On the existence of a Spine for SQG (and applications)
Abstract
based on joint work with Charles Fefferman (Princeton) and Kevin Luli (Yale).
Regularity for the Signorini problem and its free boundary
Abstract
In 1932 Signorini formulated the first variational inequality as a model of an elastic body laying on a rigid surface. In this talk we will revisit this problem from the point of view of regularity theory.
We will sketch a proof of optimal regularity and regularity of the contact set. Similar result are known for scalar equations. The proofs for scalar equations are however based on maximum principles and thus not applicable to Signorini's problem which is modelled by a system of equations.
Class invariants for quartic CM-fields
Abstract
I show how invariants of curves of genus 2 can be used for explicitly constructing class fields of
certain number fields of degree 4.
Iwasawa theory for modular forms
Abstract
he Iwasawa theory of elliptic curves over the rationals, and more
generally of modular forms, has mostly been studied with the
assumption that the form is "ordinary" at p -- i.e. that the Hecke
eigenvalue is a p-adic unit. When this is the case, the dual of the
p-Selmer group over the cyclotomic tower is a torsion module over the
Iwasawa algebra, and it is known in most cases (by work of Kato and
Skinner-Urban) that the characteristic ideal of this module is
generated by the p-adic L-function of the modular form.
I'll talk about the supersingular (good non-ordinary) case, where
things are slightly more complicated: the dual Selmer group has
positive rank, so its characteristic ideal is zero; and the p-adic
L-function is unbounded and hence doesn't lie in the Iwasawa algebra.
Under the rather restrictive hypothesis that the Hecke eigenvalue is
actually zero, Kobayashi, Pollack and Lei have shown how to decompose
the L-function as a linear combination of Iwasawa functions and
explicit "logarithm-like" series, and to modify the definition of the
Selmer group correspondingly, in order to formulate a main conjecture
(and prove one inclusion). I will describe joint work with Antonio Lei
and Sarah Zerbes where we extend this to general supersingular modular
forms, using methods from p-adic Hodge theory. Our work also gives
rise to new phenomena in the ordinary case: a somewhat mysterious
second Selmer group and L-function, which is related to the
"critical-slope L-function" studied by Pollack-Stevens and Bellaiche.
Boundary properties of graphs
Abstract
The notion of a boundary graph property is a relaxation of that of a
minimal property. Several fundamental results in graph theory have been obtained in
terms of identifying minimal properties. For instance, Robertson and Seymour showed that
there is a unique minimal minor-closed property with unbounded tree-width (the planar
graphs), while Balogh, Bollobás and Weinreich identified nine minimal hereditary
properties of labeled graphs with the factorial speed of growth. However, there are
situations where the notion of minimal property is not applicable. A typical example of this type
is given by graphs of large girth. It is known that for each particular value of k, the
graphs of girth at least k are of unbounded tree-width and their speed of growth is
superfactorial, while the limit property of this sequence (i.e., the acyclic graphs) has bounded
tree-width and its speed of growth is factorial. To overcome this difficulty, the notion of
boundary properties of graphs has been recently introduced. In the present talk, we use this
notion in order to identify some classes of graphs which are well-quasi-ordered with
respect to the induced subgraph relation.
15:45
Dense $H$-free graphs are almost $(\chi(H)-1)$-partite
Abstract
Andr\'asfai, Erdös and S\'os proved a stability result for Zarankiewicz' theorem: $K_{r+1}$-free graphs with minimum degree exceeding $(3r-4)n/(3r-1)$ are forced to be $r$-partite. Recently, Alon and Sudakov used the Szemer\'edi Regularity Lemma to obtain a corresponding stability result for the Erdös-Stone theorem; however this result was not best possible. I will describe a simpler proof (avoiding the Regularity Lemma) of a stronger result which is conjectured to be best possible.
Oblivious Routing in the $L_p$ norm
Abstract
Gupta et al. introduced a very general multi-commodity flow problem in which the cost of a given flow solution on a graph $G=(V,E)$ is calculated by first computing the link loads via a load-function l, that describes the load of a link as a function of the flow traversing the link, and then aggregating the individual link loads into a single number via an aggregation function.
We show the existence of an oblivious routing scheme with competitive ratio $O(\log n)$ and a lower bound of $\Omega(\log n/\logl\og n)$ for this model when the aggregation function agg is an $L_p$-norm.
Our results can also be viewed as a generalization of the work on approximating metrics by a distribution over dominating tree metrics and the work on minimum congestion oblivious. We provide a convex combination of trees such that routing according to the tree distribution approximately minimizes the $L_p$-norm of the link loads. The embedding techniques of Bartal and Fakcharoenphol et al. [FRT03] can be viewed as solving this problem in the $L_1$-norm while the result on congestion minmizing oblivious routing solves it for $L_\infty$. We give a single proof that shows the existence of a good tree-based oblivious routing for any $L_p$-norm.