L6

In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

The talk will be based on some of the material in the joint survey with Etienne Ghys

"Signatures in algebra, topology and dynamics"

http://arxiv.org/abs/1512.092582

In the 19th century Sturm's theorem on the number of roots of a real polynomial motivated Sylvester to define the signature of a quadratic form. In the 20th century the classification of quadratic forms over algebraic number fields motivated Witt to introduce the "Witt groups" of stable isomorphism classes of quadratic forms over arbitrary fields. Still in the 20th century the study of high-dimensional topological manifolds with nontrivial fundamental group motivated Wall to introduce the "Wall groups" of stable isomorphism classes of quadratic forms over arbitrary rings with involution. In our survey we interpreted Sturm's theorem in terms of the Witt-Wall groups of function fields. The talk will emphasize the common thread running through this developments, namely the notion of the localization of a ring inverting elements. More recently, the Cohn localization of inverting matrices over a noncommutative ring has been applied to topology in the 21st century, in the context of the speaker's algebraic theory of surgery.

We will prove an upper bound for the volume of a minimal
hypersurface in a closed Riemannian manifold conformally equivalent to
a manifold with $Ric > -(n-1)$.  In the second part of the talk we will
construct a sweepout of a closed 3-manifold with positive Ricci
curvature by 1-cycles of controlled length and prove an upper bound
for the length of a stationary geodesic net. These are joint works
with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).

Motivated by work of Borel and Serre on arithmetic groups, Bestvina and Feighn defined a bordification of Outer space; this is an enlargement of Outer space which is highly-connected at infinity and on which the action of $Out(F_n)$ extends, with compact quotient. They conclude that $Out(F_n)$ satisfies a type of duality between homology and cohomology.  We show that Bestvina and Feighn’s  bordification can be realized as a deformation retract of Outer space instead of an extension, answering some questions left open by Bestvina and Feighn and considerably simplifying their proof that the bordification is highly connected at infinity.

Wei Title: Adaptive timestep Methods for non-globally Lipschitz SDEs

Wei Abstract: Explicit Euler and Milstein methods are two common ways to simulate the numerical solutions of
SDEs for its computability and implementability, but they require global Lipschitz continuity on both
drift and diffusion coefficients. By assuming the boundedness of the p-th moments of exact solution
and numerical solution, strong convergence of the Euler-type schemes for locally Lipschitz drift has been
proved in [HMS02], including the implicit Euler method and the semi-implicit Euler method. However,
except for some special cases, implicit-type Euler method requires additional computational cost, which
is very inefficient in practice. Explicit Euler method then is shown to be divergent in [HJK11] for non-
Lipschitz drift. Explicit tamed Euler method proposed in [HJK + 12], shows the strong convergence for the
one-sided Lipschitz condition with at most polynomial growth and it is also extended to tamed Milstein
method in [WG13]. In this paper, we propose a new adaptive timestep Euler method, which shows the
strong convergence under locally Lipschitz drift and gains the standard convergence order under one-sided
Lipschitz condition with at most polynomial growth. Numerical experiments also demonstrate a better
performance of our scheme, especially for large initial value and high dimensions, by comparing the mean
square error with respect to the runtime. In addition, we extend this adaptive scheme to Milstein method
and get a higher order strong convergence with commutative noise.

Alexander Title: Functionally-generated portfolios and optimal transport

Alexander Abstract: I will showcase some ongoing research, in which I try to make links between the class of functionally-generated portfolios from Stochastic Portfolio Theory, and certain optimal transport problems.