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Program Title: Advances in High Dimensional Statistical Learning Conference
Date: 15-16 Dec 2024
Venue: Ramanujan Hall, Department of Mathematics
Program Brochure: Attached
Program Webpage: Under Construction
Thesis Defence
Speaker: PMS Sai Krishna (IIT Bombay)
Title: Exponential maps and their applications
Time, day and date: 11:30:00 AM - 12:39:00 PM, Monday, December 16
Venue: Room 216
Abstract
Exponential maps of k-domains generalize locally nilpotent derivations, and they coincide when k is of zero characteristic. They have proven to be a useful tool in approaching problems in Affine Algebraic Geometry. In the first part of the talk, we will look at some results related to locally nilpotent derivations. Next, we introduce exponential maps and define some related invariants, using which we give an algebraic characterization of the affine plane and affine three-space. The second part of the talk will be about some results related to the rigidity and triangularity of exponential maps. The last part of the talk will be about the rigidity problem, which is about the existence of a non-trivial exponential map of a k-domain. We will see some results related to the rigidity of the ring of invariants of an exponential map, and we provide a sufficient condition under which the ring of invariants of an exponential map of k^[3] is k^[2].
Title: Oddtowns, partial ovoids, and the rank-Ramsey problem
Speaker: Prof. Anurag Bishnoi, TU Delft, Netherlands
[https://anuragbishnoi.wordpress.com/]
Abstract :
What is the largest size of a family of subsets of {1, ..., m} such that
every set in the family has odd cardinality and among any three distinct
sets there is at least one pair that intersects in an even number of
elements? We'll show that this generalisation of the classic Oddtown
problem is equivalent to finding the largest size of a partial 2-ovoid in a
binary symplectic space and that of finding the largest set of nearly
orthogonal vectors in F_2^m. Moreover, we show that it's intimately linked
to the following recently introduced rank-Ramsey problem: for a given m,
find the largest n for which there exists a triangle-free graph with n
vertices whose adjacency matrix A satisfies rank(A+I) <= m. If we
consider the rank over the binary field, then the problem is equivalent to
generalized oddtowns, while over other fields it has a different behaviour.
We give new constructions of these objects, improving the state of art,
using a triangle-free Cayley graph associated with BCH codes. Moreover, by
using binary projective caps, that is, sum-free sets in F_2^n, we improve
the best construction for this rank-Ramsey problem over the reals.
Joint work with John Bamberg and Ferdinand Ihringer.