- Time:
- 11:00am - 12:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: The Capset bound of Croot-Lev-Pach and Ellenberg-Gijswijt

Abstract: A construction of Behrend from the 1940s shows that there are subsets of [N] of size N^{1-o(1)} that contain no 3-term APs (also called capsets). For a long time, it was open whether there is such a construction over F_3^n (i.e. a capset in F_3^n of size 3^{n-o(n)}). Recently, building on work of Croot, Lev and Pach, it was shown by Ellenberg and Gijswijit ( https://arxiv.org/abs/1605.09223 ) that such a construction does not exist: i.e. any capset in F_3^n can have size at most c^n for some c < 3. The construction has had several applications already in Combinatorics and Theoretical Computer Science. We will see a proof of the theorem of Ellenberg and Gijswijt

- Time:
- 11:30am
- Location:
- Room No. 216
- Description:
- Title. Rational Surface Singularities.

Title. We will prove a purely numerical criterion due to M. Artin to test

the rationality of a surface singularity. In practice this is the

criterion which is used when a rational surface singularity is being

considered.

- Time:
- 2:30pm - 3:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: Existence and regularity theory in weighted Sobolev spaces and

applications.

Abstract: My emphasis in this talk will be on functional analytical tools to

the solvability and uniqueness of solutions to the nonhomogeneous boundary

value problems, dealing with degenerate PDEs of elliptic type. My aim is to

consider possibly general class of weights. In particular, I consider the

$B_{p}$-class of weights, introduced by Kufner and Opic, which is much more

general class than the commonly studied Muckenhoupt $A_{p}$-class.

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Generalized Hamming weights of (projective) Reed-Muller codes.

Abstract: Reed-Muller codes are among the most elementary and most studied codes. Less studied, but equally elementary are their projective counterparts, the protective Reed-Muller codes. Many open questions remain about these codes. Mathematically, a very interesting question is the determination of the generalized Hamming weights. The determination of these weights is equivalent to the determination of the maximum number of common solutions to certain system of polynomial equations. In this talk, I will give an overview of recent work and developments on the theory of generalized Hamming weights of projective Reed-Muller codes. This work was carried out together with Mrinmoy Datta and Sudhir Ghorpade.

- Time:
- 3:30pm - 5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Koszul Algebras IV

- Time:
- 3:30pm
- Location:
- Ramanujan Hall
- Description:
- Speaker: Professor Vydas Cekanavicius

Vilnius University

Lithuania

Title: Infinitely Divisible Approximations for Sums of Markov-Dependent RVs.

Abstract:

We demonstrate that for discrete Markov dependent rvs, the normal approximation can be effectively replaced by compound Poisson approximation..In case of three

state Markov chain, the effect of symmetry will be estimated.

- Time:
- 2:30pm - 3:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: Jimm, a fundamental involution

Abstract: Dyer's outer automorphism of PGL(2,Z) induces an involution

of the real line, which behaves very much like a kind of modular

function. It has some striking properties: it preserves the set of

quadratic irrationals sending them to each other in a non-trivial way

and commutes with the Galois action on this set. It restricts to an

highly non-trivial involution of the set unit of norm +1 of quadratic

number fields. It conjugates the Gauss continued fraction map to the

so-called Fibonacci map. It preserves harmonic pairs of numbers

inducing a duality of Beatty partitions of N. It induces a subtle

symmetry of Lebesgue's measure on the unit interval.

On the other hand, it has jump discontinuities at rationals though its

derivative exists almost everywhere and vanishes almost everywhere. In

the talk, I plan to show how this involution arises from a special

automorphism of the infinite trivalent tree

- Time:
- 4:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Tropical Algebraic Geometry: an Introduction.

Tropical algebraic geometry is in the interface of algebraic and polyhedral geometry with applications to both these topics. We start with a gentle introduction to tropical algebraic geometry. We then focus on the tropical lifting problem and discuss recent progress. Tropical analogues of graph curves play an important role in this study.

Please note:

1. Dr. Manjunath is a faculty candidate.

2. The talk will be via skype

- Time:
- 11:00am - 12:30pm
- Location:
- Room No. 216
- Description:
- Title: Rational Singularities VI

- Time:
- 11:00am - 12:00pm
- Location:
- Ramanujan Hall
- Description:

Title: Recent developments on the Sunflower conjecture

Abstract: A sunflower with p petals is a family of sets A_1,...,A_p

such that the intersections of all pairs of distinct sets are the

same. A famous conjecture in combinatorics, called the Sunflower

conjecture, asserts a bound on the maximum size of any family of

k-sets that does not contain a p-sunflower. We review some recent work

by Ellenberg-Gijswijt and Naslund-Swain that proves a weak variant of

this conjecture due to Erdos and Szemeredi.

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall
- Description:
- Speaker: Willem H. Haemers

Tilburg University, The Netherlands

Title: Are almost all graphs determined by their spectrum?

Abstract: An important class of problems in mathematics deals with the reconstruction of a

structure from the eigenvalues of an associated matrix. The most famous such prob-

lem is: ‘Can one hear the shape of a drum?’. Here we deal with the question: ‘Which

graphs are determined by the spectrum (eigenvalues) of its adjacency matrix’? More

in particular we ask ourselves whether this is the case for almost all graphs. There

is no consensus on what the answer should be, although there is a growing number

of experts that expect it to be affirmative. In this talk we will present several re-

sults related to this question. This includes constructions of cospectral graphs and

characterizations of graphs by their spectrum. Some of these results support an

affirmative answer, some support the contrary. It will be explained why the speaker

believes that it is true.

- Time:
- 5:00pm - 6:30pm
- Location:
- Room No. 215
- Description:
- Title: Some consequences of the Riemann hypothesis for varieties over

finite fields - II

Abstract: We will talk about a result of M. Katz and W. Messing, which

says the following. From the Riemann hypothesis and the hard Lefschetz

theorem in l-adic cohomology, the corresponding facts for any Weil

cohomology follow.

- Time:
- 3:00pm - 5:00pm
- Location:
- Room No. 215
- Description:
- Title: Finiteness of homotopy groups of spheres

Abstract:

In this talk I will prove that the i-th homotopy groups of a sphere S^n are finite when i is greater than n, except in one particular case, using the Serre spectral sequence. In the first half of the talk I will give the background material needed to understand the proof.

- Time:
- 3:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: Ideals of Linear Type I

Abstract: In this talk, we study the basics of defining ideal of the Rees algebra of Ideal I and what makes the ideal to be of linear type. Further, we prove that ideals generated by a regular sequences are of linear type.

- Time:
- 2:30pm - 3:30pm
- Location:
- Ramanujan Hall
- Description:
- Title : Mixture Designs - a Review

Abstract : We introduce standard mixture models and standard mixture designs as are well-known in the literature. Some of the less known models are also introduced briefly. Next we mention about known applications of mixture experiments in agriculture, food processing and pharmaceutical studies. Then we describe the framework of exact and approximate [or, continuous] mixture designs. A broad class of research problems posed and discussed in the published literature is presented with appropriate references.

- Time:
- 3:40pm - 4:40pm
- Location:
- Ramanujan Hall
- Description:
- Speaker : Dr. Avinash Dharmadhikari, Quality Systems and Reliability, Engineering Research Centre, Tata Motors, Pune

Title: Prediction of a Warranty Cost For a Two Dimensional Policy

Abstract: Attached.

- Time:
- 11:00am - 12:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: An Extremal Problem in the study of Zero-Sum Problems

Abstract: attached.

- Time:
- 3:00pm - 5:00pm
- Location:
- Room No. 215
- Description:
- Title: Groups of Homotopy spheres

Abstract of the first talk on `Groups of homotopy spheres':

In a land-mark paper in 1956, J. Milnor showed that there are non standard differential structures on the 7-dimensional sphere. Six years later along with Kervaire, he introduced an abelian group structure on the set of equivalence classes of smooth structures on spheres of all dimension and determined these groups in several cases. We shall present some of the salient features of this work.

- Time:
- 3:30pm - 5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Ideals of linear type 2

- Time:
- 3:30pm - 5:30pm
- Location:
- Room 114
- Description:
- Title: Deligne's conjectures on critical values of L-functions

Abstract: We will explain how to attach an L-function to a motive, what

the critical points of this L-function are, and Deligne's conjectures on

the values of the L-function at critical points.

- Time:
- 2:30pm - 4:00pm
- Location:
- Room No. 215
- Description:
- h-Cobordism Thorem

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall
- Description:
- Speaker: Kathleen Shannon, Salisbury University.

Title: Pascal's Triangle, Cellular Automata and Serendipity: A Mathematical Tale

Abstract: The talk will outline the development of the PascGalois Project. Its origins are in an exercise using Pascal's Triangle and modular arithmetic. Colors are assigned to the numbers 0, 1, ..., n-1, and Pascal's Triangle modulo n is drawn. The patterns in the triangle are then related to the properties of the cyclic group Zn. The process of drawing the triangles is then generalized to non-cyclic and non-abelian groups and the new patterns are examined in light of the properties of these groups. The images can help develop visual and intuitive understanding of concepts such as subgroup closure and quotient groups. They can also be used to discuss the relationship between mathematical properties and visual aesthetics. Finally we view Pascal's Triangle as a one-dimensional cellular automata and generalize to more general initial conditions and two dimensional automata. Many of the investigations in this project have been undertaken with students in undergraduate research projects and one outgrowth of the project has been the development of a set of visualization exercises to supplement the standard undergraduate course in abstract algebra. The web site for the project is at www.pascgalois.org.

- Time:
- 3:30pm - 5:00pm
- Location:
- Room No. 215
- Description:
- Title: Values of binary quadratic forms on integer pairs

- Time:
- 3:00pm - 5:00pm
- Location:
- Room 215
- Description:
- Groups of homotopy spheres

In a land-mark paper in 1956, J. Milnor showed that there are non standard differential structures on the 7-dimensional sphere. Six years later along with Kervaire, he introduced an abelian group structure on the set of equivalence classes of smooth structures on spheres of all dimension and determined these groups in several cases. We shall present some of the salient features of this work.

This is the second talk on this topic.

- Time:
- 3:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: Counting Zeros of Multivariate Laurent Polynomials and Mixed Volumes of Polytopes

Abstract. A result of D.N. Bernstein proved in the late seventies gives an upper bound

on the number of common solutions of n multivariate Laurent polynomials in

n indeterminates in terms of the mixed volumes of their Newton polytopes.

This bound refines the classical Bezout's bound. Bernstein's Theorem has several

proofs using techniques from numerical analysis, intersection theory and tori varieties.

B. Teissier proved the theorem using intersection theory. A proof using theory of toric

varieties can be found in the book by W. Fulton on the same subject.

In this talk, I will outline an algebraic proof similar to the standard proof of Bezout's Theorem.

This proof, found in collaboration with N.V. Trung, uses basic results about Hilbert functions

of multigraded algebras first proved by van der Waerden.

- Time:
- 3:30pm - 5:00pm
- Location:
- Room No. 215
- Description:
- Title: Values of binary quadratic forms

- Time:
- 3:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: Symbolic Rees Agebra of certain monomial curves