March 2017
Public Access Category: All |

- Time:
- 3:00pm - 5:00pm
- Location:
- Room No. 215
- Description:
- Title: Borel-Weil-Bott theorem

Abstract: In this talk we shall summarize results on the structure and

representation theory of semisimple algebraic groups. This will to prepare

ground for the subsequent talks on the Borel-Weil-Bott theorem which

explains how (loosely speaking) representations of semisimple algebraic

groups can be obtained as sheaf cohomology groups associated to certain

line bundles.

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

- Time:
- 11:30am - 1:00pm
- Location:
- Room 215
- Description:
- Title: Classical Motives

Abstract: We will give some basic definitions and take a few examples

of motives. The reference is A. J. Scholl's article (1991, Seattle).

- Time:
- 11:30am - 12:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: On tricolored-sum-free sets and Green's Boolean Removal Lemma

Abstract: A tricolored-sum-free set in F_2^n is a collection of triples

{(a_i,b_i,c_i)}_{I=1..m} such that

a) for each I, a_i+b_i+c_i=0

b) If a_i+b_j+c_k = 0, then I=j=k.

The notion of a tricoloured-sum-free set generalizes the notion of a

capset to F_2^n. The basic question here is: How large can a

tricolored-sum-free set be?

We will see the following two (recent) results.

i) Kleinberg's upper bound of 6\binom{n}{n/3} for a tricolored-sum-free

set. This in conjunction with a previous result of his establishing a

lower bound of \binom{n}{n/3}2^{-\sqrt{16n/3}} gives almost asymptotically

tight results.

ii) Ben Green (in 2005) proved the following BOOLEAN REMOVAL LEMMA:

Given \epsilon>0 there exists \delta depending only on epsilon such

that the

following holds: Write N=2^n. If X,Y,Z are subsets of F_2^n if by deleting

\epsilon N elements from X,Y, Z altogether, one can eliminate all

arithmetic triangles (triples (x,y,z) with \in X,y\in Y,z\in Z such that

x+y+z=0) then there are at most \delta N^2 arithmetic triangles in

(X,Y,Z). Green's proof establishes a bound for1/(\delta) which is a tower

of 2s of length poly(1/\epsilon). We will look at a recent result of Fox

and Lovasz (junior) who obtained an almost tight bound for this

delta-epsilon dependence with \delta =O(\epsilon^{O(1)}).

- Time:
- 3:30pm - 5:00pm
- Location:
- Room 215
- Description:
- Title: Flows on homogeneous spaces

Abstract: We shall discuss the results of Marina Ratner on unipotent flows, and the techniques involved.

- Time:
- 10:30am
- Location:
- Ramanujan Hall
- Description:
- Title: The Robinson-Schensted-Knuth Algorithm for Real Matrices

Abstract: The Robinson-Schensted-Knuth (RSK) correspondence is a bijection

from the set of matrices with non-negative integer entries onto the set of

pairs of semistandard Young tableaux (SSYT) of the same shape. SSYT can be

expressed as integral Gelfand-Tsetlin patterns. We will show how Viennot's

light-and-shadows algorithm for computing the RSK correspondence can be

extended from matrices with non-negative integer entries to matrices with

non-negative real entries, giving rise to real Gelfand-Tsetlin patterns.

This real version of the RSK correspondence is piecewise-linear. Indeed,

interesting combinatorial problems count lattice points in polyhedra, and

interesting bijections are induced by volume-preserving piecewise-linear

maps.

- Time:
- 3:30pm - 5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Koszul algebras V

Abstract: In the first half of the talk, we shall recall Koszul filtation

and Grobner flag. Let R be a standard graded algebra. If R has a Koszul

filtation, then R is Koszul. If R has a Grobner flag, then R is

G-quadratic. I will mention an important result of Conca, Rossi, and

Valla: Let R be a quadratic Gorenstein algebra with Hilbert series 1 + nz

+ nz^2 + n^3. Then for n=3 and n=4, R is Koszul.

In the second half of the talk, we shall focus on class of strongly Koszul

algebras. If time permits, I will prove that Koszul algebras are preserved

under various classical constructions, in particular, under taking tensor

products, Segre products, fibre products and Veronese subrings.

- Time:
- 11:00am
- Location:
- Ramanujan Hall
- Description:
- Title : Homotopy groups of highly connected manifolds

Abstract : We shall discuss a new method of computing (integral) homotopy

groups of certain manifolds in terms of the homotopy groups of spheres. The

techniques used in this computation also yield formulae for homotopy groups

of connected sums of sphere products and CW complexes of a similar type. In

all the families of spaces considered here, we verify a conjecture of J. C.

Moore. This is joint work with Somnath Basu.

- Time:
- 2:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: On perfect classification for Gaussian processes

Abstract: We study the problem of discriminating Gaussian processes by analyzing the behavior of the underlying probability measures in an infinite-dimensional space. Motivated by singularity of a certain class of Gaussian measures, we first propose a data based transformation for the training data. For a J class classification problem, this transformation induces complete separation among the associated Gaussian processes. The misclassification probability of a component-wise classifier when applied on this transformed data asymptotically converges to zero. In finite samples, the empirical classifier is constructed and related theoretical properties are studied.

This is a joint work with Juan A. Cuesta-Albertos.

- Time:
- 2:10pm
- Location:
- Ramanujan Hall
- Description:
- Time: 2:15-3:15

Title: On tricolored-sum-free sets and Green's Boolean Removal Lemma

(Part II, continued from last week)

Abstract: A tricolored-sum-free set in F_2^n is a collection of triples

{(a_i,b_i,c_i)}_{I=1..m} such that

a) for each I, a_i+b_i+c_i=0

b) If a_i+b_j+c_k = 0, then I=j=k.

The notion of a tricoloured-sum-free set generalizes the notion of a

capset to F_2^n. The basic question here is: How large can a

tricolored-sum-free set be?

We will see the following two (recent) results.

i) Kleinberg's upper bound of 6\binom{n}{n/3} for a tricolored-sum-free

set. This in conjunction with a previous result of his establishing a

lower bound of \binom{n}{n/3}2^{-\sqrt{16n/3}} gives almost asymptotically

tight results.

ii) Ben Green (in 2005) proved the following BOOLEAN REMOVAL LEMMA:

Given \epsilon>0 there exists \delta depending only on epsilon such

that the

following holds: Write N=2^n. If X,Y,Z are subsets of F_2^n if by deleting

\epsilon N elements from X,Y, Z altogether, one can eliminate all

arithmetic triangles (triples (x,y,z) with \in X,y\in Y,z\in Z such that

x+y+z=0) then there are at most \delta N^2 arithmetic triangles in

(X,Y,Z). Green's proof establishes a bound for1/(\delta) which is a tower

of 2s of length poly(1/\epsilon). We will look at a recent result of Fox

and Lovasz (junior) who obtained an almost tight bound for this

delta-epsilon dependence with \delta =O(\epsilon^{O(1)}).

- Time:
- 3:30pm - 5:00pm
- Location:
- Room 215
- Description:
- Title: Flows on homogeneous spaces

Abstract: We shall continue the discussion on the results of Marina Ratner on unipotent flows, and the techniques involved.

- Time:
- 2:10pm - 3:10pm
- Location:
- Ramanujan Hall
- Description:
- Time: 2.15-3.15

Title: Error Correction and List Decoding for Reed Solomon Codes

Abstract:

In this talk, we will have a look at three results, starting with the following.

[Berlekamp-Welch] Given a univariate polynomial function over F_q

with data corruption at t < q/2 points, we can recover the function

completely if the degree of the function is sufficiently low.

A generalization of the above is as follows, where instead of

'recovering' the function, we find all its 'close approximates'.

[Madhu Sudan] Given data points (x_i,y_i), i \in [n], and parameters

k and t, we can list all polynomials with degree at most k, which

satisfy at least t data points.

This result can further be generalized as follows.

[Madhu Sudan] Given data points (x_i,y_i) with weights w_i, i \in

[n], and parameters k and W, we can list all polynomials with degree

at most k such that the sum of weights of data points satisfied by the

polynomial is at least W.

The last two results provide list-decoding of Reed-Solomon codes.

- Time:
- 4:00pm - 5:00pm
- Location:
- Room 113
- Description:
- Title: Asymptotics of the number of points of symplectic lattices in subsets of Euclidean spaces

Abstract: It is well known that a "good" large subset of the Euclidean space contains approximately as many lattice points as its volume. This need not hold for a general subset. On the other hand, a classical theorem of Siegel asserts that for any subset of positive measure, the "average" number of points (in an appropriate sense) of a general unimodular lattice contained in it, equals the measure of the set. In place of the average over the entire space of lattices one may also ask for analogous results for smaller subclasses. In a recent work with Jayadev Athreya, we explored this issue, with some modifications that place the problem in perspective, for the case of symplectic lattices, viz. lattices (in even-dimensional spaces) obtained from the standard lattice under symplectic transformations. In this talk I shall describe the overall asymptotics in this case, together with the historical background of the results and techniques involved.

- Time:
- 3:30pm - 5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: A new proof of Zariski's Theorem about Complete ideals in two-dimensional regular local rings.

Abstract: Zariski's first paper in algebra written in 1938 proved among many other results that product

of complete ideals is complete in the polynomial ring $K[X,Y]$ where $K$ is an algebraically

closed field of characteristic zero. This was generalised to two-dimensional regular local rings

in Appendix 5 of Zariski-Samuel's classic "Commutative Algebra". We will present a new proof

of this theorem using a formula of Hoskin-Deligne about co-length of a zero-dimensional

complete ideal in a two-dimensional regular local ring in terms of quadratic transforms of

$R$ birationally dominating $R.$

- Time:
- 10:10am - 11:00am
- Location:
- Ramanujan Hall
- Description:
- Time 10:15-11:00

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

Abstract: http://www.math.iitb.ac.in/~seminar/colloquium/arup-bose-30-march-17.pdf

- Time:
- 3:00pm
- Location:
- Room 215
- Description:
- 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.