Tue 2014-02-11, 15:30. Room: LH6 Speaker: Subramani M Title: Z[\sqrt{14}] is Euclidean. Abstract: Let K be a real quadratic number field and O[K] be the ring of algebraic integers. We prove that if O[K] is a P.I.D in K and contains a set of two "admissible" primes then O[K] is Euclidean and deduce that Z[\sqrt{14}] is Euclidean. This result is from Malcolm Harper: Z[\sqrt{14}] is Euclidean, Canad. J. Math., 2004 ***** Tue 2014-02-18, 15:30. Room: LH6 Speaker: Sudipta Kolay Title: Generalized Schoenflies theorem Abstract: The Jordan Curve Theorem states that any simple closed curve decomposes the 2-sphere (or equivalently the Euclidean plane) into two connected components and is their common boundary. There is a strengthening of the Jordan curve theorem, called the Schoenflies theorem, which states that the closure of either connected component in the 2 sphere is a 2-cell. While the first statement is true in higher dimensions, the latter is not. However, under the additional hypothesis that the n-1 sphere is embedded in the n sphere in a locally flat way, then the closure of either connected component is an n-cell. This result is called the Generalized Schoenflies theorem, and was proved independently by Morton Brown and Barry Mazur. In this talk, I will describe the proof of due to Morton Brown. ***** Tue 2014-02-25, 15:30. Room: LH6 Midterm week. No talk ***** Tue 2014-03-04, 15:30. Room: LH6 Speaker: Aneesh Karthik Title: Enumerative Combinatorics and the Secret of Chambers Abstract: Partitioning of Euclidean and projective spaces is a source of many interesting enumerative problems in combinatorial geometry. In particular, it deals with determining the number of pieces into which a certain geometric set is divided by given subsets. In this talk I shall address the counting problem in three contexts, namely, the Euclidean plane, the projective plane and the 2-torus. I will start with an introduction to arrangements of lines in a plane and talk about some of their enumerative aspects. Then I will focus on the class of projective arrangements and mention some hot questions in this area. Finally I will describe the current joint work with Priyavrat Deshpande regarding arrangements of toric lines. ***** Tue 2014-03-11, 15:30. Room: LH6 No Talk ***** Tue 2014-03-18, 15:30. Room: LH6 Speaker: Kuldeep Saha Title: Lickorish-Wallace theorem Abstract: The aim of this talk will be to prove the Lickorish-Wallace theorem. We shall define Heegaard splitting and other necessary constructions for a closed 3-manifold and try to discuss some examples. Then we shall prove the main theorem and discuss some of its consequences. ***** Tue 2014-03-25, 15:30. Room: LH6 Speaker: Mitra Koley Title: Representation of Temperley-Lieb algebra. Abstract: We will first define Temperley-Lieb diagram followed by Temperley-Lieb algebra by generators and relations. Then we will classify all irreducible representation of semisimple Temperley-Lieb algebra. Finally we will see some applications of Temperley-Lieb algebra. ***** Tue 2014-04-01, 15:30. Room: LH6 Speaker: Narasimha Chary Title: Introduction to Schubert varieties. Abstract: Let Gr(r,n) be the Grassmannian. We define Schubert varieties in Gr(r,n). Let H^*(Gr(r,n), Z) be the cohomology ring of Gr(r,n), [X] be the Schubert class corresponding to the Schubert variety X. We will prove ( Basis Theorem): Considered additively, H^*(Gr(r,n), Z) is free Abelian group and the Schubert classes [X] forms a basis. We discuss the structure of this cohomology ring H^*(Gr(r,n), Z) namely, the Giambelli's determinantal formula and Pieri's formula. Using this we give solution to a enumerative problem. Finally, we define Schubert varieties for general reductive group. ***** Tue 2014-04-08, 15:30. Room: LH6 Speaker: Samir Kumar Hazra Title: Invariant rings and the Noether bound. Abstract: We will discuss basic properties of invariant rings of finite groups. We will give a proof of the Noether bound for degrees of generators of an invariant ring. We will also discuss some examples in the modular case. ***** Tue 2014-04-15, 15:30. Room: LH6 Speaker: Vinay Kumaraswamy Title: Bounded gaps between primes Abstract: In this talk I will sketch a proof of the following theorem due to Yitang Zhang: Lim inf n\to \infty p_{n+1} - p_n \leq 7,000,000, where p_n is the nth prime number. ***** Tue 2014-04-23, 15:30. Room: LH6 (Unusual day) Speaker: Sayantan Roy Title: Riemann-Roch Theorem. Abstract: We will define divisors and canonical divisors on X(a non singular model of an irreducible projective curve C followed by the vector space L(D) and genus of the curve. Then we will prove for any divisor D, l(D)=deg(D)+1-g+l(W-D)where l(D) is the dimension of the vector space L(D),g is the genus of the curve,deg(D) is the degree of the divisor D and W is a canonical divisor on X.