L3

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.


The points in question can be found on any semi-abelian surface over an

elliptic curve with complex multiplication. We will show that they provide

counter-examples to natural expectations in a variety of fields : Galois

representations (following K. Ribet's initial study from the 80's),

Lehmer's problem on heights, and more recently, the relative analogue of

the Manin-Mumford conjecture. However, they do support Pink's general

conjecture on special subvarieties of mixed Shimura varieties.

A topological space $(X,\tau)$ is submaximal if $\tau$ is the maximal element of $[{\tau}_{s}]$. Submaximality was first defined and characterized by Bourbaki. Since then, some mathematicians presented several characterizations of submaximal spaces.

In this paper, we will attempt to develop the concept of submaximality and offer some new results. Furthermore, some results concerning $\alpha$-scattered space will be obtained.

As with conventional Higgs bundles, calculating Betti numbers of twisted Higgs bundle moduli spaces through Morse theory requires us to

study holomorphic chains. For the case when the base is P^1, we present a recursive method for constructing all the possible stable chains of a given type and degree by representing a family of chains by a quiver. We present the Betti numbers when the twists are O(1) and O(2), the latter of which coincides with the co-Higgs bundles on P^1. We offer some open questions. In doing so, we mention how these numbers have appeared elsewhere recently, namely in calculations of Mozgovoy related to conjectures coming from the physics literature (Chuang-Diaconescu-Pan).

I'll describe an approach to perturbative quantum field theory
which is philosophically similar to the deformation quantization approach
to quantum mechanics. The algebraic objects which appear in our approach --
factorization algebras -- also play an important role in some recent work
in topology (by Francis, Lurie and others).  This is joint work with Owen
Gwilliam.

We review the relation between the geometry of Kleinian singularities and Dynkin diagrams of types ADE, recalling in particular the construction of a braid group action of type A, D, or E on the derived category of coherent sheaves on the minimal resolution of a Kleinian singularity. By work of Seidel-Thomas, this action was known to be faithful in type A. We extend this faithfulness result to types ADE, which provides the missing ingredient for completing Bridgeland's description of spaces of stability conditions for certain triangulated categories associated to Kleinian singularities. Our main tool is the Garside normal form for braid group elements. This project is joint work with Hugh Thomas from the University of New Brunswick.

Zilber constructed an exponential field B, which is conjecturally isomorphic to the complex exponential field. He did so by giving axioms in an infinitary logic, and showing there is exactly one model of those axioms. Following a suggestion of Zilber, I will give a different list of axioms satisfied by B which, under a number-theoretic conjecture known as CIT, describe its complete first-order theory