Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The class of Lorentzian polynomials contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. I will give several combinatorial applications. No specific background will be needed to enjoy the talk. Joint work with Petter Brändén (https://arxiv.org/abs/1902.03719).
Computer Science and Discrete Mathematics (CSDM)
A linear matrix is a matrix whose entries are linear forms in some indeterminates $t_1,\dots, t_m$ with coefficients in some field $F$. The commutative rank of a linear matrix is obtained by interpreting it as a matrix with entries in the function field $F(t_1,\dots,t_m)$, and is directly related to the central PIT (polynomial identity testing) problem. The
Randomness dispersers are an important tool in the theory of pseudorandomness, with numerous applications. In this talk, we will consider one-bit strong dispersers and show their connection to erasure list-decodable codes and Ramsey graphs.
Given an arbitrary graph, we show that if we are allowed to modify (say) 1% of the edges then it is possible to obtain a much smaller regular partition than in Szemeredi's original proof of the regularity lemma. Moreover, we show that it is impossible to improve upon the bound we obtain.
In this talk, I will give an overview on how PCPs, combined with cryptographic tools,
are used to generate succinct and efficiently verifiable proofs for the correctness of computations.
I will focus on constructing (computationally sound) *succinct* proofs that are *non-interactive*
(assuming the existence of public parameters) and are *publicly verifiable*.
In particular, I will focus on a recent result with Omer Paneth and Lisa Yang,
where we show how to construct such proofs for all polynomial time computations,
What is the largest number of projections onto k coordinates guaranteed in every family of m binary vectors of length n? This fundamental question is intimately connected to important topics and results in combinatorics and computer science (Turan number, Sauer-Shelah Lemma, Kahn-Kalai-Linial Theorem, and more), and is wide open for most settings of the parameters. We essentially settle the question for linear k and sub-exponential m.
Based on joint work with Noga Alon and Noam Solomon.