Let \(R={\cal O}_{{\bf C},0}\) be the ring of power series
convergent in a neighborhood of zero in the complex plane. Every
scheme \(\cal X\) of finite type over \(R\) defines a complex
analytic space \({\cal X}^h\) over an open disc \(D\) of small...
Feynman categories are a new universal categorical framework for
generalizing operads, modular operads and twisted modular operads.
The latter two appear prominently in Gromov-Witten theory and in
string field theory respectively. Feynman categories...
To any bounded family of \(\mathbb F_\ell\)-linear representations
of the etale fundamental of a curve \(X\) one can associate
families of abstract modular curves which, in this setting,
generalize the `usual' modular curves with level \(\ell\)...
I will report on joint work with Nick Sheridan concerning
structural aspects of mirror symmetry for Calabi-Yau manifolds. We
show (i) that Kontsevich's homological mirror symmetry (HMS)
conjecture is a consequence of a fragment of the same...
Let \(k\) be an algebraically closed field and let \(c:C\rightarrow
X\times X\) be a correspondence. Let \(\ell \) be a prime
invertible in \(k\) and let \(K\in D^b_c(X, \overline {\mathbb
Q}_\ell )\) be a complex. An action of \(c\) on \(K\) is by...
We consider the problem of defining cylindrical contact
homology, in the absence of contractible Reeb orbits, using
"classical" methods. The main technical difficulty is failure of
transversality of multiply covered cylinders. One can fix
this...
I will begin with a brief introduction to the deformation theory
of Galois representations and its role in modularity lifting. This
will motivate the study of local deformation rings and more
specifically flat deformation rings. I will then discuss...
Let \chi be a primitive real character. We first establish a
relationship between the existence of the Landau-Siegel zero of
L(s,\chi) and the distribution of zeros of the Dirichlet L-function
L(s,\psi), with \psi belonging to a set \Psi of...
In this lecture I will explain the moment-weight inequality, and
its role in the proof of the Hilbert-Mumford numerical criterion
for $\mu$-stability. The setting is Hamiltonian group actions on
closed Kaehler manifolds. The major ingredients are...
