Schur-Frobenius inversion formula the generalised resolvent on the resonant inclusion. An effective description of the original family of transmission problems. A time-dispersive effective formulation in the whole space. An example of the effective formulae for a specific cell geometry. Band gaps and “metamaterial” properties.

Periodic media with resonant components (“high contrast composites”). Gelfand transform and direct integral: a reduction of the full-space problem to a family of “transmission” problems on the period cell. A reformulation in terms of the M-operator on the interface. Diagonalisation of the M-operator on the nonresonant component: Steklov eigenvalue problem.

Spectral boundary-value problems: boundary triples and the corresponding M-operators (“Dirichlet-to-Neumann maps”). Their role in the quantitative analysis of degenerate problems. Krein formula for a generalised Robin problem.

An overview of the mathematical theory of homogenisation as a toolbox for the analysis of multiscale problems. Wave propagation: resonant and nonresonant regimes. Non-resolvent estimates, time dispersion, and metamaterials: amotivation for a novel homogenisation principle.

We will see a two variable zeta function associated with a graph due to Dino Lorenzini. I will mainly spend time discussing divisor theory on graphs that arises in the definition of this zeta function.

Given the lack of progress on complexity lower bounds, it is natural to ask whether they are hard to prove, in some formal sense. I will begin by briefly describing the classical incompleteness results of Godel and Chaitin, and posing the question for whether there are analogues of these results in complexity theory. I will then introduce the finitistic framework of propositional proof complexity, where we are interested in the existence of polynomial size proofs verifiable in polynomial time. I will explain what it means to prove circuit complexity or proof complexity lower bounds in this framework. Finally, I will describe a strong complexity conjecture for which it can be shown unconditionally that there are no feasible propositional proofs, in a certain technical sense.

Esnault and Srinivas proved that as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of a smooth proper surface over a finite field $\F_q$ is taken in the span of the Neron-Severi group inside of of $\ell$-adic cohomology. In this talk we will discuss some analogous questions in higher dimensions motivated by their results and techniques.

Tensors are higher dimensional analogues of matrices. But unlike matrices, there is still so much we don't know about their fundamental algebraic properties. For example, for rank-r matrices we know that the determinants of all (r+1)-minors of the matrix furnish a generating set for the ideal of all relations among the entries of such matrices, but for general rank-r tensors we have almost no idea what polynomials generate their ideals. Moreover the entire minimal free resolution of the ideal in the matrix case is know in terms of representation theory (Lascoux, Eagon-Northocott, Weyman, and others), but relatively little is known in the tensor case, (not even the length of the resolution). I'll present evidence toward a conjecture on arithmetic Cohen-Macaulay-ness that would generalize the Eagon-Hochster result in the matrix case. I'll also highlight recent work with Raicu and Sam where we compute precise vanishing and non-vanishing of the syzygies of rank-1 tensors.

We will see a two variable zeta function associated with a graph due to Dino Lorenzini. I will mainly spend time discussing divisor theory on graphs that arises in the definition of this zeta function.