Past

We will go through the GIT construction of the moduli space of quiver representations. Concentrating on examples (probably the cases of Hilbert schemes of points of $\mathbb{C}^{2}$ and $\mathbb{C}^{3}$) we will try to give an idea of why this methods became relevant in modern (algebraic) geometry.

No prerequisites required, experts would probably get bored.

By classical results of Thomason, the Grothendieck group of a

triangulated category classifies the triangulated subcategories. More

precisely, there is a bijective correspondence between the set of

triangulated subcategories and the set of subgroups of the Grothendieck

group. In this talk, we extend Thomason's results to "higher"

triangulated categories, namely the recently introduced n-angulated

categories. This is joint work with Marius Thaule.

We will discuss the numerical simulation of acoustic wave propagation with localized inhomogeneities. To do this we will apply a standard finite element method (FEM) in space and explicit time-stepping in time on a finite spatial domain containing the inhomogeneities. The equations in the exterior computational domain will be dealt with a time-domain boundary integral formulation discretized by the Galerkin boundary element method (BEM) in space and convolution quadrature in time.

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We will give the analysis of the proposed method, starting with the proof of a positivity preservation property of convolution quadrature as a consequence of a variant of the Herglotz theorem. Combining this result with standard energy analysis of leap-frog discretization of the interior equations will give us both stability and convergence of the method. Numerical results will also be given.

Speaker: Tigran Atoyan\\

Title: A revised approach to hedging and pricing\\

Abstract:\\

After a brief review of the classical option pricing framework, we present a motivating example on the evaluation of hedging P&L using a simplistic strategy which does very well in practice. We then present preliminary results about a relatively unknown approach called business time hedging. Some applications of the latter approach to pricing certain derivative products as well as future research directions in this topic are discussed.\\

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Speaker: Sean Ledger\\

Title: Stochastic Evolution Equations in Portfolio Credit Modelling\\

Abstract:\\

I shall present an infinite-dimension structural model for a large portfolio of credit risky assets. As the number of assets approaches infinity we obtain a limiting system with a density process. I shall outline the properties of this density process and how one can use the SPDE satisfied by this process to estimate the loss function of the portfolio. Extensions to the model shall be onsidered, including contagion effects and Lévy noise. Finally I shall present some of the numerical testing for these models.\\

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Speaker: Peter Spoida\\

Title: Robust Pricing and Hedging of the Barrier Option with a Finite Number of Intermediate Law Constraints\\

Abstract:\\

We propose a robust superhedging strategy for simple barrier options, consisting of a portfolio of calls with different maturities and a self-financing trading strategy. The superhedging strategy is derived from a pathwise inequality. We illustrate how a stochastic control ansatz can provide a good guess for finding such strategies. By constructing a worst-case model, we demonstrate that this superhedge is the cheapest possible. Our construction generalizes the Skorokhod embedding obtained by Brown, Hobson and Rogers (2001). The talk is based on joint work with Pierre Henry-Labordere, Jan Obloj and Nizar Touzi.

We establish the exact boundary controllability of nodal profile for general first order quasi linear hyperbolic systems in 1-D. And we apply the result in a tree-like network with general nonlinear boundary conditions and interface conditions. The basic principles of choosing the controls and getting the controllability are given.

Wave-particle duality is a quantum behaviour usually assumed to have no possible counterpart in classical physics. We revisited this question when we found that a droplet bouncing on a vibrated bath could become self-propelled by its coupling to the surface waves it excites. A dynamical wave-particle association is thus formed.Through several experiments we addressed the same general question. How can a localized and discrete droplet have a common dynamics with a continuous and spatially extended wave? Surprisingly several quantum-like behaviors emerge; a form of uncertainty and a form of quantization are observed. I will show that both properties are related to the "path memory" contained in the wave field. The relation of this experiment with the pilot-wave models proposed by de Broglie and by Bohm for quantum mechanics will be discussed.

I will explain a construction of a group acting on a rooted tree, related to the Grigorchuk group. Those groups have exponential growth, at least under certain circumstances. I will also show how it can be seen that any two elements fulfil a non-trivial relation, implying the absence of non-cyclic free subgroups.

I will give a brief report on some the topics discussed at the workshop "Golod-Shafarevich groups and rank gradient" that took place this August in Vienna. I will focus on results involving rank gradient.

There is a well-acknowledged analogy between mapping class
groups and lattices in higher rank groups. I will discuss to which
extent does Margulis's superrigidity hold for mapping class groups:
examples, very partial results and questions.

Counting nodal curves in linear systems $|L|$ on smooth projective surfaces $S$ is a problem with a long history. The G\"ottsche conjecture, now proved by several people, states that these counts are universal and only depend on $c_1(L)^2$, $c_1(L)\cdot c_1(S)$, $c_1(S)^2$ and $c_2(S)$. We present a quite general definition of reduced Gromov-Witten and stable pair invariants on S. The reduced stable pair theory is entirely computable. Moreover, we prove that certain reduced Gromov-Witten and stable pair invariants with many point insertions coincide and are both equal to the nodal curve counts appearing in the Göttsche conjecture. This can be seen as version of the MNOP conjecture for the canonical bundle $K_S$. This is joint work with R. P. Thomas.

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. By contrast to the previous literature, we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis and, under

minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience adjusted market depth.

When ice is raised to a temperature above its usual melting temperature
of 273 K, small cylindrical discs of water form within the bulk of the
ice. Subsequent internal melting of the ice causes these liquid discs to
grow radially outwards. However, many experimentalists have observed
that the circular interface of these discs is unstable and eventually
the liquid discs turn into beautiful shapes that resemble flowers or
snowflakes. As a result of their shape, these liquid figures are often
called liquid snowflakes. In this talk I'll discuss a simple
mathematical model of liquid snowflake formation and I'll show how a
combination of analytical and numerical methods can yield much insight
into the dynamics which govern their growth.

Given a form $F(x)$, the circle method is frequently used to provide an asymptotic for the number of representations of a fixed integer $N$ by $F(x)$. However, it can also be used to prove results of a different flavor, such as showing that almost every number (in a certain sense) has at least one representation by $F(x)$. In joint work with Roger Heath-Brown, we have recently considered a 2-dimensional version of such a problem. Given two quadratic forms $Q_1$ and $Q_2$, we ask whether almost every integer (in a certain sense) is simultaneously represented by $Q_1$ and $Q_2$. Under a modest geometric assumption, we are able to prove such a result if the forms are in $5$ variables or more. In particular, we show that any two such quadratic forms must simultaneously attain prime values infinitely often. In this seminar, we will review the circle method, introduce the idea of a Kloosterman refinement, and investigate how such "almost all" results may be proved.

Abstract: By using the Skorohod equation we derive an
iteration procedure which allows us to solve a class of reflected backward
stochastic differential equations with non-linear resistance induced by the
reflected local time. In particular, we present a new method to study the
reflected BSDE proposed first by El Karoui et al. (1997).

Abstract: In the talk we first introduce the level set equation for the evolution by mean curvature flow, explaining the main difference between the standard Euclidean case and the horizontal evolution.

Then we will introduce a stochastic representation formula for the viscosity solution of the level set equation related to the value function of suitable associated stochastic controlled ODEs which are motivated by a concept of intrinsic Brownian motion in Carnot-Caratheodory spaces.

In an incomplete continuous-time securities market with uncertainty generated by Brownian motions, we derive closed-form solutions for the equilibrium interest rate and market price of risk processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously on a finite time-interval in a money market account and a single risky security. Besides establishing the existence of an equilibrium, our main result shows that if the investors' unspanned income has stochastic countercyclical volatility, the resulting equilibrium can display both lower interest rates and higher risk premia compared to the Pareto efficient equilibrium in an otherwise identical complete market. This is joint work with Peter Ove Christensen.

• Matt Webber - ‘Stochastic neural field theory’
• Yohan Davit - ‘Multiscale modelling of deterministic problems with applications to biological tissues and porous media’
• Patricio Farrell - ‘An RBF multilevel algorithm for solving elliptic PDEs’

I shall present a very general class of virtual elements in a structure, ultraimaginaries, and analyse their model-theoretic properties.

The use of formal mathematical models in sociology started in the 1940s and attracted mathematicians such as Frank Harary in the 1950s. The idea is to take the rather intuitive ideas described in social theory and express these in formal mathematical terms. Social network analysis is probably the best known of these and it is the area which has caught the imagination of a wider audience and has been the subject of a number of popular books. We shall give a brief over view of the field of social networks and will then look at three examples which have thrown up problems of interest to the mathematical community. We first look at positional analysis techniques and give a formulation that tries to capture the notion of social role by using graph coloration. We look at algebraic structures, properties, characterizations, algorithms and applications including food webs. Our second and related example looks at core-periphery structures in social networks. Our final example relates to what the network community refer to as two-mode data and a general approach to analyzing networks of this form. In all cases we shall look at the mathematics involved and discuss some open problems and areas of research that could benefit from new approaches and insights.

The derivatives of PDE models are key ingredients in many

important algorithms of computational science. They find applications in

diverse areas such as sensitivity analysis, PDE-constrained

optimisation, continuation and bifurcation analysis, error estimation,

and generalised stability theory.

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These derivatives, computed using the so-called tangent linear and

adjoint models, have made an enormous impact in certain scientific fields

(such as aeronautics, meteorology, and oceanography). However, their use

in other areas has been hampered by the great practical

difficulty of the derivation and implementation of tangent linear and

adjoint models. In his recent book, Naumann (2011) describes the problem

of the robust automated derivation of parallel tangent linear and

adjoint models as "one of the great open problems in the field of

high-performance scientific computing''.

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In this talk, we present an elegant solution to this problem for the

common case where the original discrete forward model may be written in

variational form, and discuss some of its applications.

This talk will discuss the notion of a Nahm transform in differential geometry, as a way of relating solutions to one differential equation on a manifold, to solutions of another differential equation on a different manifold. The guiding example is the correspondence between solutions to the Bogomolny equations on $\mathbb{R}^3$ and Nahm equations on $\mathbb{R}$. We extract the key features from this example to create a general framework.

Nematic elastomers are rubbery elastic solids made of cross-linked polymeric chains with embedded nematic mesogens. Their mechanical behaviour results from the interaction of electro-optical effects typical of nematic liquid crystals with the elasticity of a rubbery matrix. We show that the geometrically linear counterpart of some compressible models for these materials can be justified via Gamma-convergence. A similar analysis on other compressible models leads to the question whether linearised elasticity can be derived from finite elasticity via Gamma-convergence under weak conditions of growth (from below) of the energy density. We answer to this question for the case of single well energy densities.

We discuss Ogden-type extensions of the energy density currently used to model nematic elastomers, which provide a suitable framework to study the stiffening response at high imposed stretches.

Finally, we present some results concerning the attainment of minimal energy for both the geometrically linear and the nonlinear model.

The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves $W^*(X)$ on a smooth algebraic variety $X$ over a finite field, computing the cristalline cohomology of $X$. I am going to present a non-commutative generalization of this: even for a non-commutative ring $A$, one can define a functorial "Hochschild-Witt complex" with homology $WHH^*(A)$; if $A$ is commutative, then $WHH^i(A)=W^i(X)$, $X = Spec A$ (this is analogous to the isomorphism $HH^i(A)=H^i(X)$ discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

I shall present a geometric property valid in many Hrushovski
amalgamation constructions, relative CM-triviality, and derive
consequences on definable groups: modulo their centre they are already
products of groups interpretable in the initial theories used for the
construction. For the bad field constructed in this way, I shall
moreover classify all interpretable groups up to isogeny.

The study of free groups and their automorphisms has a long pedigree, going back to the work of Nielsen and Dehn in the early 20th century, but in many ways the subject only truly reached maturity with the introduction of Outer Space by Culler and Vogtmann in the “Big Bang” of 1986. In this (non-expert) talk, I will walk us through the construction of Outer Space and some related complexes, and survey some group-theoretic applications.

We prove that the rank gradient vanishes for mapping class groups, Aut(Fn) for all n, Out(Fn), n > 2 and any Artin group whose underlying graph is connected. We compute the rank gradient and verify that it is equal to the first L2-Betti number for some classes of Coxeter groups.

We study the problem of covering R^d by overlapping translates of a convex polytope, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov and Minkowski, and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile R^d. By contrast, for k ≥ 2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational, centrally symmetric polytope, and if P has centrally symmetric facets, then P must k-tile R^d for some positive integer k.

I will explain an approach to study DT theory of toric CY 3-folds using $L_\infty$ algebras. Based on the construction of strong exceptional collection of line bundles on Fano toric stack of dimension two, we realize any bounded families of sheaves on local surfaces support on zero section as critical sets of the Chern-Simons functions. As a consequence of this construction, several interesting properties of DT invariants on local surfaces can be checked.

Abstract: We consider an implicit finite difference
scheme on uniform grids in time and space for the Cauchy problem for a second
order parabolic stochastic partial differential equation where the parabolicity
condition is allowed to degenerate. Such equations arise in the nonlinear
filtering theory of partially observable diffusion processes. We show that the
convergence of the spatial approximation can be accelerated to an arbitrarily
high order, under suitable regularity assumptions, by applying an extrapolation
technique.