Thu, October 10, 2019
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October 2019
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9:00am [9:00am] Charu Goel : IIIT Nagpur
CACAAG seminar I. Speaker: Charu Goel. Affiliation: IIIT Nagpur. Date and Time: Thursday 10 October, 9:00 am - 10:00 am. Venue: Ramanujan Hall, Department of Mathematics. Title: The analogue of Hilbert’s 1888 Theorem for even symmetric forms Abstract: Abstract. Hilbert in 1888 studied the inclusion Pn,2d ⊇ Σn,2d, where Pn,2d and Σn,2d are respectively the cones of positive semidefinite forms and sum of squares forms of degree 2d in n variables. He proved that: “Pn,2d = Σn,2d if and only if n = 2,d = 1, or (n,2d) = (3,4)”. In order to establish that Σn,2d Pn,2d for all the remaining pairs, he demonstrated that Σ3,6 P3,6, Σ4,4 P4,4, thus reducing the problem to these two basic cases. In 1976, Choi and Lam considered the same inclusion for symmetric forms and claimed that Hilbert’s characterisation above remains unchanged. They demonstrated that establishing the strict inclusion reduces to show it just for the basic cases (3,6),(n,4)n≥4. In this talk, we will explain the algebraic geometric ideas behind these reductions and how we extended these methods to investigate the above inclusion for even symmetric forms. We will present our leading tool a “Degree Jumping Principle”, an analogue of reduction to basic cases and construction of explicit counterexamples for the basic pairs. As a complete analogue of Hilbert’s theorem for even symmetric forms, we establish that “an even symmetric n-ary 2d-ic psd form is sos if and only if n = 2 or d = 1 or (n,2d) = (n,4)n≥3 or (n,2d) = (3,8)”. This is a joint work with S. Kuhlmann and B. Reznick.

3:00pm [3:30pm] Sudarshan Gurjar: IIT Bombay
Commutative Algebra seminar. Speaker: Sudarshan Gurjar. Affiliation: IIT Bombay. Date and Time: Thursday 10 October, 3:30 pm - 5:00 pm. Venue: Room 215, Department of Mathematics. Title: Introduction to Vector Bundles. Abstract: I will introduce the subject of algebraic vector bundles on projective varieties in this talk. Vector bundles are used in commutative algebra in several contexts. For example, they provide geometric interpretation of tight closure of an ideal. They were used to show that tight closure does not commute with localization. A subtle notion of the semistability of vector bundles plays an important role in this subject. I will try to explain the relevance of this notion and discuss some examples. This talk will be a prequel to a talk by Prof. Nitin Nitsure on October 14.