Title: Bisecting and D-secting families for hypergraphs
Abstract: Let n be any positive integer, [n]:={1,2,...,n}, and suppose
$D\subset\{-n,-n+1,..,-1,0,1,...,n}$. Let F be a family of
subsets of [n]. A family F' of subsets of [n] is said to be
D-secting for F if for every A in the family F, there exists a subset A'
in F' such that $|A\cap A'|-|A\cap ([n]\setminus A')| = i$, for some $i\in
D$. A D-secting family F' of F, where D = {-1,0,1}, is a bisecting family
ensuring the existence of a subset $A'\in F'$ such that $|A\cap
A'|\in{\lfloor |A|/2\rfloor, \lceil |A|/2\rceil\}$ for each $A\in F$. We
consider the problem of determining minimal D-secting families F' for
certain families F and some related questions.
This is based on joint work with Rogers Mathew, Tapas Mishra, and
Sudebkumar Prashant Pal.
Time:
11:00am-11:50am
Location:
Ramanujan Hall
Description:
Time 11.00 AM -11.45 AM
Title: Ramanujan's Master theorem for radial sections of line bundles over the real hyperbolic space
Abstract: Ramanujan's master theorem states that under suitable
conditions, the Mellin transfrom of an alternating power series provides
an interpolation formula for the coefficients of this power series.
Ramanujan applied this theorem to compute several definite integrals and
power series and this explains why it is referred as "Master Theorem". In
this talk we shall try to explain its analogue for radial sections of line bundles over the real hyperbolic space.
This a joint work (in progress) with Prof. Swagato K Ray.
Time:
12:00pm-12:50pm
Location:
Ramanujan Hall
Description:
Time 12.00 noon -12.45 PM
Title: Riemann-Roch, Alexander Duality and Free Resolutions.
Abstract: The Riemann-Roch theorem is fundamental to algebraic geometry. In 2006, Baker and Norine discovered an analogue of the Riemann-Roch theorem for graphs. This theorem is not a mere analogue but has concrete relations with its algebro-geometric counterpart. Since its conception this topic has been explored in different directions, two significant directions are i. Connections to topics in discrete geometry and commutative algebra ii. As a tool to studying linear series on algebraic curves. We will provide a glimpse of these developments. Topics in commutative algebra such as Alexander duality and minimal free resolutions will make an appearance. This talk is based on my dissertation and joint work with i. Bernd Sturmfels, ii. Frank-Olaf Schreyer and John Wilmes and iii. an ongoing work with Alex Fink.
Time:
2:10pm-3:10pm
Location:
Ramanujan Hall
Description:
Time 2.15-3.15
Title : Labeling the complete bipartite graphs with no simple zero cycles
Abstract : Suppose we want to label the edges of the complete bipartite graph K_{n,n} with elements of F_2^d in such a way that the sum of labels over any simple cycle is nonzero. What is the smallest possible value of d be for such a labeling to exist?
It was proved by Gopalan et. al. that log^2(n) \leq d \leq nlog(n). Kane, Lovett and Rao recently proved that d is in fact linear in n. In particular we have n/2-2 \leq d < 6n.
Upper bound is established by explicit construction while lower bound is obtained by bounding the size of independent sets in certain Cayley graphs of S_n.
Time:
4:00pm-5:00pm
Location:
Ramanujan Hall
Description:
Speaker: Prof. Arup Bose.
Title: Large sample behaviour of high dimensional autocovariance matrices with application
Title: Compact forms of spaces of constant negative (sectional) curvature.
Abstract: One knows that any compact riemann surface of genus > 2 carries
a riemanniann metric of constant curvature. In higher dimension even the
existence of compact manifolds of constant negative curvature is by no
means that abundant. In this lecture we will show how arithmetic enables us
to construct such manifolds in every dimension greater than equal to 2.
Time:
3:30pm-5:00pm
Location:
Ramanujan Hall
Description:
Title: The Hoskin-Deligne formula for the co-length of a complete ideal in
2-dimensional regular local ring.
Abstract: We shall present a simple proof due to Vijay Kodiyalam.
This proof makes use of the fact that transform of a complete ideal
in a two-dimensional regular local ring R in a quadratic transform of R
is again complete. It also uses a structure theorem, due to Abhyankar,
of two-dimensional regular local rings birationally dominating R.