School of Mathematics
Nonnegative k-Sums, Fractional Covers, and Probability of Small Deviations
Three Knot-Theoretic Perspectives on Algebra
Legendrian Invariants in Rational Homology Spheres
Limit Theorems for Theta Sums and Applications
Concentration Compactness Method for Global Well-Posedness for the Energy-Supercritical Nonlinear Wave Equation
Pseudo-Holomorphic Curves and Approximations of Zero Entropy Hamiltonian Systems by Periodic Ones
Stability and Instability of Near-Integrable Hamiltonian Systems
Existence of Small Families of t-wise Independent Permutations and t-Designs Via Local Limit Theorems
We show existence of rigid combinatorial objects that previously were not known to exist. Specifically, we consider two families of objects:
1. A family of permutations on n elements is t-wise independent if it acts uniformly on tuples of t elements. Constructions of small families of t-wise independent permutations are known only for \( t=1,2,3 \) . We show that there exist small families of t-wise independent permutations for all t , whose size is \( n^{O(t)} \) .
Workshop on Sheaf-Theoretic Methods in Symplectic Topology
Serre's Conjectures on the Number of Rational Points of Bounded Height
JOINT IAS/PU NUMBER THEORY SEMINAR
We give a survey of recent results on conjectures of Heath-Brown and Serre on the asymptotic density of rational points of bounded height. The main tool in the proofs is a new global determinant method inspired by the local real and p-adic determinant methods of Bombieri-Pila and Heath-Brown.