# Video Lectures

### Graphs as geometric objects

Nathan Linial

It may seem quite obvious that graphs carry a lot of geometric structure.  Don't we learn in algorithm classes how to solve all-pairs-shortest-paths, minimum spanning trees etc.?  However, in this talk, I will try to impress on you the idea that...

### Moments and bounds for L-functions of large degree

We will discuss recent results concerning the problem of establishing rigorous moment estimates and subconvex bounds for L-functions of large degree.

### Moments of large families of Dirichlet L-functions

Vorrapan Chandee

Sixth and higher moments of L-functions are important and challenging problems in analytic number theory. In this talk, I will discuss my recent joint works with Xiannan Li, Kaisa Matom\"aki and Maksym Radziw\il\l on an asymptotic formula of the...

### Sums of certain arithmetic functions over 𝔽q[T] and non-unitary distributions

In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb{F}_q[T]$ to certain...

### The Ruelle invariant and convexity in higher dimensions

I will explain how to construct the Ruelle invariant of a symplectic cocycle over an arbitrary measure preserving flow. I will provide examples and computations in the case of Hamiltonian flows and Reeb flows (in particular, for toric domains). As...

### The symbolic language of Ethiopian crosses: Explorations through form and ritual

Maria Evangelatou

Ethiopia is unique in the world for the incomparable prominence of the cross in the life of its Orthodox Christian population. Crosses of unparalleled intricacy and sophistication are extensively used in religious and magic rituals, as well as in...

### The recipe for moments of L-functions and characteristic polynomials of random matrices

Sieg Baluyot

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith formulated a 'recipe' that leads to precise conjectures for the asymptotic behavior of integral moments of various families of $L$-functions. They also proved exact formulas for moments of...

### Half-Isolated Zeros and Zero-Density Estimates

Kyle Pratt

We introduce a new zero-detecting method which is sensitive to the vertical distribution of zeros of the zeta function. This allows us to show that there are few ‘half-isolated’ zeros. If we assume that the zeros of the zeta function are restricted...

### Negative moments of the Riemann zeta function

Alexandra Florea

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can...

### RMT statistics in number theory and in quantum chaos

Montgomery's pair correlation conjecture ushered a new paradigm into the theory of the Riemann zeta function, that of the occurrence of Random Matrix Theory statistics, as developed in part by Dyson, into the theory. A parallel development was the...