We outline what we expect from a good cohomology theory and introduce some of the most common cohomology theories. We go on to discuss what properties each should encode and detail attempts to fit them into a common framework. We build evidence for this viewpoint through several worked number theoretic examples and explain how many of the key conjectures in number theory fit into this theory of motives.

This talk will be on general amalgamation theory, covering ground from the 1950s to original research, with applications and examples from many different areas of mathematics and ranging from classical results to open problems.

The character variety of a manifold is a moduli space of representations of its fundamental group into some fixed gauge group.  In this talk I will outline the construction of a fully extended topological field theory in dimension 4, which gives a uniform functorial quantization of the character varieties of low-dimensional manifolds, when the gauge group is reductive algebraic (e.g. $GL_N$).

I'll focus on important examples in representation theory arising from the construction, in genus 1:  spherical double affine Hecke algebras (DAHA), difference-operator q-deformations of the Grothendieck-Springer sheaf, and the construction of irreducible DAHA modules mimicking techniques in classical geometric representation theory.  The general constructions are joint with David Ben-Zvi, Adrien Brochier, and Noah Snyder, and applications to representation theory of DAHA are joint with Martina Balagovic and Monica Vazirani.

In this talk on joint work with REBECCA BELLOVIN we discuss the “local ε-isomorphism” conjecture of Fukaya and Kato for (crystalline) families of G_{Q_p}-representations. This can be regarded as a local analogue of the global Iwasawa main conjecture for families, extending earlier work of Kato for rank one modules, of Benois and Berger for crystalline representations with respect to the cyclotomic extension as well as of Loeffler, Venjakob and Zerbes for crystalline representations with respect to abelian p-adic Lie extensions of Q_p. Nakamura has shown Kato’s - conjecture for (ϕ,\Gamma)-modules over the Robba ring, which means in particular only after inverting p, for rank one and trianguline families. The main ingredient of (the integrality part of) the proof consists of the construction of families of Wach modules generalizing work of Wach and Berger and following Kisin’s approach via a corresponding moduli space.
 

The study of 3-manifolds is founded on the strong connection between algebra and topology in dimension three. In particular, the sine qua non of much of the theory is the Loop Theorem, stating that for any embedding of a surface into a 3-manifold, a failure to be injective on the fundamental group is realised by some genuine embedding of a disc. I will discuss this theorem and give a proof of it.