Description

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.

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.

Description

Ramanujan Hall, Department of Mathematics

Date

Fri, October 11, 2019

Start Time

4:30pm-5:30pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

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Updated

Wed, October 9, 2019 10:06am IST