|
 |
Schedule:
- Tuesday April 16th:
11 am: Don Zagier
1.45 pm: Jonathan Pridham
3 pm: No talk, see below (*)
- Wednesday April 17th:
10 am: Robert Laugwitz
11.30 am: Netan Dogra
2 pm: Oren Ben-Bassat
3.30 pm: Amnon Besser
- Thursday April 18th:
10 am: Emily Cliff
11.30 am: Jack Shotton
2 pm: Ulrike Tillmann
3.30 pm: Minhyong Kim
6 pm: Picnic!
- Friday April 19th:
9.20 am: Emily Cliff
10 am: Kevin McGerty
11.30 am: Jakob Blaavand
2 pm: Yakov Kremnitzer
3.30 pm: Dominic Joyce
(*) This gap in our schedule is entirely intentional. It will allow participants to attend Don Zagier's talk on the workshop Higher Structures in Topology and Number Theory, also held at the Mathematical Institute. This is in addition to his talk on Deformation Week earlier that day.
Abstracts:
- Oren Ben-Bassat: Deformations of schemes and stacks
We will discuss some of the theory of deformations and non-commutative aspects related to the Hochschild complex.
- Amnon Besser: Deformations and multiple zeta values.
Multiple zeta values are a simple generalization of the values of the Riemann zeta function at positive integers. They make numerous appearances in number theory. The interesting relations that they satisfy are related to the structure of the Galois group of the rationals and have been studied from many points of view. Their relation with deformations is via the relation with Quantum groups (Drinfeld associators) and the moduli spaces of marked genus 0 curves.
- Jakob Blaavand: Introduction to connections and curvature
This is a gentle introduction to the world of connections, curvature, holomorphic bundles and anti-self-dual connections. In the last part of the talk, we discuss the moduli space of instantons on a compact Kähler surface, its tangent space, and its dimension. This moduli space (or perhaps a variant thereof) will appear in Joyce's talk.
- Emily Cliff: Introduction to Symmetric Operads
We give several equivalent definitions of an operad and explore some examples. We consider the notion of an algebra over an operad in terms of these definitions, and introduce the idea of a free algebra over an operad. These talks are intended to provide the background necessary to follow Ulrike Tillmann's discussion of TQFTs, as well as Kobi Kremnizer's introduction to Koszul duality on Friday afternoon.
- Netan Dogra: Iterated integrals and applications
We will give a very naive introduction to the theory of iterated integrals, and discuss their interpretation as periods of mixed Hodge structure. In the case of the mixed Hodge structure associated to the pro-unipotent motivic fundamental group of the projective plane minus three points, this interpretation is important in the relating zeta values and multi-zeta values to arithmetic.
- Dominic Joyce: Approaches to moduli problems
We will contrast differential-geometric approaches to moduli problems (typically involving elliptic operators, nonlinear Fredholm maps between Banach spaces, the Implicit Function Theorem for Banach spaces) with algebro-geometric approaches to moduli problems (moduli functors, and functors represented by a scheme or stack), and discuss how the two approaches can be connected for the example of moduli of holomorphic vector bundles on a projective complex manifold.
- Minhyong Kim: Geometric families in Diophantine geometry
We discuss the use of families of objects in the proofs of some key theorems of Diophantine geometry: the Mordell conjecture, Fermat's last theorem, and some cases of the conjecture of Birch and Swinnerton-Dyer.
- Kobi Kremnitzer: Koszul duality
We will discuss the relation between deformation theory and Koszul duality as explained by Hinich. There are different paths through this, but we will focus on the approach via operads.
- Robert Laugwitz: q-Deformation and Braided Monoidal Categories
We argue that braided monoidal categories of comodules give a natural framework to study quantum groups. The quantum groupsand the quantum coordinate rings
can be obtained from constructions in these categories. This is the case of comodules over a lattice. More generally, the lattice can be replaced by other groups giving, for example, a setting to include rational Cherednik algebras into this picture.
- Kevin McGerty: Noncommutative resolutions, deformations, and rational Cherednik algebras
Classically, the singularities of algebraic varieties can be studied by deformations: familes of varieties which are generically smooth but possess the singular variety as a "special fibre", or by resolutions: smooth varieties possessing a map to the singular variety with compact fibres which is an isomorphism on the smooth locus. Recently there has been much interest in noncommutative analogues of these structures: one can consider noncommutative deformations of the functions on a singular variety, and, slightly less obviously, noncommutative resolutions. We will outline these notions in the context of symplectic singularities and gives examples coming from the theory of rational Cherednik algebras.
- Jonathan Pridham: Deformations from an algebraic perspective
Deformation theory in algebraic geometry looks at infinitesimal neighbourhoods in moduli spaces. This involves a functorial interpretation and a notion of fat points. I will illustrate these with several examples, and maybe explain how derived deformations arise.
- Jack Shotton: Deformations of Galois representations
Mazur realised that deformation rings are a natural way of putting Galois representations into families. After introducing the notion of a Galois representation, I will present some of the foundational results in their deformation theory, with examples. Time permitting, I will discuss some more advanced results of Kisin on the local deformation rings at p.
- Ulrike Tillmann: Operads: from loop spaces to TQFT and back.
I will start with the origins of operad theory in the theory of loop spaces in the 1970s, explain their connection to topological quantum field theory which lead to the renaissance of operads in the 1990s, and discuss some results and applications linking the two.
- Don Zagier: Deformations of Picard-Fuchs equations, modular forms, and the holomorphic anomaly equation.