Student talks: AustMS practice

Ivo Vekemans: The slice filtration and spectral sequence

A Postnikov tower from classical algebraic topology is built via localisation functors from a filtered sequence of localising subcategories $\tau_{\geq n}$ each generated by spheres $S^k$ for $k \geq n$. Similarly, in the equivariant setting slice towers are built via localising functors from the slice filtration of localising subcategories generated by suitable representation spheres. In this talk I will introduce the slice tower and its spectral sequence (by contrast, the spectral sequence of a Postnikov tower is boring, ie. stable on the first page).

Edmund Heng: Braid group action on triangulated categories

It is known that braid groups act on the curves of punctured disk and through this, Thurston gave a classification theory of braid groups by looking at the dynamics of this action. By a paper written by Khovanov and Seidel, these curves on the punctured disk can be viewed as objects in the homotopy category of bounded chain complexes of projective modules over the zig-zag algebra. Furthermore, the braid group acts on it just as how it acts on curves.

In this talk I’ll describe the recipe to relate the geometrical objects (curves) to the algebraic objects (complexes) and how this gives us an action on a triangulated category.

Dominic Weiller: Algebras in Braided Tensor Categories and Embedded Surfaces

Given a monoidal category, the algebra objects, their bimodules and 
bimodule intertwiners form a bicategory. If we start with a braided 
tensor category (BTC), the braiding produces a monoidal product of 
algebras making the category of algebras, bimodules and intertwiners 
into a tricategory with a single 0-morphism (also called a monoidal 
bicategory). Using the theory of surface diagrams for Gray categories 
with duals, one expects to produce an `extended TQFT of embedded 
surfaces' given a special Frobenius algebra in a BTC.

We will consider a non-extended version and see how to obtain invariants 
of embedded surfaces up to isotopy by drawing `fine enough' trivalent 
graphs on a surface and choosing a special Frobenius algebra. 
Interesting examples must come from non-symmetrically braided categories 
as algebras in symmetric tensor categories give boring invariants. For 
example, picking the $E6$ algebra in $A_{11}$ (a non-symmetric BTC) we 
can distinguish the unlinked union of two tori from two concentrically 
nested tori.