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.