L4

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Morita Frobenius numbers were introduced by Kessar in 2004 in the context of Donovan’s Conjecture in block theory. I will present the latest results of a project in which we aim to calculate the Morita Frobenius numbers of the blocks of quasi-simple finite groups. I will also discuss the importance of a recent result of Bonnafe-Dat-Rouquier for our methods, and explain the relationship between Morita Frobenius numbers and Donovan’s Conjecture. 

Instanton bundles were introduced by Atiyah, Drinfeld, Hitchin and Manin in the late 1970s as the holomorphic counterparts, via twistor
theory, to anti-self-dual connections (a.k.a. instantons) on the sphere S^4. We will revise some recent results regarding some of the basic
geometrical features of their moduli spaces, and on its possible degenerations. We will describe the singular loci of instanton sheaves,
and how these lead to new irreducible components of the moduli space of stable sheaves on the projective space.

We construct triangle equivalences between singularity categories of
two-dimensional cyclic quotient singularities and singularity categories of
a new class of finite dimensional local algebras, which we call Knörrer
invariant algebras. In the hypersurface case, we recover a special case of Knörrer’s equivalence for (stable) categories of matrix factorisations.
We’ll then explain how this led us to study Ringel duality for
certain (ultra strongly) quasi-hereditary algebras.
This is based on joint work with Joe Karmazyn.

I shall report on recent progress, in Australia and Germany, on the empirical evaluation of special values of L-functions by minors of period matrices whose elements include Feynman integrals from diagrams with up to 20 loops. Previously such relations were known only for diagrams with up to 6 loops.
 

The projects include melting of the paste, fluid flow, particle segregation, baking, shrinkage and material properties of baked electrode.

We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations. We also numerically compute a special class of such solutions, namely those possessing icosahedral symmetry. We overcome several difficulties along the way. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly under-determined mid-plane deformation. We then use well known group-theoretic selection techniques combined with global bifurcation methods to obtain our results.