Past

We shall report on the use of algebraic geometry for the calculation of Feynman amplitudes (work of Bloch, Brown, Esnault and Kreimer). Or how to combine Grothendieck's motives with high energy physics in an unexpected way, radically distinct from string theory.

TBA

Heike Gramberg - Flagellar beating in trypanosomes

Robert Whittaker - High-Frequency Self-Excited Oscillations in 3D Collapsible Tube Flows

Micron-sized bacteria or algae operate at very small Reynolds numbers.

In this regime, inertial effects are negligible and, hence, efficient

swimming strategies have to be different from those employed by fish

or bigger animals. Mathematically, this means that, in order to

achieve locomotion, the swimming stroke of a microorganism must break

the time-reversal symmetry of the Stokes equations. Large ensembles of

bacteria or algae can exhibit rich collective dynamics (e.g., complex

turbulent patterns, such as vortices or spirals), resulting from a

combination of physical and chemical interactions. The spatial extent

of these structures typically exceeds the size of a single organism by

several orders of magnitude. One of our current projects in the Soft

and Biological Matter Group aims at understanding how the collective

macroscopic behavior of swimming microorganisms is related to their

microscopic properties. I am going to outline theoretical and

computational approaches, and would like to discuss technical

challenges that arise when one tries to derive continuum equations for

these systems from microscopic or mesoscopic models.

There is a widespread use of mathematical tools in finance and its

importance has grown over the last two decades. In this talk we

concentrate on optimization problems in finance, in particular on

numerical aspects. In this talk, we put emphasis on the mathematical problems and aspects, whereas all the applications are connected to the pricing of derivatives and are the

outcome of a cooperation with an international finance institution.

As one example, we take an in-depth look at the problem of hedging

barrier options. We review approaches from the literature and illustrate

advantages and shortcomings. Then we rephrase the problem as an

optimization problem and point out that it leads to a semi-infinite

programming problem. We give numerical results and put them in relation

to known results from other approaches. As an extension, we consider the

robustness of this approach, since it is known that the optimality is

lost, if the market data change too much. To avoid this effect, one can

formulate a robust version of the hedging problem, again by the use of

semi-infinite programming. The numerical results presented illustrate

the robustness of this approach and its advantages.

As a further aspect, we address the calibration of models being used in

finance through optimization. This may lead to PDE-constrained

optimization problems and their solution through SQP-type or

interior-point methods. An important issue in this context are

preconditioning techniques, like preconditioning of KKT systems, a very

active research area. Another aspect is the preconditioning aspect

through the use of implicit volatilities. We also take a look at the

numerical effects of non-smooth data for certain models in derivative

pricing. Finally, we discuss how to speed up the optimization for

calibration problems by using reduced order models.

In 2008, Juhasz published the following result, which was proved using sutured Floer homology.

Let $K$ be a prime, alternating knot. Let $a$ be the leading coefficient of the Alexander polynomial of $K$. If $|a|

TBA

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.