Series
School of Mathematics Colloquium
School of Mathematics Colloquium
Permanent Link
Series Type
Event Series
Description
Associated Organization(s)
Associated Organization(s)
Publication Search Results
Now showing
1 - 6 of 6
-
ItemDetecting gerrymandering with mathematical rigor( 2020-02-06) Pegden, WesleyIn recent years political parties have more and more expertly crafted political districtings to favor one side or another, while at the same time, entirely new techniques to detect and measure these efforts are being developed. I will discuss a rigorous method which uses Markov chains---random walks---to statistically assess gerrymandering of political districts without requiring heuristic validation of the structures of the Markov chains which arise in the redistricting context. In particular, we will see two examples where this methodology was applied in successful lawsuits which overturned district maps in Pennsylvania and North Carolina.
-
ItemAn introduction to KAM theory II: The twist theorem( 2019-03-30) de la Llave, RafaelThe KAM (Kolmogorov Arnold and Moser) theory studies the persistence of quasi-periodic solutions under perturbations. It started from a basic set of theorems and it has grown into a systematic theory that settles many questions. The basic theorem is rather surprising since it involves delicate regularity properties of the functions considered, rather subtle number theoretic properties of the frequency as well as geometric properties of the dynamical systems considered. In these lectures, we plan to cover a complete proof of a particularly representative theorem in KAM theory. The first lecture covered all the prerequisites (analysis, number theory and geometry). In this second lecture we will present a complete proof of Moser's twist map theorem (indeed a generalization to more dimensions). The proof also lends itself to very efficient numerical algorithms. If there is interest and energy, we will devote a third lecture to numerical implementations.
-
ItemAn introduction to KAM theory I: the basics( 2019-03-29) de la Llave, RafaelThe KAM (Kolmogorov Arnold and Moser) theory studies the persistence of quasi-periodic solutions under perturbations. It started from a basic set of theorems and it has grown into a systematic theory that settles many questions. The basic theorem is rather surprising since it involves delicate regularity properties of the functions considered, rather subtle number theoretic properties of the frequency as well as geometric properties of the dynamical systems considered. In these lectures, we plan to cover a complete proof of a particularly representative theorem in KAM theory. In the first lecture we will cover all the prerequisites (analysis, number theory and geometry). In the second lecture we will present a complete proof of Moser's twist map theorem (indeed a generalization to more dimensions). The proof also lends itself to very efficient numerical algorithms.
-
ItemLocally decodable codes and arithmetic progressions in random settings( 2018-11-09) Gopi, Sivakanth(1) A set D of natural numbers is called t-intersective if every positive upper density subset A of natural numbers contains a (t+1)-length arithmetic progression (AP) whose common differences is in D. Szemeredi's theorem states that the set of all natural numbers is t-intersective for every t. But there are other non-trivial examples like {p-1: p prime}, {1^k,2^k,3^k,\dots} for any k etc. which are t-intersective for every t. A natural question to study is at what density random subsets of natural numbers become t-intersective? (2) Let X_t be the number of t-APs in a random subset of Z/NZ where each element is selected with probability p independently. Can we prove precise estimates on the probability that X_t is much larger than its expectation? (3) Locally decodable codes (LDCs) are error correcting codes which allow ultra fast decoding of any message bit from a corrupted encoding of the message. What is the smallest encoding length of such codes? These seemingly unrelated problems can be addressed by studying the Gaussian width of images of low degree polynomial mappings, which seems to be a fundamental tool applicable to many such problems. Adapting ideas from existing LDC lower bounds, we can prove a general bound on Gaussian width of such sets which reproves the known LDC lower bounds and also implies new bounds for the above mentioned problems. Our bounds are still far from conjectured bounds which suggests that there is plenty of room for improvement. If time permits, we will discuss connections to type constants of injective tensor products of Banach spaces (or chernoff bounds for tensors in simpler terms). Joint work with Jop Briet.
-
ItemLarge girth approximate Steiner triple systems( 2018-09-28) Warnke, LutzIn 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.) We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6-o(1))n^2 triples and girth larger than \ell. The process iteratively adds random triples subject to the constraint that the girth remains larger than \ell. Our result is best possible up to the o(1)-term, which is a negative power of n. Joint work with Tom Bohman.
-
ItemNew Applications of the Polynomial Method to Problems in Combinatorics(Georgia Institute of Technology, 2016-10-13) Croot, ErnestErnest Croot will discuss some new applications of the polynomial method to some classical problems in combinatorics, in particular the Cap-Set Problem. The Cap-Set Problem is to determine the size of the largest subset A of F_p^n having no three-term arithmetic progressions, which are triples of vectors x,y,z satisfying x+y=2z. I will discuss an analogue of this problem for Z_4^n and the recent progress on it due to myself, Seva Lev and Peter Pach; and will discuss the work of Ellenberg and Gijswijt, and of Tao, on the F_p^n version (the original context of the problem).