Thursday, October 1: Preperation meeting, in this meeting we prepare the semester schedule. Thursday, October 8: Speaker: Ibrahim Saleh, KSU Title: Fomin- Zelevensky Positivity conjecture Abstract: "Fomin and Zelevinsky have conjectured that every cluster variable is a positive element, i.e. it can be written as a Laurent polynomial with variables from any other cluster with positive integers coefficients. The conjecture has been proofed in so many interesting cases. A combinatorial use of the conjecture will be discussed." ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- Thursday, October 15- Time:4:30-5:20 Place: CW 122 Speaker: Ibrahim Saleh, KSU Title: Fomin- Zelevensky Positivity conjecture II Abstract: I will continue my talk from last week, ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Thursday, October 22- Time: 4:30-5:20 pm Place: CW 122 Speaker: Zhaobing Fan, KSU Title: Quiver variety and geometric construction of quantum group Abstract: "Quiver variety is introduced. Geometric construction of quantum group and the integrable highest weight module are given by defining convolution in Borel-Moore homology of the Lagrangian subvariety. This talk is based on Nakajima's paper, Quiver varieties and Kac-Moody algebras, Duke Math. J.91(1998),515-560". --------------------------------------------------------------------------------------------------------- Thursday, October 29- cancled -------------------------------------------------------------------------------------------------------------------------------------------- Thursday, November 5- Place: CW 122 Time:4:30 pm Speaker: Naeem Muhammad Ahmed, KSU Subject: U-,SU-Bordism Theory Abstract: We will continue our summer discussion of oriented cobordism theory. According to Milnor and Novikov, there is a bordism theory corresponding to every stable classical group. For instance, the oriented cobordism theory corresponds to special orthogonal group. As evident from the names, U- and SU-bordism theories correspond to unitary and special unitary groups. A lot of the basic ideas of several bordism theories are analogous, we will start with the general theory of oriented bordism of the pair of spaces and consider it as a model or prototype for the other theories. Contrary to the summer these talks would be more like seminars than lectures. In other words, on the popular demand of summer audience we will normally keep from going into proofs to keep a better track of framework of general theory. We will fill in the details like spectral theory, classifying spaces, certain theory of G-bundles and associated fiber bundles, etc. These talks would be helpful to the students currently taking topology class: the bordism theory is a kind of homology theory, so there would be a chance to see and get better hands on the ideas of homology; As there is an exact sequence of homology modules associated to a subcomplex of a chain complex, the spectral sequence is a sequence of exact sequences of homology modules associated to an increasing sequence of subcomplexes of a chain complex. It is clearly a tool to compute homology, which can also be used to study homotopy groups and cohomology. Our first talk would be just a warm up: We will start with a brief map of what we did over the summer- pretty much the Thom bordism theory. Then we will define a covariant functor called the oriented bordism functor on the pair of spaces and maps between them. We will talk about Eilenberg-Steenrod axioms of generalized homology theory and probably their consequences, depending upon the time. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- Thursday, November 12- Thursday, November 19- Thursday, November 26- Thrusday, December 4- |