# TTT98: Leicester

University of Leicester.

Supported by the London Mathematical Society.

## Date

11 March 2016

## Speakers

### Scott Balchin (Leicester)

Crossed simplicial group constructions

Crossed simplicial group constructions

Abstract: Crossed simplicial groups were introduced by Loday and Fiedorowicz (and independently by Krasauskas under the name of skew-simplicial sets). These are categories which extend the simplicial category and allow us to have natural group actions on each level of the simplicial structure. One of the most recognised crossed simplicial groups is Connes cyclic category.

In this talk we will look at extending the nerve and bar constructions of groups to a general crossed simplicial group, and if time allows, discuss some possible avenues of research involving crossed simplicial groups.

### Carmelo Di Natale (Newcastle)

Hodge theory and deformations of affine cones of subcanonical projective varieties

Hodge theory and deformations of affine cones of subcanonical projective varieties

Abstract: This is a joint work with E Fatighenti and D Fiorenza. We investigate the relation between the Hodge theory of a smooth subcanonical n-dimensional projective variety X and the deformation theory of the affine cone A_X over X.

We start by identifying H^{n−1,1}_{prim}(X) as a distinguished graded component of the module of first order deformations of A_X, and later on we show how to identify the whole primitive cohomology of X as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over X.

In the particular case of a projective smooth hypersurface X we recover Griffiths' isomorphism between the primitive cohomology of X and certain distinguished graded components of the Milnor algebra of a polynomial defining X.

### Andreas Holmstrom (Ålesund)

An elementary approach to motives

An elementary approach to motives

Abstract: We explain a general strategy for constructing explicit models of Grothendieck rings of Tannakian categories. Applied to categories of motives, this gives a very explicit description of various Grothendieck rings of motives, including operations such as exterior powers, symmetric powers, Adams operations, suspensions, and Tate twists.

Somewhat surprisingly, it turns out that many deep statements about motives make sense in this completely elementary framework. This project grew out of an attempt to teach motives to high-school students in Norway.