Abstract: In this talk we will introduce the notions of divisor, meromorphic functions and meromorphic forms on a Compact Riemann Surface. We will state the Riemann Roch theorem and derive several interesting consequences of it. For example this gives us the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. We will interpret the theorem as a statement about Euler characteristic and explore possible generalizations of this theorem in the context of algebraic geometry. If time permits we shall see a sketch of a proof of Riemann-Roch.

Abstract: Abstract: Newton showed that, if a polynomial \(p(t)=\sum_{i=0}^n a_i t^i\) has only real roots, then the coefficient sequence \((a_0,a_1,\ldots,a_n)\) satisfies the inequalities \(a_i^2\geq a_{i-1}a_{i+1}\). This implies, in particular, that the sequence is (up to sign) unimodal.

In 1968, Ronald Read conjectured that the coefficients of the chromatic polynomial of a graph form a (sign-alternating) unimodal sequence. Soon afterwards, Rota, Heron and Welsh proposed a much more daring conjecture: that the coefficients of the characteristic polynomial of a matroid form a sign-alternating log-concave sequence.

In a sequence of recent papers, June Huh, then Huh with Eric Katz, and finally Huh, Katz and Karim Adiprasito proved the Rota-Heron-Welsh conjecture. First for matroids realizable in characteristic zero, then over any field, and most recently for matroids without linear realizations. The methods in each case make use of or are inspired by inequalities in algebraic geometry.

My objective is to give a gentle introduction to their program.