Abstract: We all know what a multiplication in a group is, but what is meant by a comultiplication, the dual of a multiplication? To introduce this concept we generalize groups to group objects in a category and then consider the concept in the associated dual category. We give three important examples: (1) co-H-spaces and cogroups in the homotopy category, (2) free groups in the category of groups, and (3) Hopf algebras in the category of graded algebras. This talk will be largely expository and should be accessible to graduate students.
Sponsor: Larry Gerstein
Abstract: Modular representation theory studies representations of finite groups on vector spaces over a field of positive characteristic p. Contrary to the complex case, "modular" representations are not completely reducible. This makes their classification a daunting and, in fact, for most finite groups an impossible task. I shall start the talk with a quick introduction to the representation theory followed by a "baby" example of a cyclic group Z/p. The failure of complete reducibility is already evident in this case, but representations can still be classified in terms of Jordan canonical forms. Then I'll discuss an approach to the study of modular representations via local Jordan forms which makes an extensive use of the representation theory of Z/p. This will lead, in particular, to various geometric constructions associated to modular representations, such as support varieties and vector bundles. Despite the generality of the theory, it is quite interesting even for products of cyclic groups which I'll use throughout the talk as my "toddler" examples. Based on joint works with E. Friedlander and J. Carlson, and also with D. Benson.
Sponsor: Ken Goodearl
Abstract: Over the last 10 years the question of which manifolds do or do not admit positive curvature has been studied extensively under the presence of a large symmetry group. We will review some of the results, discuss a recent new example of positive curvature, and an obstruction as well.
Sponsor: Guofang Wei
Abstract: Density functional theory, for which Walter Kohn (UCSB physics) shared the 1998 Nobel prize in chemistry, is routinely used in many branches of science to understand and even predict the properties of molecules and materials. In recent years, applied mathematicians have been improving the efficiency of numerical algorithms used in DFT calculations. But the underlying theory itself continues to pose significant challenges in analysis, as recognized in the ground-breaking work of Lieb and others. A recent extension to time-dependent phenomena has become popular, because it allows calculation of electronic excited states, but raises similar challenges that have yet to be addressed. I will discuss these in a style accessible to a general math audience.
Sponsor: Carlos Garcia-Cervera
Abstract: The Hanna Neumann Conjecture (HNC) is an easy-to-formulate question coming from group theory, it asserts a specific bound on the rank of intersections of finitely generated subgroups in free groups. HNC has been open since 1957. I will discuss several points of view on this conjecture, generalizations of its statement, submultiplicativity, relations of HNC to graph theory, geometry, topology, analysis, ring theory. Then I will give a brief outline a recent (one-month-old) proof of HNC and pose some general questions.
Sponsor: Jon McCammond