The slopes of modular forms are the $p$-adic valuations of the eigenvalues of the Hecke operators $T_p$. The study of slopes plays an important role in understanding the geometry of the eigencurves, introduced by Coleman and Mazur.

The study of the slope began in the 1990s when Gouvea and Mazur computed many numerical data and made several interesting conjectures. Later, Buzzard, Calegari, and other people made more precise conjectures by studying the space of overconvergent modular forms. Recently, Bergdall and Pollack introduced the ghost conjecture that unifies the previous conjectures in most cases. The ghost conjecture states that the slope can be predicted by an explicitly defined power series. We prove the ghost conjecture under a certain mild technical condition. In the pre-talk, I will explain an example in the quaternionic setting which was used as a testing ground for the proof.

This is joint work with Ruochuan Liu, Liang Xiao, and Bin Zhao.

The q-Canonical Commutation Relation (q-CCR) is an interpolation between the CCR and the CAR with a parameter q. In the 1990s, M. Bożejko and R. Speicher found that the q-CCR is represented on the q-Fock space. The q-Gaussians are realized as the field operators with the vacuum state, which forms a non-commutative distribution. 

On the other hand, a conjugate system is a notion of free probability introduced by D. Voiculescu. This carries important information about a non-commutative distribution of given operators and has many implications for the generated von Neumann algebra.  

In this talk, I will present a concrete formula for the conjugate system for the q-Gaussians. This talk is based on the joint work with R. Speicher.

We will discuss various constructions of modular forms in the context of classical and exceptional theta correspondences. 

Geometric numerical integrators are numerical methods used to solve ordinary and partial differential equations that preserve geometric properties of the underlying dynamical systems. These methods are designed to accurately approximate the trajectories of the systems while conserving important physical or mathematical properties such as energy, momentum, symplecticity, or volume. In this talk, I will be talking about two classes of geometric numerical integrators both of which suffer from parasitic instabilities namely G-symplectic general linear methods for Hamiltonian systems and Variational Integrators for degenerate Lagrangian systems. I will also discuss the strategies to control the effect of parasitism in these methods.

How many essentially distinct ways are there to arrange $n$ convex sets in $\mathbb{R}^d$? Here, `essentially distinct' means with different intersection pattern'. We discuss this question both in the dimension $1$, where it amounts to counting the interval graphs, and in higher dimenions. Based on the joint works with Amzi Jeffs.  Plain text abstract: How many essentially distinct ways are there to arrange n convex sets in R^d? Here, `essentially distinct' means `with different intersection pattern'. We discuss this question both in the dimension 1, where it amounts to counting the interval graphs, and in higher dimenions. Based on the joint works with Amzi Jeffs.

The $C_2$-equivariant stable stems were first studied by Bredon and Landweber, in the 1960s. From the start, it was clear that these groups exhibited certain periodic behavior closely related to James periodicity for stunted projective spaces. This was made more explicit and extensively applied to computations by Araki and Iriye in the late 1970s / early 1980s. In the past decade, this phenomena has been lifted to $\mathbb{R}$-motivic homotopy theory under the guise of "$\tau$-periodicity", and plays a central role in Behrens and Shah's work relating $\mathbb{R}$-motivic and $C_2$-equivariant homotopy theory.

In this talk, I will review some of the above story, and then explain how similar periodic phenomena occurs in $G$-equivariant stable homotopy theory for an arbitrary finite group $G$.

The classical Kakeya needle problem asks: what is the smallest set in the plane inside which a unit-length needle can be translated and rotated through a full 360-degree turn? In this talk, we will show how to construct such sets of arbitrarily small measures (known as Kakeya sets). We will then describe some applications to several important problems in analysis where this seemingly innocuous rotation property can lead to very counterintuitive and pathological results.

The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on a compact manifold of negative curvature become equidistributed as the eigenvalue tends to infinity. In the talk, I will discuss recent work on this problem for arithmetic quotients of the three-dimensional hyperbolic space. I will discuss our key result that Hecke eigenfunctions cannot concentrate on certain proper submanifolds. Joint work with Lior Silberman.

We consider {0,1}-matrices. For matrices A and B, we say A contains B as a configuration if there is a submatrix of A that is a column and row permutation of B. For instance, if A and B are incidence matrices of graphs G and H respectively, then A contains B as a configuration if and only if G contains H as a subgraph. In this talk, we study some extremal problems for matrices and hypergraphs (set systems).

The birational classification of foliated surface is pretty much complete, thanks to the work of Brunella, Mendes, McQuillan. The next obvious step in this endeavour, in analogy with the classical case of projective varieties and log pairs, is to construct moduli spaces for foliated varieties (starting from the general type case). The first question to ask, on the road towards constructing such a moduli space, is how to show that foliated varieties of fixed Kodaira dimension are bounded, that is, they come in finitely many algebraic families ― provided, of course, that we fix certain appropriate numerical invariants. It turns out that, to best answer this question, rather than working with the canonical divisor of a foliation it is better to consider linear systems of the form $|nK_X + mK_F|, n,m >0, as those encode a lot of the positivity features that classically the canonical divisor of a projective variety displays.
In this talk, I will introduce this framework and explain how this approach leads to answering the question about boundedness for foliated surfaces. Time permitting, I will address also what happens when we try to construct a moduli functor, or rather, what we have been finding out, so far. This talk features joint work with C. Spicer, work with J. Pereira, and work in progress with M. McQuillan and C. Spicer.