In this talk, I will review recent results about how the use of linear and nonlinear flows has been key to prove functional inequalities and qualitative properties for their extremal functions. I will also explain how from these inequalities and their best constants, optimal spectral estimates can be obtained for Schrodinger operators. This is a topic which is at the crossroads of nonlinear analysis and probability, with implications in differential geometry and potential applications in modelling in physics and biology.

Let X=(x_1, ... ,x_n) be a vector of distinct positive integers. The n x n matrix with ij-th entry equal to gcd(x_i,x_j), the greatest common divisor of x_i and x_j, is called the GCD matrix on X. By a surprising result of Beslin and Ligh (1989), all GCD matrices are positive definite. In this talk, we will discuss new characterizations of the GCD matrices satisfying the stronger property of being totally nonnegative (i.e., all their minors are nonnegative). Joint work with Lucas Wu (U. Delaware).

In a sequence of seminal results in the 80's, Kaltofen showed that if an n-variate polynomial of degree poly(n) can be computed by an arithmetic circuit of size poly(n), then each of its factors can also be computed an arithmetic circuit of size poly(n). In other words, the complexity class VP (the algebraic analog of P) of polynomials, is closed under taking factors. A fundamental question in this line of research, which has largely remained open is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, constant depth arithmetic circuits or the complexity class VNP (the algebraic analog of NP) of polynomials, are closed under taking factors. In addition to being fundamental questions on their own, such 'closure results' for polynomial factorization play a crucial role in the understanding of hardness randomness tradeoffs for algebraic computation. I will talk about the following two results, whose study was motivated by these questions. 1. The class VNP is closed under taking factors. This proves a conjecture of B{\"u}rgisser. 2. All factors of degree at most poly(log n) of polynomials with constant depth circuits of size poly(n) have constant (a slightly larger constant) depth arithmetic circuits of size poly(n). This partially answers a question of Shpilka and Yehudayoff and has applications to hardness-randomness tradeoffs for constant depth arithmetic circuits. Based on joint work with Chi-Ning Chou and Noam Solomon.