Abstract: In this talk I will discuss the interplay between the local and

the global invariants in modular representation theory with a focus on the

first Hochschild cohomology $\mathrm{HH}^1(B)$ of a block algebra $B$. In

particular, I will show the compatibility between $r$-integrable

derivations

and stable equivalences of Morita type. I will also show that if

$\mathrm{HH}^1(B)$ is a simple Lie algebra such that $B$ has a unique

isomorphism class of simple modules, then $B$ is nilpotent with an

elementary abelian defect group $P$ of order at least 3. The second part

is joint work with M. Linckelmann.

# Past Forthcoming Seminars

Abstract: I will describe how the ADE preprojective algebras appear in

certain Conformal Field Theories, namely SU(2) WZW models, and explain

the generalisation to the SU(3) case, where 'almost CY3' algebras appear.

Abstract: In this talk, we will introduce new affine algebraic varieties

for algebras given by quiver and relations. Each variety contains a

distinguished element in the form of a monomial algebra. The properties

and characteristics of this monomial algebra govern those of all other

algebras in the variety. We will show how amongst other things this gives

rise to a new way to determine whether an algebra is quasi-hereditary.

This is a report on joint work both with Ed Green and with Ed Green and

Lutz Hille.

Abstract: This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.

The classical McKay correspondence relates the geometry of so-called

Kleinian surface singularities with the representation theory of finite

subgroups of SL(2,C). M. Auslander observed an algebraic version of this

correspondence: let G be a finite subgroup of SL(2,K) for a field K whose

characteristic does not divide the order of G. The group acts linearly on

the polynomial ring S=K[x,y] and then the so-called skew group algebra

A=G*S can be seen as an incarnation of the correspondence. In particular

A is isomorphic to the endomorphism ring of S over the corresponding

Kleinian surface singularity.

Our goal is to establish an analogous result when G in GL(n,K) is a finite

subgroup generated by reflections, assuming that the characteristic

of K does not divide the order of the group. Therefore we will consider a

quotient of the skew group ring A=S*G, where S is the polynomial ring in n

variables. We show that our construction yelds a generalization of

Auslander's result, and moreover, a noncommutative resolution of the

discriminant of the reflection group G.

Abstract: Joint work with Carlson, Grodal, Nakano. In this talk we will

present some recent results on an 'important' class of modular

representations for an 'important' class of finite groups. For the

convenience of the audience, we'll briefly review the notion of an

endotrivial module and present the main results pertaining endotrivial

modules and finite reductive groups which we use in our ongoing work.

I shall describe recent work with Srikanth Iyengar, Henning

Krause and Julia Pevtsova on the representation theory and cohomology

of finite group schemes and finite supergroup schemes. Particular emphasis

will be placed on the role of generic points, detection of projectivity

for modules, and detection modulo nilpotents for cohomology.

Topological field theories (TFT's) are physical theories depending only on the topological properties of spacetime as opposed to also depending on the metric of spacetime. This talk will introduce topological field theories, and the work of Freed and Hopkins on how a class of TFT's called "invertible" TFT's describe certain states of matter, and are classified by maps of spectra. Constructions of field theories corresponding to specific maps of spectra will be described.

This half-day research workshop will address issues at the intersection between network science, matrix theory and mathematical physics.

Network science is producing a wide range of challenging research problems that have diverse applications across science and engineering. It is natural to cast these research challenges in terms of matrix function theory. However, in many cases, closely related problems have been tackled by researchers working in statistical physics, notably quantum mechanics on graphs and quantum chaos. This workshop will discuss recent progress that has been made in both fields and highlight opportunities for cross-fertilization. While focusing on mathematical, physical and computational issues, some results will also be presented for real data sets of relevance to practitioners in network science.

Fibrillation is a chaotic, turbulent state for the electrical signal fronts in the heart. In the ventricle it is fatal if not treated promptly. The standard treatment is by an electrical shock to reset the cardiac state to a normal one and allow resumption of a normal heart beat.

The fibrillation wave fronts are organized into scroll waves, more or less analogous to a vortex tube in fluid turbulence. The centerline of this 3D rotating object is called a filament, and it is the organizing center of the scroll wave.

The electrical shock, when turned on or off, creates charges at the conductivity discontinuities of the cardiac tissue. These charges are called virtual electrodes. They charge the region near the discontinuity, and give rise to wave fronts that grow through the heart, to effect the defibrillation. There are many theories, or proposed mechanisms, to specify the details of this process. The main experimental data is through signals on the outer surface of the heart, so that simulations are important to attempt to reconstruct the electrical dynamics within the interior of the heart tissue. The primary electrical conduction discontinuities are at the cardiac surface. Secondary discontinuities, and the source of some differences of opinion, are conduction discontinuities at blood vessel walls.

In this lecture, we will present causal mechanisms for the success of the virtual electrodes, partially overlapping, together with simulation and biological evidence for or against some of these.

The role of small blood vessels has been one area of disagreement. To assess the role of small blood vessels accurately, many details of the modeling have been emphasized, including the thickness and electrical properties of the blood vessel walls, the accuracy of the biological data on the vessels, and their distribution though the heart. While all of these factors do contribute to the answer, our main conclusion is that the concentration of the blood vessels on the exterior surface of the heart and their relative wide separation within the interior of the heart is the factor most strongly limiting the significant participation of small blood vessels in the defibrillation process.

Since the pioneering work of Hodgkin and Huxley , we know that electrical signals propagate along a nerve fiber via ions that flow in and out of the fiber, generating a current. The voltages these currents generate are subject to a diffusion equation, which is a reduced form of the Maxwell equation. The result is a reaction (electrical currents specified by an ODE) coupled to a diffusion equation, hence the term reaction diffusion equation.

The heart is composed of nerve fibers, wound in an ascending spiral fashion along the heart chamber. Modeling not individual nerve fibers, but many within a single mesh block, leads to partial differential equation coupled to the reaction ODE.

As with the nerve fiber equation, these cardiac electrical equations allow a propagating wave front, which normally moves from the bottom to the top of the heart, giving rise to contractions and a normal heart beat, to accomplish the pumping of blood.

The equations are only borderline stable and also allow a chaotic, turbulent type wave front motion called fibrillation.

In this lecture, we will explain the 1D traveling wave solution, the 3D normal wave front motion and the chaotic state.

The chaotic state is easiest to understand in 2D, where it consists of spiral waves rotating about a center. The 3D version of this wave motion is called a scroll wave, resembling a fluid vortex tube.

In simplified models of reaction diffusion equations, we can explain much of this phenomena in an analytically understandable fashion, as a sequence of period doubling transitions along the path to chaos, reminiscent of the laminar to turbulent transition.