Past

I will introduce the theory of cluster categories after Amiot and Plamondon. For a quiver with a potential, the cluster category is defined as the quotient of the category of perfect dg-modules by the category of dg-modules with finite dimensional cohomologies. We can show the existence of the equivalence in the first talk as an application of the cluster category. I will also propose a definition of a counting invariant for each element in the cluster category.

The line graph operation, in which the edges of one graph are taken as the vertices of a new graph with adjacency preserved, is arguably the most interesting of graph transformations. In this survey, we will begin looking at characterisations of line graphs, focusing first on results related to our set of nine forbidden subgraphs. This will be followed by a discussion of some generalisations of line graphs, including our investigations into the Krausz dimension of a graph G, defined as the minimum, over all partitions of the edge-set of G into complete subgraphs, of the maximum number of subgraphs containing any vertex (the maximum in Krausz's characterisation of line graphs being 2).

The line graph operation, in which the edges of one graph are taken as the vertices of a new graph with adjacency preserved, is arguably the most interesting of graph transformations.  In this survey, we will begin looking at characterisations of line graphs, focusing first on results related to our set of nine forbidden subgraphs. This will be followed by a discussion of some generalisations of line graphs, including our investigations into the Krausz dimension of a graph G, defined as the minimum, over all partitions of the edge-set of G into complete subgraphs, of the maximum number of subgraphs containing any vertex (the maximum in Krausz's characterisation of line graphs being 2).

We develop a theory of optimal stopping problems under ambiguity in continuous time. Using results from (backward) stochastic calculus, we characterize the value function as the smallest (nonlinear) supermartingale dominating the payoff process. For Markovian models, we derive a Hamilton–Jacobi–Bellman equation involving a nonlinear drift term that describes the agent’s ambiguity aversion. We show how to use these general results for search problems and American Options.

Let $(Q',w')$ be a quiver with a potential given by successive mutations from a quiver with a potential $(Q,w)$. Then we have an equivalence of the derived categories of dg-modules over the Ginzburg dg-algebras satisfying the following condition: a simple module over the dg-algebra for $(Q',w')$ is either concentrated on degree 0 or concentrated on degree 1 as a dg-module over the

dg-algebra for $(Q,w)$. As an application of this equivalence, I will give a description of the space of stability conditions.

Abstract: Nonlinear models have been widely employed to characterize the

underlying structure in a time series. It has been shown that the

in-sample fit of nonlinear models is better than linear models, however,

the superiority of nonlinear models over linear models, from the

perspective of out-of-sample forecasting accuracy remains doubtful. We

compare forecast accuracy of nonlinear regime switching models against

classical linear models using different performance scores, such as root

mean square error (RMSE), mean absolute error (MAE), and the continuous

ranked probability score (CRPS). We propose and investigate the efficacy

of a class of simple nonparametric, nonlinear models that are based on

estimation of a few parameters, and can generate more accurate forecasts

when compared with the classical models. Also, given the importance of

gauging uncertainty in forecasts for proper risk assessment and well

informed decision making, we focus on generating and evaluating both point

and density forecasts.

Keywords: Nonlinear, Forecasting, Performance scores.

By means of a series of examples (Korteweg-de Vries equation, non-

linear stochastic heat equations and Navier-Stokes equation) we will show how it is possible to apply rough path ideas in the study of the Cauchy problem for PDEs with and without stochastic terms.

We study a generalized version of the signaling processoriginally introduced and studied by Argiento, Pemantle, Skyrms and Volkov(2009), which models how two interacting agents learn to signal each other andthus create a common language.

We show that the process asymptotically leads to the emergence of a graph ofconnections between signals and states which has the property that nosignal-state correspondance could be associated both to a synonym and aninformational bottleneck.

TBA

After hep-th/0909.0483

I have reconstructed multiple palaeoecological records from sites across the British Isles; this work has resulted in detailed time series that demonstrate changes in vegetation, herbivore density, nitrogen cycling, fire levels and air temperature across an 8,000 year time span covering the end of the last glacial period. The aim of my research is to use statistics to infer the relationships between vegetation changes and changes in the abiotic and biotic environment in which they occurred. This aim is achieved by using a model-fitting and model-selection method whereby sets of ordinary differential equations (ODE) are ‘fitted’ to the time series data via maximum likelihood estimation in order to find the model(s) that provide the closest match to the data. Many of the differential equation models that I have used in this study are well established in the theoretical ecology literature (i.e. plant – resource dynamics and plant – herbivore dynamics); however, there are no existing ODE models of fire or temperature dynamics that were appropriate for my data. For this workshop, I will present the palaeoecological data that I collected along with the models that I have chosen to work with (including my first attempt at models for fire and temperature dynamics) and I hope to get your feedback on these models and suggestions for other useful modelling methods that could be used to represent these dynamics.

We study upper bounds on topological complexity of sets definable in o-minimal structures over the reals. We suggest a new construction for approximating a large class of definable sets, including the sets defined by arbitrary Boolean combinations of equations and inequalities, by compact sets.

Those compact sets bound from above the homotopies and homologies of the approximated sets.

The construction is applicable to images under definable maps.

Based on this construction we refine the previously known upper bounds on Betti numbers of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae, and prove similar new upper bounds, individual for different Betti numbers, for their images under arbitrary continuous definable maps.

Joint work with A. Gabrielov.

If an ideal elastic spring is greatly stretched, it will develop large stresses. However, solid biological tissues are able to grow without developing such large stresses. This is because the cells within such tissues are able to lay down new fibres and remove old ones, fundamentally changing the mechanical structure of the tissue. In many ways, this is analogous to classical plasticity, where materials stretched beyond their yield point begin to flow and the unloaded state of the material changes. Unfortunately, biological tissues are not closed systems and so we are not able to use standard plasticity techniques where we require the flow to be mass conserving and energetically passive.

In this talk, a general framework will be presented for modelling the changing zero stress state of a biological tissue (or any other material). Working from the multiplicative decomposition of the deformation gradient, we show that the rate of 'desired' growth can represented using a tensor that describes both the total rate of growth and any directional biases. This can be used to give an evolution equation for the effective strain (a measure of the difference between the current state and the zero stress state). We conclude by looking at a perhaps surprising application for this theory as a method for deriving the constitutive laws of a viscoelastic fluid.

Saddle-point problems occur frequently in liquid crystal modelling. For example, they arise whenever Lagrange multipliers are used for the pointwise-unit-vector constraints in director modelling, or in both general director and order tensor models when an electric field is present that stems from a constant voltage. Furthermore, in a director model with associated constraints and Lagrange multipliers, together with a coupled electric-field interaction, a particular ''double'' saddle-point structure arises. This talk will focus on a simple example of this type and discuss appropriate numerical solution schemes.

This is joint work with Eugene C. Gartland, Jr., Department of Mathematical Sciences, Kent State University.

TBA

We will consider the monodromy action on mod 2 cohomology for SL(2) Hitchin systems. We will study Copeland's approach to the subject and use his results to compute the monodromy action on mod 2 cohomology. An interpretation of our results in terms of geometric properties of fixed points of a natural involution on the moduli space is given.

The background for the multitarget tracking problem is presented

along with a new framework for solution using the theory of random

finite sets. A range of applications are presented including

submarine tracking with active SONAR, classifying underwater entities

from audio signals and extracting cell trajectories from biological

data.

\ \ In 1939 Rademacher derived a conditionally convergent series expression for the modular j-invariant, and used this expression---the first Rademacher sum---to verify its modular invariance. We may attach Rademacher sums to other discrete groups of isometries of the hyperbolic plane, and we may ask how the automorphy of the resulting functions reflects the geometry of the group in question.

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\ \ In the case of a group that defines a genus zero quotient of the hyperbolic plane the relationship is particularly striking. On the other hand, of the common features of the groups that arise in monstrous moonshine, the genus zero property is perhaps the most elusive. We will illustrate how Rademacher sums elucidate this phenomena by using them to formulate a characterization of the discrete groups of monstrous moonshine.

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\ \ A physical interpretation of the Rademacher sums comes into view when we consider black holes in the context of three dimensional quantum gravity. This observation, together with the application of Rademacher sums to moonshine, amounts to a new connection between moonshine, number theory and physics, and furnishes applications in all three fields.

Extending work of Klyachko, we give a combinatorial description of pure equivariant sheaves on a nonsingular projective toric variety X and use this description to construct moduli spaces of such sheaves. These moduli spaces are explicit and combinatorial in nature. Subsequently, we consider the moduli space M of all Gieseker stable sheaves on X and describe its fixed point locus in terms of the moduli spaces of pure equivariant sheaves on X. As an application, we compute generating functions of Euler characteristics of M in case X is a toric surface. In the torsion free case, one finds examples of new as well as known generating functions. In the pure dimension 1 case using a conjecture of Sheldon Katz, one obtains examples of genus zero Gopakumar-Vafa invariants of the canonical bundle of X.

The notion of a boundary graph property is a relaxation of that of a

minimal property. Several fundamental results in graph theory have been obtained in

terms of identifying minimal properties. For instance, Robertson and Seymour showed that

there is a unique minimal minor-closed property with unbounded tree-width (the planar

graphs), while Balogh, Bollobás and Weinreich identified nine minimal hereditary

properties of labeled graphs with the factorial speed of growth. However, there are

situations where the notion of minimal property is not applicable. A typical example of this type

is given by graphs of large girth. It is known that for each particular value of k, the

graphs of girth at least k are of unbounded tree-width and their speed of growth is

superfactorial, while the limit property of this sequence (i.e., the acyclic graphs) has bounded

tree-width and its speed of growth is factorial. To overcome this difficulty, the notion of

boundary properties of graphs has been recently introduced. In the present talk, we use this

notion in order to identify some classes of graphs which are well-quasi-ordered with

respect to the induced subgraph relation.

Let $\phi$ be a convex, $C^1$-function and consider the functional: $$ (1)\qquad \mathcal{F}(\bf u)=\int_{\Omega} \phi (|\nabla \bf u|) \,dx $$ where $\Omega\subset \mathbb{R}^n$ is a bounded open set and $\bf u: \Omega \to \mathbb{R}^N$. The associated Euler Lagrange system is $$ -\mbox{div} (\phi' (|\nabla\bf u|)\frac{\nabla\bf u}{|\nabla\bf u|} )=0 $$ In a fundamental paper K.~Uhlenbeck proved everywhere $C^{1,\alpha}$-regularity for local minimizers of the $p$-growth functional with $p\ge 2$. Later on a large number of generalizations have been made. The case $1

{\bf Theorem.} Let $\bfu\in W^{1,\phi}_{\loc}(\Omega)$ be a local minimizer of (1), where $\phi$ satisfies suitable assumptions. Then $\bfV(\nabla \bfu)$ and $\nabla \bfu$ are locally $\alpha$-Hölder continuous for some $\alpha>0$.

We present a unified approach to the superquadratic and subquadratic $p$-growth, also considering more general functions than the powers. As an application, we prove Lipschitz regularity for local minimizers of asymptotically convex functionals in a $C^2$ sense.

We consider the time average of the (renormalized) current fluctuation field in one-dimensional weakly asymmetric simple exclusion.

The asymmetry is chosen to be weak enough such that the density fluctuation field still converges in law with respect to diffusive scaling. Remark that the density fluctuation field would evolve on a slower time scale if the asymmetry is too strong and that then the current fluctuations would have something to do with the Tracy-Widom distribution. However, the asymmetry is also chosen to be strong enough such that the density fluctuation field does not converge in law to an infinite-dimensional Ornstein-Uhlenbeck process, that is something non-trivial is happening.

We will, at first, motivate why studying the time average of the current fluctuation field helps to understand the structure of this non-trivial scaling limit of the density fluctuation field and, second, show how one can replace the current fluctuation field by a certain functional of the density fluctuation field under the time average. The latter result provides further evidence for the common belief that the scaling limit of the density fluctuation field approximates the solution of a Burgers-type equation

Abstract: The goal of this talk is to discuss threeproblems on fractional and related stochastic fields, in which wavelet methodshave turned out to be quite useful.

  The first problemconsists in constructing optimal random series representations of Lévyfractional Brownian field; by optimal we mean that the tails of the seriesconverge to zero as fast as possible i.e. at the same rate as the l-numbers.Note in passing that there are close connections between the l-numbers of aGaussian field and its small balls probabilities behavior.

  The secondproblem concerns a uniform result on the local Hölder regularity (the pointwiseHölder exponent) of multifractional Brownian motion; by uniform we mean thatthe result is satisfied on an event with probability 1 which does not depend onthe location.

  The third problemconsists in showing that multivariate multifractional Brownian motion satisfiesthe local nondeterminism property. Roughly speaking, this property, which wasintroduced by Berman, means that the increments are asymtotically independentand it allows to extend to general Gaussian fields many results on the localtimes of Brownian motion.

TBA

The viability of a market impact model is usually considered to be equivalent to the absence of price manipulation strategies in the sense of Huberman & Stanzl (2004). By analyzing a model with linear instantaneous, transient, and permanent impact components, we discover a new class of irregularities, which we call transaction-triggered price manipulation strategies. Transaction-triggered price manipulation is closely related to the non-existence of measure-valued solutions to a Fredholm integral equation of the first kind. We prove that price impact must decay as a convex decreasing function of time to exclude these market irregularities along with standard price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest. Joint work with Aurélien Alfonsi, Jim Gatheral, and Alla Slynko.

Based on hep-th/0912.3249 by Arkani-Hamed et. al..

The significance of the effects of non-healing wounds has been the topic of many research papers and lectures during the last 25 years. Efforts have been made to understand the effects of long-standing venous hypertension, diabetes, the prevalence of wounds in such conditions with as well as the difficulties faced in managing such wounds with some success. Successful efforts to define standard care regimes have also been made. However, attempts to introduce innovative therapy have been much less successful. Is this merely because we have not understood the intricacies of the problem? And would system based modelling be an untried technique?

Venous ulcers are the majority of lower extremity wounds, and a clinical challenge. A previously developed model of venous ulcers permits some understanding of why compression bandaging is successful but fails to accommodate complications such as exudate and infection. Could this experimental model be improved by system based modelling?

Chronic wounds need to be modelled however the needs for such models should be examined in order that the outcome permits advances in our thinking as well in clinical management.