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Monday, February 26, 2018
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8:00am [11:30am]G. Arunkumar Description: Speaker: G. Arunkumar Date & Time : Monday Feb 26, at 11:30am Venue: Ramanujan Hall Title: Chromatic polynomials and Lie algebras Abstract: In this talk, I will prove a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. I will show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. As an application, using the combinatorics of Lyndon words, we construct a basis for the root spaces corresponding to these roots and determine the Hilbert series in the case when all simple roots are imaginary. In last ten minutes, We will talk about chromatic discriminant of a graph: The absolute value of the coefficient of q in the chromatic polynomial of a graph G is known as the chromatic discriminant of G and is denoted $\alpha(G)$. We start with a brief survey on many interesting algebraic and combinatorial interpretations of $\alpha(G)$. We use two of these interpretations (in terms of acyclic orientations and spanning trees) to give two bijective proofs for a recurrence formula of $\alpha(G)$ which comes from the Peterson recurrence formula for root multiplicities of Kac-Moody algebras. [2:00pm]Ronnie Sebastian Description: CACAAG (Combinatorial Aspects of Commutative Algebra and Algebraic Geometry) seminar Speaker: Ronnie Sebastian Date & Time : 26th February, 2pm Venue : Ramanujan Hall Abstract: This talk will be based on the following elementary and nice exposition http://www.math.stonybrook.edu/~roblaz/Reprints/Green.Laz.Simple.Pf.Petri.pdf Using some simple facts about projective space, cohomology, cohomology of line bundles on projective space, we shall prove the following theorems: 1. Noether's theorem - Projective normality of the canonical embedding of non-hyperelliptic curves. 2. Petri's -theorem - Let X be a smooth and projective curve of genus g \geq 5. Assume that X carries a line bundle A of degree g-1 with h^0(A)=2. Further assume that both A and \Omega_X\otimes A^* are generated by their global sections. Then the homogeneous ideal of X in its canonical embedding is generated by degree 2 elements.