Date and Time: Monday 07 October, 5:15 pm - 6:15 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Hypocoercivity.
Abstract: The purpose of hypocoercivity is to obtain rates for solutions
of non-purely diffusive equations, in asymptotic regimes. This is a very
useful technique for kinetic equations. After reviewing some easy results
based on hypo-ellipticity, the lecture will focus on linear kinetic
equations without regularizing effects and the L2 hypocoercivity method.
Some motivations will be introduced, with a toy model. The core of the
lecture will be a theoretical result based on a joint work with C. Mouhot
and C. Schmeiser. Initially intended for systems with compactness or
confinement in position space and simple local equilibira, the method has
been extended to various local equilibria in velocities and non-compact
situations in positions. It is also flexible enough to include non-local
transport terms associated with Poisson coupling. Some recent results rely
on various, deep functional inequalities. An application to the linearized
Vlasov-Poisson-Fokker-Planck system will also be briefly presented.
Time:
10:15am-11:15am
Location:
Ramanujan Hall, Department of Mathematics
Description:
Combinatorics seminar
Speaker: Sudeep Stephen.
Affiliation: National University of Singapore.
Date and Time: Wednesday 09 October, 10:15 am - 11:15 am.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Zero Forcing in Graphs.
Abstract: Fo a two-colouring of the vertex set of a simple graph G = (V,E), consider the following colour-change rule: a red vertex is converted to blue if it is the only red neighbour of some blue vertex. A vertex set S ⊆ V is called zero-forcing if, starting with the vertices in S blue and the vertices in the complement V \ S red, all the vertices can be converted to blue by repeatedly applying the colour-change rule. The minimum cardinality of a zero-forcing set for the graph G is called the zero-forcing number of G, denoted by Z(G). This concept was introduced by the AIM Minimum Rank –Special Graphs Work Group in [1] as a tool to bound the minimum rank of matrices associated with the graph G. In this talk, I shall give an overview of the zero forcing problem along with some of the results that we have obtained during my Ph.D candidature. To conclude, I shall state few open problems that I intend to tackle along with my mentors. References [1] AIM Minimum Rank –Special Graphs Work Group. Zero forcing sets and the minimum rank of graphs. Linear Algebra and its Applications, 428(7):16281648, 2008.
Time:
4:00pm-5:00pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
Mathematics Colloquium.
Speaker: Charu Goel.
Affiliation: IIIT Nagpur.
Date and Time: Wednesday 09 October, 4:00 pm - 5:00 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Nonnegative Polynomials and Sums of Squares
Abstract:
Abstract. Sums of squares representations of polynomials is of fundamental importance in real algebraic geometry and goes back to the 1888 seminal paper of Hilbert. The main theorem in his paper is a full characterisation of all pairs (n,d) for which every nonnegative polynomial of a fixed degree d in a given number of variables n is a sum of squares of polynomials. Ninety years later, Choi and Lam asserted that this characterisation remains unchanged for symmetric forms. In this talk first some key observations and problems related to Hilbert’s theorem will be discussed. We then complete the above assertion of Choi-Lam. Along the way, we shall also discuss briefly how test sets for positivity of symmetric polynomials play an important role in establishing this assertion.
Time:
9:00am-10:00am
Location:
Ramanujan Hall, Department of Mathematics
Description:
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.
Time:
3:30pm-5:00pm
Location:
Room No. 215 Department of Mathematics
Description:
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.
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.