finite element methods are numerical methods that approximate solutions to pdes using functions on a mesh representing the problem domain. discontinuous-petrov galerkin methods are a class of finite element methods that are aimed at achieving stability of the petrov-galerkin finite element approximation through a careful selection of the associated trial and test spaces. in this talk, i will present dpg theorems as they apply to linear problems, and then approaches for those theorems in the case of non-linear problems, as well as suggest further approaches to non-linear problems.

for
any open riemann surface $x$ admitting green functions, suita asked about the precise relations between the bergman kernel and the logarithmic capacity. it was conjectured that the gaussian curvature of the suita metric is bounded from above by $-4$, and moreover
the curvature is identically equal to $-4$ if and only if $x$ is conformally equivalent to the unit disc less a (possible) polar set. after the contributions made by b\l{}ocki
and guan & zhou, we provide an alternative and simplified proof of the equality part in suita's conjecture. our proof combines the ohsawa-takegoshi extension theorem and the plurisubharmonic variation properties of bergman kernels.

i will describe recent joint work with c. mantoulidis in which we study the properties of bounded morse index solutions to the allen--cahn equation on 3-manifolds. one consequence of our work is that a generic riemannian 3-manifold contains an embedded minimal surface with morse index p, for each positive integer p.

let $x$ be an elliptically fibered k3 surface with a fixed $su(n)$ bundle $\mathcal{e}$. i will talk about degenerations of connections on $\mathcal{e}$ that are hermitian-yang-mills with respect to a collapsing family of ricci flat metrics. this can be thought of as a vector bundle analog of the degeneration of ricci flat metrics studied by gross-wilson and gross-tosatti-zhang. i will show that under some mild conditions on the bundle, the restriction of the connections to a generic elliptic fiber converges to a flat connection. i will also talk about some ongoing work on strengthening this result. this is based on joint work with adam jacob and yuguang zhang.

branching brownian motion is a prototype of a disordered system and a toy model for spin glasses and log-correlated fields. it also has an exact duality relation with the fkpp equation, a well-known reaction diffusion equation. in this talk, i will present recent results (obtained with michel pain) on the fluctuations of the gibbs measure at the critical temperature. by gibbs measure i mean here the measure whose atoms are the positions of the particles, weighted by their gibbs weight. it is known that this gibbs measure, after a suitable scaling, converges to a deterministic measure. we prove a non-standard central limit theorem for the integral of a function against the gibbs measure, for a large class of functions. the possible limits are 1-stable laws with arbitrary asymmetry parameter depending on the function. in particular, the derivative martingale and the usual additive martingale satisfy such a central limit theorem with, respectively, a totally asymmetric and a cauchy distribution.

in recent years, popa’s deformation/rigidity theory has lead to a wealth of classification
and rigidity results for von neumann algebras given by countable groups and their actions on
measure spaces. in this talk, i will present the first rigidity and classification results for von
neumann algebras given by locally compact groups and their actions. i establish that the crossed
product von neumann algebra has a unique cartan subalgebra, when the action is
free and probability measure preserving and the group is a connected simple lie group of real rank one, or
a group acting properly on a tree. from this, i deduce a w*-strong rigidity result for irreducible
actions of products of such groups. i also establish that the group von neumann algebra of such
groups are strongly solid. more generally, our results hold for locally compact groups that are
non-amenable, weakly amenable and belong to ozawa’s class s. this is joint work with arnaud
brothier and stefaan vaes.

classical modular curves associated to gl(2) are moduli spaces of
elliptic curves with additional structure. taking advantage of the
analogy between number fields and function fields, drinfeld modules (of
rank 2) were introduced as a good analogue of elliptic curves. while
there are no shimura varieties associated to the general linear group
gl(n) for n$>$2, the situation is sharply different over function fields.
the drinfeld modular variety for gl(n) is a moduli space of drinfeld
modules of rank n (with auxiliary level structure). it is an affine
scheme of dimension n-1. in this talk, i will explain how analogues of
well-established theories due to hida and coleman in the classical
p-adic context extend to drinfeld modular varieties and their associated
modular forms.

joint with g. rosso (montr\'eal).

i will talk about two problems, completely different from
each other and both quite fun, that turn out to be related to
the riemann hypothesis. one of them concerns the universal
limiting position (after rescaling) of the zeros of polynomials
belonging to a rather general class, with as an application
a weak version of an old conjecture of polya that in its strong
version would imply the riemann hypothesis. the other gives
an equivalence between the generalized riemann hypothesis and
a statement about the growth rate of the determinants of certain
matrices whose entries are elementary cotangent sums, with an
unexpected appearance of quantum modular forms as a byproduct.

in a pioneering paper, pila and zannier showed how one can
prove arithmetic results (the manin-mumford conjecture) using
transcendental methods (the ax-lindemann conjecture). their approach has
since been greatly developed, and is a major ingredient in the
andre-oort conjecture for shimura varieties as well as the more
general zilber-pink conjecture, that serves as a sort of flagship for
the field of unlikely intersections. we'll explain this story and
present a new result (joint with pila and mok)
proving a general transcendence theorem known as ax-schanuel for
arbitrary shimura varieties.