In joint work with A. Dhillon and Y. Yan, we investigate the behaviour of tilting sheaves under pushforward by a finite Galois morphism. We determine conditions under which such a pushforward of a tilting sheaf is a tilting sheaf. We then produce some examples of Severi Brauer flag varieties and arithmetic toric varieties in which our method produces a tilting sheaf, adding to the list of positive results in the literature. We also produce some counterexamples to show that such a pushfoward need not be a tilting sheaf.

I shall give a quick introduction to Yang-Mills equations and self-duality as they appear both in mathematics and high energy physics.

Following Chapter 2 of the HoTT Book, I will continue presenting the main properties of the Identity Type.

I will (re)state and prove estimates that relate the super-level sets of the diagonal Bergman kernel on a pseudoconvex tube domain with the floating bodies of its convex base.

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: 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.

A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This implies the above mentioned conjectures and their generalization to arbitrary matroids.

Joint work with Karim Adiprasito and Eric Katz.