The Jacobian (or sandpile group) is an algebraic invariant of a graph that plays a similar role to the class group from number theory. There are multiple recent results controlling the sizes of these groups in Galois towers of graphs that mimic the classical results in Iwasawa theory, though the connection to the values of the Ihara zeta function often requires some adjustment. In this talk we will give a new way to view the Jacobian of a graph that more directly centers the edges of the graph, construct a module over the relevant Iwasawa algebra that nearly corresponds to the interpolated zeta function, and discuss where the discrepancy comes from.

Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.

Given a family of algebraic varieties over an irreducible scheme, a natural question to ask is what type of properties of the generic fiber, and how do those properties extend to other fibers. For example, the Hilbert irreducibility theorem states that a dominant map from an irreducible variety X defined over a number field to some projective space which is generically of degree d provides a Zariski dense set of degree d points on X. One can also get quantitative estimates for size of the complement which does not carry the generic property.

We will explore this topic from an arithmetic point of view by looking at several scenarios. For instance, suppose we have a 1-dimensional family of pairs of elliptic curves over a number field,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. One may ask how does the property of "being (non-)isogenous" extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most B, such that the two elliptic curves at the specializations are isogenous? 

The cokernel of a random p-adic matrix can be used to study the distribution of objects that arise naturally in number theory. For example, Cohen and Lenstra suggested a conjectural distribution of the p-parts of the ideal class groups of imaginary quadratic fields. Friedman and Washington proved that the distribution of the cokernel of a random p-adic matrix is the same as the Cohen–Lenstra distribution. Recently, Wood generalized the result of Friedman–Washington by considering a far more general class of measure on p-adic matrices. In this talk, we explain a further generalization of Wood’s work. This is joint work with Dong Yeap Kang and Jungin Lee.

Complex Hodge theory equips the rational singular cohomology of smooth projective variety with a Hodge filtration, and period maps measure how these filtrations vary in families. Period maps are highly transcendental in nature, and the transcendence properties of these maps reflect interesting aspects of the geometry; for example a theorem of Cohen and Shiga-Wolfhart show that a complex abelian variety A has CM if and only both A and the Hodge filtration are defined over a number field.

In this talk, we describe a similar situation in p-adic Hodge theory, and discuss joint work with Sean Howe that proves an analogue of the theorem of Cohen and Shiga-Wolfhart. The first part of the talk will contain a brief introduction to Hodge theory, so no previous experience with Hodge theory (complex or p-adic) is required!