L3

In the theory of dislocations one is naturally led to consider energies of “line tension” type concentrated on lines. These lines may have a local vector-valued multiplicity, and the energy may depend on this multiplicity and on the orientation of the line. In the two-dimensional case this problem reduces to the classical problem of energies defined on partitions which arises in the sharp-interface models for phase transitions. 

I will introduce the main results concerning functionals in the calculus of variations that are defined on partitions. Such partitions are nicely characterized as level sets of function with bounded variations with a discrete set of values.  In this setting I will recall the characterization of the lower semicontinuity and the relaxation formula, which gives rise to the notion of BV-ellipticity. The case of dislocations in a three-dimensional crystal requires a formulation in the setting of 1-rectifiable currents with multiplicity in a lattice. In this context I will describe the main results and some examples of interest, in which relaxation is necessary and can be characterized.

One of the things sustainability applications have in common with industrial applications is their close connection with decision-making and policy. We will discuss how a decision-support viewpoint may inspire new mathematical questions. For example, the concept of resilience (of ecosystems, food systems, communities, economies, etc) is often described as the capacity of a system to withstand disturbance and retain its functional characteristics. This has several familiar mathematical interpretations, probing the interaction between transient dynamics and noise. How does a focus on resilience change the modeling, dynamical and policy questions we ask? I look forward to your ideas and discussion.

The talk is on impacts, penetrations and lift-offs involving bodies and fluids, with applications that range from aircraft and ship safety and our tiny everyday scales of splashing and washing, up to surface movements on Mars. Several studies over recent years have addressed different aspects of air-water effects and fluid-body interplay theoretically. Nonlinear interactions and evolutions are key here and these are to be considered in the presentation. Connections with experiments will also be described.

Molecular modelling has become a valuable tool and is increasingly part of the standard methodology of chemistry, physics, engineering and biology. The power of molecular modelling lies in its versatility: as potential energy functions improve, a broader range of more complex phenomena become accessible to simulation, but much of the underlying methodology can be re-used. For example, the Verlet method is still the most popular molecular dynamics scheme for constant energy molecular dynamics simulations despite it being one of the first to be proposed for the purpose.

One of the most important challenges in molecular modelling remains the computation of averages with respect to the canonical Gibbs (constant temperature) distribution, for which the Verlet method is not appropriate. Whereas constant energy molecular dynamics prescribes a set of equations (Newton's equations), there are many alternatives for canonical sampling with quite different properties. The challenge is therefore to identify formulations and numerical methods that are robust and maximally efficient in the computational setting.

One of the simplest and most effective methods for sampling is based on Langevin dynamics which mimics coupling to a heat bath by the incorporation of random forces and an associated dissipative term. Schemes for Langevin dynamics simulation may be developed based on the familiar principle of splitting. I will show that the invariant measure ('long term') approximation may be strongly affected by a simple re-ordering of the terms of the splitting. I will describe a transition in weak numerical order of accuracy that occurs (in one case) in the t->infty limit.

I will also entertain some more radical suggestions for canonical sampling, including stochastic isokinetic methods that enable the use of greatly enlarged timesteps for expensive but slowly-varying force field components.

Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. I will present a novel methodology which allows us to quantify multi-modal gene expression distributions and single cell power spectra in gene regulatory networks. The method is based on an extension of the linear noise approximation; in particular we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. I will demonstrate the applicability of our approach to several examples and discuss some new dynamical characteristics e.g., how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks and how genetic oscillators can display concerted noise-induced bimodality and noise-induced oscillations.

In Parkinson’s disease, increased power of oscillations in firing rate has been observed throughout the cortico-basal-ganglia circuit. In

particular, the excessive oscillations in the beta range (13-30Hz) have been shown to be associated with difficulty of movement initiation. However, on the basis of experimental data alone it is difficult to determine where these oscillations are generated, due to complex and recurrent structure of the cortico-basal-ganglia-thalamic circuit. This talk will describe a mathematical model of a subset of basal-ganglia that is able to reproduce experimentally observed patterns of activity. The analysis of the model suggests where and under which conditions the beta oscillations are produced.

Nesetril and Ossona de Mendez introduced a new notion of convergence of graphs called FO convergence. This notion can be viewed as a unified notion of convergence of dense and sparse graphs. In particular, every FO convergent sequence of graphs is convergent in the sense of left convergence of dense graphs as studied by Borgs, Chayes, Lovasz, Sos, Szegedy, Vesztergombi and others, and every FO convergent sequence of graphs with bounded maximum degree is convergent in the Benjamini-Schramm sense.

FO convergent sequences of graphs can be associated with a limit object called modeling. Nesetril and Ossona de Mendez showed that every FO convergent sequence of trees with bounded depth has a modeling. We extend this result

to all FO convergent sequences of trees and discuss possibilities for further extensions.

The talk is based on a joint work with Martin Kupec and Vojtech Tuma.

The Tutte polynomial of a graph $G$ is a two-variable polynomial $T(G;x,y)$, which encodes much information about~$G$. The number of spanning trees in~$G$, the number of acyclic orientations of~$G$, and the partition function of the $q$-state Potts model are all specialisations of the Tutte polynomial. Jackson and Sokal have studied the sign of the Tutte polynomial, and identified regions in the $(x,y)$-plane where it is ``essentially determined'', in the sense that the sign is a function of very simple characteristics of $G$, e.g., the number of vertices and connected components of~$G$. It is natural to ask whether the sign of the Tutte polynomial is hard to compute outside of the regions where it is essentially determined. We show that the answer to this question is often an emphatic ``yes'': specifically, that determining the sign is \#P-hard. In such cases, approximating the Tutte polynomial with small relative error is also \#P-hard, since in particular the sign must be determined. In the other direction, we can ask whether the Tutte polynomial is easy to approximate in regions where the sign is essentially determined. The answer is not straightforward, but there is evidence that it often ``no''. This is joint work with Leslie Ann Goldberg (Oxford).

Let $A$ be a finite-dimensional $K$-algebra over an algebraically closed field $K$. Denote by $\Omega_A$ the syzygy operator on the category $\mod A$ of finite-dimensional right $A$-modules, which assigns to a module $M$ in $\mod A$ the kernel $\Omega_A(M)$ of a minimal projective cover $P_A(M) \to M$ of $M$ in $\mod A$. A module $M$ in $\mod A$ is said to be periodic if $\Omega_A^n(M) \cong M$ for some $n \geq 1$. Then $A$ is said to be a periodic algebra if $A$ is periodic in the module category $\mod A^e$ of the enveloping algebra $A^e = A^{\op} \otimes_K A$. The periodic algebras $A$ are self-injective and their module categories $\mod A$ are periodic (all modules in $\mod A$ without projective direct summands are periodic). The periodicity of an algebra $A$ is related with periodicity of its Hochschild cohomology algebra $HH^{*}(A)$ and is invariant under equivalences of the derived categories $D^b(\mod A)$ of bounded complexes over $\mod A$. One of the exciting open problems in the representation theory of self-injective algebras is to determine the Morita equivalence classes of periodic algebras.

We will present the current stage of the solution of this problem and exhibit prominent classes of periodic algebras.

We investigate symmetric quotient algebras of symmetric algebras,

with an emphasis on finite group algebras over a complete discrete

valuation ring R with residue field of positive characteristic p. Using elementary methods, we show that if an

ordinary irreducible character of a finite group gives

rise to a symmetric quotient over R which is not a matrix algebra,

then the decomposition numbers of the row labelled by the character are

all divisible by p. In a different direction, we show that if is P is a finite

p-group with a cyclic normal subgroup of index p, then every ordinary irreducible character of P gives rise to a

symmetric quotient of RP. This is joint work with Shigeo Koshitani and Markus Linckelmann.