Mon, 13 Nov 2017

15:45 - 16:45

Lie-Butcher series and rough paths on homogeneous manifolds I+II

(NTNU Trondheim)

Abstract: Butcher’s B-series is a fundamental tool in analysis of numerical integration of differential equations. In the recent years algebraic and geometric understanding of B-series has developed dramatically. The interplay between geometry, algebra and computations reveals new mathematical landscapes with remarkable properties. 

The shuffle Hopf algebra,  which is fundamental in Lyons’s groundbreaking work on rough paths,  is based on Lie algebras without additional properties.  Pre-Lie algebras and the Connes-Kreimer Hopf algebra are providing algebraic descriptions of the geometry of Euclidean spaces. This is the foundation of B-series and was used elegantly in Gubinelli’s theory of Branched Rough Paths. 
Lie-Butcher theory combines Lie series with B-series in a unified algebraic structure based on post-Lie algebras and the MKW Hopf algebra, which is giving algebraic abstractions capturing the fundamental geometrical properties of Lie groups, homogeneous spaces and Klein geometries. 

In these talks we will give an introduction to these new algebraic structures. Building upon the works of Lyons, Gubinelli and Hairer-Kelly, we will present a new theory for rough paths on homogeneous spaces built upon the MKW Hopf algebra.

Joint work with: Charles Curry and Dominique Manchon

Tue, 11 Oct 2016

Categorical matrix factorizations

Petter Bergh
(NTNU Trondheim)

We define categorical matrix factorizations in a suspended additive category, 
with respect to a central element. Such a factorization is a sequence of maps 
which is two-periodic up to suspension, and whose composition equals the 
corresponding coordinate map of the central element. When the category in 
question is that of free modules over a commutative ring, together with the 
identity suspension, then these factorizations are just the classical matrix 
factorizations. We show that the homotopy category of categorical matrix 
factorizations is triangulated, and discuss some possible future directions. 
This is joint work with Dave Jorgensen.

Thu, 17 Oct 2013

16:45 - 17:45

Coxeter groups, path algebras and preprojective algebras

Idun Reiten
(NTNU Trondheim)
To a finite connected acyclic quiver Q there is associated a path algebra kQ, for an algebraically closed field k, a Coxeter group W and a preprojective algebra. We discuss a bijection between elements of the Coxeter group W and the cofinite quotient closed subcategories of mod kQ, obtained by using the preprojective algebra. This is taken from a paper with Oppermann and Thomas. We also include a related result by Mizuno in the case when Q is Dynkin.
Thu, 25 Oct 2012

14:00 - 15:00

Generation times in certain representation theoretic triangulated categories

Johan Steen
(NTNU Trondheim)
A triangulated category admits a strong generator if, roughly speaking,

every object can be built in a globally bounded number of steps starting

from a single object and taking iterated cones. The importance of

strong generators was demonstrated by Bondal and van den Bergh, who

proved that the existence of such objects often gives you a

representability theorem for cohomological functors. The importance was

further emphasised by Rouquier, who introduced the dimension of

triangulated categories, and tied this numerical invariant to the

representation dimension. In this talk I will discuss the generation

time for strong generators (the least number of cones required to build

every object in the category) and a refinement of the dimension which is

due to Orlov: the set of all integers that occur as a generation time.

After introducing the necessary terminology, I will focus on categories

occurring in representation theory and explain how to compute this

invariant for the bounded derived category of the path algebras of type

A and D, as well as the corresponding cluster categories.

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