Thu, March 16, 2017
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8:00am [2:10pm] Niranjan Balachandran 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)}). [3:30pm] S.G. Dani 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.