Fri, February 24, 2017
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3:00pm [3:00pm] Prof. A. R Shastri
Groups of homotopy spheres In a land-mark paper in 1956, J. Milnor showed that there are non standard differential structures on the 7-dimensional sphere. Six years later along with Kervaire, he introduced an abelian group structure on the set of equivalence classes of smooth structures on spheres of all dimension and determined these groups in several cases. We shall present some of the salient features of this work. This is the second talk on this topic.

[3:30pm] J. K. Verma
Title: Counting Zeros of Multivariate Laurent Polynomials and Mixed Volumes of Polytopes Abstract. A result of D.N. Bernstein proved in the late seventies gives an upper bound on the number of common solutions of n multivariate Laurent polynomials in n indeterminates in terms of the mixed volumes of their Newton polytopes. This bound refines the classical Bezout's bound. Bernstein's Theorem has several proofs using techniques from numerical analysis, intersection theory and tori varieties. B. Teissier proved the theorem using intersection theory. A proof using theory of toric varieties can be found in the book by W. Fulton on the same subject. In this talk, I will outline an algebraic proof similar to the standard proof of Bezout's Theorem. This proof, found in collaboration with N.V. Trung, uses basic results about Hilbert functions of multigraded algebras first proved by van der Waerden.