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.