We will discuss Raynaud's classical theory on Néron models of Jacobians of curves, and mention some tropical aspects of the theory that help us understand modular curves from a modern non-Archimedean viewpoint. There will be an annoyingly large number of examples illustrating the key principles throughout.
We describe a framework for defining and classifying TQFTs via
surgery. Given a functor
from the category of smooth manifolds and diffeomorphisms to
finite-dimensional vector spaces,
and maps induced by surgery along framed spheres, we give a set of axioms
that allows one to assemble functorial coboridsm maps.
Using this, we can reprove the correspondence between (1+1)-dimensional
TQFTs and commutative Frobenius algebras,
and classify (2+1)-dimensional TQFTs in terms of a new structure, namely
split graded involutive nearly Frobenius algebras
endowed with a certain mapping class group representation. The latter has
not appeared in the literature even in conjectural form.
This framework is also well-suited to defining natural cobordism maps in
Heegaard Floer homology.
Vector bundles over a compact manifold can be defined via transition
functions to a linear group. Often one imposes
conditions on this structure group. For example for real vector bundles on
may ask that all
transition functions lie in the special orthogonal group to encode
orientability. Commutative K-theory arises when we impose the condition
that the transition functions commute with each other whenever they are
simultaneously defined.
We will introduce commutative K-theory and some natural variants of it,
and will show that they give rise to new generalised
cohomology theories.
This is joint work with Adem, Gomez and Lind building on previous work by
Adem, F. Cohen, and Gomez.
Consider a nonsingular projective variety $X$ defined by a system of $R$ forms of the same degree $d$. The circle method proves the Hasse principle and Manin's conjecture for $X$ when $\text{dim}X > C(d,R)$. I will describe how to improve the value of $C$ when $R$ is large. I use a technique for estimating mean values of exponential sums which I call a ``moat lemma". This leads to a novel and intriguing system of auxiliary inequalities.
Counterexamples to Vaught's Conjecture regarding the number of countable
models of a theory in a logical language, may felicitously be studied by investigating a tree
of types of different arities and belonging to different languages. This
tree emerges from a category of topological spaces, and may be studied as such, without
reference to the original logic. The tree has an intuitive character of absoluteness
and of self-similarity. We present theorems expressing these ideas, some old and some new.
I will explain an approach to Hamiltonian reduction using derived
symplectic geometry. Roughly speaking, the reduced space can be
presented as an intersection of two Lagrangians in a shifted symplectic
space, which therefore carries a natural symplectic structure. A slight
modification of the construction gives rise to quasi-Hamiltonian
reduction. This talk will also serve as an introduction to the wonderful
world of derived symplectic geometry where statements that morally ought
to be true are indeed true.