린 (Seoul National University)
Title
Basic representation theory of vertex operator algebras
Abstract
In this lecture, I will introduce basic notions related to vertex operator algebras and their representations. In particular, relations between representations of Kac-Moody Algebras and affine vertex algebras will be discussed via the concept of Zhu's algebras. If time allows, I will mention recent results on representations of W-algebras, which are Hamiltonian reductions of affine vertex algebras.
송아림 (Seoul National University)
Title
Affine W-algebra and its generators
Abstract
We introduce the definition of affine W-algebra associated with principal nilpotent element via BRST cohomology. We then find its explicit generators in some cases.
우르노 아끼또 (Seoul National University) (Preparatory seminar for "Auslander-Reiten transformation and τ-tilting theory")
Title
Algebras and modules
Abstract
In this talk, we introduce the notations and terminology we use on algebras and modules, and we briefly recall some of the basic facts from module theory. The notions of projective cover, injective envelope, and basic algebra will be explained.
이강산 (Seoul National University)
Title
Examples of vertex operator algebras and their modules
Abstract
In this talk I will explain properties of Virasoro algebra and vertex operator algebra. Also I will explain construction of their modules which is goal of our talk. We can achieve same goal starting from affine Lie algebras. Directly comparing these two important processes is help to understand vertex operator algebras and their modules. If time allows, detail calculations will be explained including Segal-Sugawara construction about finite simple Lie algebra.
이민 (University of Bristol)
Title
Effective equidistribution of rational points on the expanding horospheres
Abstract
The study of dynamics on homogeneous spaces has found a remarkable range of applications in seemingly unrelated areas such as combinatorics, number theory and mathematical physics over the last four decades. Then a new line of research is emerging, that of making those results effective: giving rates of convergence for equidistribution results in homogeneous dynamics. For these lectures, we study the distribution of rational points on the expanding horospheres with the methods from the theory of automorphic forms and Fourier analysis.
이석형 (Princeton University)
Lecture 1 Title
Parametrizing rings of low rank: cubic ring example
Lecture 1 Abstract
In 2001 Bhargava established parametrizations of rings of low rank (rank up to 5) in series of works called "Higher Composition Laws", which gave inspirations for several related results in parametrizing arithmetic objects. In this talk, we will discuss the case of parametrization of cubic rings (Delone-Faddeev correspondence), together with related concepts including geometric construction and resolvent forms.
Lecture 2 Title
Counting rings of low rank: cubic ring example
Lecture 2 Abstract
The main problem of arithmetic statistics is to count the the number of arithmetic objects with bounded invariants, and counting rings having bounded discriminant was the first to be considered. I will introduce results on counting cubic forms by Davenport and Heilbronn (1971) and its refinement by Bhargava and Shankar, (2012) focusing on methodology of counting lattice points.
이현세 (Seoul National University) (Preparatory seminar for "Auslander-Reiten transformation and τ-tilting theory")
Title
Quivers and algebras
Abstract
A finite dimensional algebra can be presented by path algebras of a finite quiver, which is a vector spaces of paths where two paths are multiplicated by concatenation. In this lecture, we will see how the radical and idempotents of an algebra are represented under the presentation.
임선희 (Seoul National University)
Title
Homogeneous dynamics and Diophantine approximation
Abstract
In these lectures, we will start by a short review on the relation between geodesic flow on the modular surface and the classical Gauss map. We will then explain Dani correspondence, a correspondence allowing us to solve certain problems in Diophantine approximation using dynamics on homogeneous spaces. We will then see how to obtain Hausdorff dimension of certain badly approximable vectors in Diophantine approximation and finish up by presenting some recent work on the problem.
허태혁 (Seoul National University)
Title
Representation of quivers (Preparatory seminar for "Auslander-Reiten transformation and τ-tilting theory")
Abstract
A representation of a quiver is a collection of vector space s and linear maps associated with the vertices and arrows of the quiver. It is known that a finite dimensional algebra is isomorphic to a path algebra of a quiver or its quotient algebra bounded by an admissible ideal. In this talk, we construct a category equivalence between the category of quivers and the category of finite dimensional algebras. Moreover, we describe special modules of quivers; simple, projective, injective, indecomposable modules.
황준호 (MIT)
Title
Finiteness criteria for surface group representations
Abstract
First, we give a brief expository account of Selberg's lemma that a finitely generated linear group in characteristic zero is virtually torsion-free. Second, using this as an ingredient, we describe a proof of the following theorem (joint with Anand Patel and Ananth Shankar): a semisimple rank two complex representation of a positive-genus surface group has finite image if and only if all simple loops have finite order image. The proof will require basic topology and basic algebraic number theory. Finally, time permitting, we sketch applications of our theorem to certain problems in arithmetic of differential equations and on mapping class group dynamics.
Alexander Molev (University of Sydney)
Title
Symmetrization map, Casimir elements and Sugawara operators
Abstract
The canonical symmetrization map is a g-module isomorphism between the symmetric algebra S(g) of a finite-dimensional Lie algebra g and its universal enveloping algebra U(g). This implies that the images of g-invariants in S(g) are Casimir elements. For each simple Lie algebra g of classical type we consider basic g-invariants arising from the characteristic polynomial of the matrix of generators. We calculate the Harish-Chandra images of the corresponding Casimir elements. By using counterparts of the symmetric algebra invariants for the associated affine Kac-Moody algebras, we obtain new formulas for generators of the centers of the affine vertex algebras at the critical level.