Fri, October 11, 2019
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4:00pm [4:30pm] Mrinal Kumar : Computer Science Department, IIT Bombay
CACAAG seminar II. Speaker: Mrinal Kumar. Affiliation: Computer Science Department, IIT Bombay. Date and Time: Friday 11 October, 4:30 pm - 5:30 pm. Venue: Ramanujan Hall, Department of Mathematics. Title: Some applications of the Polynomial Method in Combinatorics. Abstract: In the next couple of lectures, we will see some applications of the so called Polynomial Method to problems in Combinatorics. We will focus on the following three applications: 1. Joints Problem: For a set L of lines in R^3, a point p in R^3 is said to be a joint in L if there are at least three non-coplanar lines in L which pass through p. We will discuss a result of Guth and Katz who showed an upper bound on the maximal number of joints in an arrangement of N lines. 2. Lower bounds on the size of Kakeya sets over finite fields: For a finite field F, a Kakeya set is a subset of F^n that contains a line in every direction. We will discuss a result of Dvir showing a lower bound of C_n*q^n on the size of any Kakeya set over F^n, where C_n only depends on n and F is a finite field of size q. 3. Upper bounds on the size of 3-AP free sets over finite fields: We will then move on to discuss a recent result of Ellenberg and Gijswijt who showed that if F is a finite field with three elements, and S is a subset of of F^n such that S does not that does not contain three elements in an arithmetic progression, then |S| is upper bounded by c^n for a constant c < 3.