Universal categories.

Universal categories.

Abstract: Universal constructions are ubiquitous in mathematics. For example, the polynomial ring is uniquely characterized by a universal property for commutative rings. Other examples include free monoids, free groups, and tensor algebras. In this mainly expository talk we will discuss an analogous, but somewhat less well known, concept on the level of categories. In particular, we will see how one can define categories that are determined by universal properties. Examples include the Temperley-Lieb category (the free monoidal category on a self-dual object), the Brauer category (the free symmetric monoidal category on a self-dual object), and the oriented Brauer category (the free symmetric monoidal category on a pair of dual objects). We will discuss intuitive diagrammatic descriptions of these categories and how these universal constructions allow one to easily find deep symmetries in a wide range of categories.
 

Special Location: STEM 201

Twisted foldings, moment graphs, and their applications to geometry.

Twisted foldings, moment graphs, and their applications to geometry.

Abstract: In this talk, we will explore the construction of twisted quadratic folding for finite crystallographic root systems. Twisted folding of a root system is a generalized notion of its classical counterpart, and it allows us to ``fold" some of those root systems that were not foldable in the classical setup, such as root systems of type $E_8$ and some of its subsystems, into smaller root systems. However, one twist occurs here - it is that unlike classical foldings, the outcome, though finite, may not necessarily be crystallographic. In the second part of the talk, we bring another key thread into our picture, namely the notion of a moment graph for finite Coxeter groups and their structure algebras and observe how the folding of a root system induces a map on the level of their structure algebras. In the last part of the talk, we will touch some of the dictionaries established between sheaves over moment graphs and the geometry of the corresponding flag varieties, and indicate how the twisted folding of a root system could apply to the study of geometric questions. The content of this talk comprises the startup of my PhD research project. It is expository, and will be delivered with descriptive examples throughout.
 

Rost multipliers of Kronecker tensor products.

Rost multipliers of Kronecker tensor products.

Abstract: We extend the techniques employed by Garibaldi to construct a map SP_2nxSP_2m-->Spin_4nm for all values of n and m. We then show how, depending on the parities of n and m, this map induces injections between central quotients, for example PSP_2nxPSP_2m-->HSpin_4nm when n and m are not both odd. Additionally, we calculate the Rost multipliers for each map we have constructed.
 

Locally nilpotent derivations and rationality.

Special location : STE J0106

Locally nilpotent derivations and rationality.

Abstract: Let X be an affine algebraic variety over a field k of characteristic zero and let B = k[X] be the coordinate algebra of X. If B admits “many” locally nilpotent derivations D : B —> B, then does it follow that X is a rational variety? I will talk about the history of this question and some recent results.
 

Polynomial equations over finite fields coming from a formula for analytic rank of twisted Carlitz modules.

Polynomial equations over finite fields coming from a formula for analytic rank of twisted Carlitz modules.

Abstract: Drinfeld modules (Carlitz modules are their particular cases) are analogs of elliptic curves over functional fields. There are notions of their twists, L-functions and analytic rank (order of 0 of L-functions at some points). It turns out that the sets of twists whose analytic rank is greater than some fixed number, is an algebraic variety over a finite field. Polynomials defining these varieties come from some resultantal-type matrices. Some problems of further research will be presented. It is not necessary to know the theory of Drinfeld modules: all relevant definitions will be done.
 

Existence of an irreducible representation of a semisimple algebraic group with a prescribed highest weight.

Junior Algebra Seminar

Existence of an irreducible representation of a semisimple algebraic group with a prescribed highest weight.

Abstract: In this talk, I will present the standard proof for the existence of an irreducible representation of a semisimple algebraic group with a prescribed highest weight when the base field is algebraically closed. We will revise the key notions associated to semisimple algebraic groups and their representations (root system, dominant weight, highest weight etc.) at the beginning, state the key preliminary results, then discuss the proof using them. This talk is for anybody who is interested in the study of algebraic groups.
 

Classification of a-2 finite Coxeter groups.

Classification of a-2 finite Coxeter groups.

Abstract: In 1985, Lusztig defined a function a : W --> N for any Coxeter group W by using the Kazhdan-Lusztig basis of the Hecke algebra of W . The a-function has important connections with the cell representation theory of W and its Hecke algebra, but is usually difficult to compute directly. However, it is known that an element has a-value 1 if and only it it is a non-identity element with a unique reduced word, and that W contains finitely many elements of a-value 1 if and only the Coxeter diagram of W satisfy certain conditions. In this talk, we describe a similar classification, in terms of Coxeter diagrams, of all Coxeter groups with finitely many elements of a-value 2. We will show that elements of a-value 2 are fully-commutative in the sense of Stembridge, and our main tools for the classification include Viennot’s heaps and certain so-called star operations on Coxeter groups. (Joint work with Richard Green.)
 

Localized Landweber-Novikov operations on generalized cohomology.

Localized Landweber-Novikov operations on generalized cohomology.

Abstract: Cohomological operations on generalized cohomology theories (e.g. Steenrod, Adams, Landweber-Novikov) have been extensively studied during the past decade (Brosnan, Levine, Merkurjev, Vishik). They turned out to be extremely useful in generating interesting rational cycles in higher codimension (e.g. idempotents or $0$-cycles on twisted flag varieties), hence, in computing various geometric invariants of torsors (incompressibility, canonical dimension, torsion, motivic decomposition type, etc.). In the present talk, we explain how to extend the Landweber-Novikov operations on algebraic cobordism to the setup of equivariant generalized cohomology theories via the localization techniques of Kostant-Kumar. The operations we obtain we call localized operations. These operations can be viewed as operations on global sections of the so called structure sheaves on moment graphs (corresponding to arbitrary Coxeter groups). They satisfy several natural properties, e.g. they commute with characteristic map and restrict to usual Landweber-Novikov operations.