The category {\bf FI} and its variants have been of great interest recently. Being a finitely generated {\bf FI}-module implies many desirable properties about sequences of symmetric group representations, in particular representation stability. We define a new combinatorial category analogous to {\bf FI} for the 0-Hecke algebra, denoted by $\mathcal{H}$, indexing sequences of representations of $H_n(0)$ as $n$ varies under suitable compatibility conditions. We then provide examples of $\mathcal{H}$-modules and use these to discuss some properties finitely generated $\mathcal{H}$-modules possess, including a new form of representation stability and eventually polynomial growth.

A landmark result by Dykema in 1993 classified free products of finite-dimensional von Neumann algebras equipped with tracial states. In 1997, Shlyakhtenko constructed the almost periodic free Araki-Woods factors, a natural non-tracial analogue to free group factors. He asked whether free products of finite-dimensional von Neumann algebras with respect to non-tracial states can be described in terms of free Araki-Woods factors. In this talk, I will answer Shlyakhtenko's question in the affirmative, therefore providing a complete classification of free products of finite dimensional von Neumann algebras. This is joint work with Brent Nelson.

We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\infty,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory.

The interesting part of the cohomology of the theta divisor $D$ of an abelian fivefold $A$ shares numerical properties with the Lie algebra $E_6$. We define 27 surfaces inside $D$, one for each realisation of $A$ as a Prym variety, and explain how they generate a sublattice of $H^4(D, \mathbb{Z})$ isomorphic to the root lattice of $E_6$. This gives an effective proof of the Hodge conjecture for the theta divisor.

An alternative to selecting an integer uniformly from $1$ to $N$ and letting $N$ go to infinity is to select an
integer according to the Riemann zeta distribution: the probability of selecting $n$ is $1/\zeta(s)n^s$, and
letting $s$ go to $1$. We will explain several results that arise naturally due to the multiplicative property of this distribution.

We study the effective behavior of a Brownian motion in both one and two dimensional comb like domains. This problem arises in a variety of physical situations such as transport in tissues, and linear porous media. We show convergence to a limiting process when when both the spacing between the teeth, and the probability of entering a tooth vanish at the same rate. This limiting process exhibits an anomalous diffusive behavior, and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. At the PDE level, this leads to equations that have fractional time derivatives and are similar to the Bassett differential equation.

Fix an integer $N$ and a prime $(p,N)=1$ where $p>3$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $T_p(f) = 0$ is bounded independently of $k$, where $T_p$ is the Hecke operator.

Consider the Bergman kernel associated to the tensor power of a positive line bundle on a compact Kähler manifold. We will present our work on its near-diagonal asymptotic and off-diagonal decay properties. This is joint work with H. Hezari and Z. Lu.

Epitaxial growth is an important physical process for forming solid films or other nano-structures. It occurs as atoms, deposited from above, adsorb and diffuse on a crystal surface. Modeling the rates that atoms hop and break bonds leads in the continuum limit to degenerate 4th-order PDE that involve exponential nonlinearity and the p-Laplacian with p=1, for example. We discuss a number of analytical results for such models, some of which involve subgradient dynamics for Radon measure solutions and a new notion of weak solutions.

Malle's conjecture can be thought of as a
generalization of the inverse Galois problem, which asks for every
finite group $G$, is there a number field $K$ such that their Galois
group over $\mathbb{Q}$ is isomorphic to $G$? Although open, this
question is widely believed to be true, and Malle went further to
predict the asymptotics of how many number fields there are with a
given Galois group that only depended on the group structure of $G$
and the degree of the number field. In this talk, we will discuss the
history as well as recent results and techniques surrounding these
conjectures.

(joint with B.Bakker and Y.Brunebarbe) One
very fruitful way of studying complex algebraic varieties is by
forgetting the underlying algebraic structure, and just thinking of
them as complex analytic spaces. To this end, it is a natural and
fruitful question to ask how much the complex analytic structure
remembers. One very prominent result is Chows theorem, stating that
any closed analytic subspace of projective space is in fact
algebraic. A notable consequence of this result is that a compact
complex analytic space admits at most one algebraic structure - a
result which is false in the non-compact case. This was generalized
and extended by Serre in his famous GAGA paper using the language of
cohomology.

We explain how we can extend Chows theorem and in fact all of GAGA to
the non-compact case by working with complex analytic structures that
are 'tame' in the precise sense defined by o-minimality. This leads to
some very general 'algebraization' theorems, which can be used to
obtain new results in Hodge Theory. In particular, we use this
technology to prove a conjecture of Griffiths on the algebraicity and
quasi-projectivity of images of period maps. As prerequisities for
this talk, it would be helpful to have gone through a first year
course in Algebraic Geometry, covering in particular the theory of
sheaves.