We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropical geometry).
The Green-Griffiths conjecture from 1979 says that every projective algebraic variety $X$ of general type contains a certain proper algebraic subvariety $Y$ such that all nonconstant entire holomorphic curves in $X$ must lie inside $Y$. In this talk we explain that for projective hypersurfaces of degree $d>dim(X)^6$ this is the consequence of a positivity conjecture in global singularity theory.
A graph is $\chi$-bounded with a function $f$ is for all induced subgraph H of G, we have $\chi(H) \le f(\omega(H))$. Here, $\chi(H)$ denotes the chromatic number of $H$, and $\omega(H)$ the size of a largest clique in $H$. We will survey several results saying that excluding various kinds of induced subgraphs implies $\chi$-boundedness. More precisely, let $L$ be a set of graphs. If a $C$ is the class of all graphs that do not any induced subgraph isomorphic to a member of $L$, is it true that there is a function $f$ that $\chi$-bounds all graphs from $C$? For some lists $L$, the answer is yes, for others, it is no.
A decomposition theorems is a theorem saying that all graphs from a given class are either "basic" (very simple), or can be partitioned into parts with interesting relationship. We will discuss whether proving decomposition theorems is an efficient method to prove $\chi$-boundedness.
The main aim is to incorporate the nonlinear term into non-Markovian Master equations for a continuous time random walk (CTRW) with non-exponential waiting time distributions. We derive new nonlinear evolution equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems of KPP-type.
We find an explicit expression for the speed of a propagating front in the case of subdiffusive transport.
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.
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.
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.