14:00
14:00
Complete-market stochastic volatility models (Joint seminar with OMI)
Abstract
securities. While this is easy to show in a finite-state setting, getting a satisfactory theory in
continuous time has proved highly problematic. The goal is however worth pursuing since it would
provide arbitrage-free dynamic models for the whole volatility surface. In this talk we describe an
approach in which all prices in the market are functions of some underlying Markov factor process.
In this setting general conditions for market completeness were given in earlier work with J.Obloj,
but checking them in specific instances is not easy. We argue that Wishart processes are good
candidates for modelling the factor process, combining efficient computational methods with an
adequate correlation structure.
Hausdorff dimension and complexity of Kleinian groups
Abstract
In this talk I'll give a general presentation of my recent work that a purely loxodromic Kleinian group of Hausdorff dimension<1 is a classical Schottky group. This gives a complete classification of all Kleinian groups of dimension<1. The proof uses my earlier result on the classification of Kleinian groups of sufficiently small Hausdorff dimension. This result in conjunction to another work (joint with Anderson) provides a resolution to Bers uniformization conjecture. No prior knowledge on the subject is assumed.
A backward stochastic differential equation approach to singular stochastic control
Abstract
Singular stochastic control problems ae largely studied in literature.The standard approach is to study the associated Hamilton-Jacobi-Bellman equation (with gradient constraint). In this work, we use a different approach (BSDE:Backward stochastic differntial equation approach) to show that the optimal value is a solution to BSDE.
The advantage of our approach is that we can study this kind of singular stochastic control with path-dependent coefficients
14:15
Obstructions to positive scalar curvature via submanifolds of different codimension
Abstract
Question: Given a smooth compact manifold $M$ without boundary, does $M$
admit a Riemannian metric of positive scalar curvature?
We focus on the case of spin manifolds. The spin structure, together with a
chosen Riemannian metric, allows to construct a specific geometric
differential operator, called Dirac operator. If the metric has positive
scalar curvature, then 0 is not in the spectrum of this operator; this in
turn implies that a topological invariant, the index, vanishes.
We use a refined version, acting on sections of a bundle of modules over a
$C^*$-algebra; and then the index takes values in the K-theory of this
algebra. This index is the image under the Baum-Connes assembly map of a
topological object, the K-theoretic fundamental class.
The talk will present results of the following type:
If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has
non-trivial index, what conditions imply that $M$ does not admit a metric of
positive scalar curvature? How is this related to the Baum-Connes assembly
map?
We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$),
Engel and new generalizations. Moreover, we will show how these results fit
in the context of the Baum-Connes assembly maps for the manifold and the
submanifold.
Well-posedness and regularizing properties of stochastic Hamilton-Jacobi equations
Abstract
We consider fully nonlinear parabolic equations of the form $du = F(t,x,u,Du,D^2 u) dt + H(x,Du) \circ dB_t,$ which can be made sense of by the Lions-Souganidis theory of stochastic viscosity solutions. I will first recall the ideas of this theory, and will discuss more recent developments (including the use of rough path theory in this context). In the second part of my talk, I will explain how in the case where $H(x,Du)=|Du|^2$, the solution $u$ may enjoy better regularity properties than the solution to the unperturbed equation, which can be measured by (a pair of) solutions to a reflected SDE. Based on joint works with P. Friz, B. Gess, P.L. Lions and P. Souganidis.
Obstructions to positive scalar curvature via submanifolds of different codimension
Abstract
We want to discuss a collection of results around the following Question: Given a smooth compact manifold $M$ without boundary, does $M$ admit a Riemannian metric of positive scalar curvature?
We focus on the case of spin manifolds. The spin structure, together with a chosen Riemannian metric, allows to construct a specific geometric differential operator, called Dirac operator. If the metric has positive scalar curvature, then 0 is not in the spectrum of this operator; this in turn implies that a topological invariant, the index, vanishes.
We use a refined version, acting on sections of a bundle of modules over a $C^*$-algebra; and then the index takes values in the K-theory of this algebra. This index is the image under the Baum-Connes assembly map of a topological object, the K-theoretic fundamental class.
The talk will present results of the following type:
If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has non-trivial index, what conditions imply that $M$ does not admit a metric of positive scalar curvature? How is this related to the Baum-Connes assembly map?
We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$), Engel and new generalizations. Moreover, we will show how these results fit in the context of the Baum-Connes assembly maps for the manifold and the submanifold.
Black Holes and Higher Derivative Gravity
Abstract
Eigenvectors of Tensors
Abstract
Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems. We present an introduction that emphasizes algebraic and geometric aspects
14:15
The Weak Constraint Formulation of Bayesian Inverse Problems
Abstract
Inverse problems arise in many applications. One could solve them by adopting a Bayesian framework, to account for uncertainty which arises from our observations. The solution of an inverse problem is given by a probability distribution. Usually, efficient methods at hand to extract information from this probability distribution involves the solution of an optimization problem, where the objective function is highly nonconvex. In this talk, we explore a reformulation of inverse problems, which helps in convexifying the objective function. We also discuss a method to sample from this probability distribution.
The impact of geometry and diffusion barriers in cell polarity and receptor dynamics
The de Rham algebra of a point in affine space
Abstract
Following the notes and an article of B. Bhatt, we shall compute the de Rham algebra of the immersion of the zero-section of affine space over Z/p^nZ.
This talk is part of the workshop on Beilinson's approach to p-adic Hodge theory.
Unanticipated interaction loops involving autonomous systems
Abstract
We are entering a world where unmanned vehicles will be common. They have the potential to dramatically decrease the cost of services whilst simultaneously increasing the safety record of whole industries.
Autonomous technologies will, by their very nature, shift decision making responsibility from individual humans to technology systems. The 2010 Flash Crash showed how such systems can create rare (but not inconceivably rare) and highly destructive positive feedback loops which can severely disrupt a sector.
In the case of Unmanned Air Systems (UAS), how might similar effects obstruct the development of the Commercial UAS industry? Is it conceivable that, like the high frequency trading industry at the heart of the Flash Crash, the algorithms we provide UAS to enable autonomy could decrease the risk of small incidents whilst increasing the risk of severe accidents? And if so, what is the relationship between probability and consequence of incidents?
10:00
(Strongly) quasihereditary algebras
Abstract
Quasihereditary algebras are the 'finite' version of a highest weight category, and they classically occur as blocks of the category O and as Schur algebras.
They also occur as endomorphism algebras associated to modules endowed with special filtrations. The quasihereditary algebras produced in these cases are very often strongly quasihereditary (i.e. their standard modules have projective dimension at most 1).
In this talk I will define (strongly) quasihereditary algebras, give some motivation for their study, and mention some nice strongly quasihereditary algebras found in nature.
17:30
Analytic properties of zeta functions and model theory
Abstract
A hyperkähler metric on the cotangent bundle of a complex reductive group
Abstract
Abstract: A hyperkähler manifold is a Riemannian manifold $(M,g)$ with three complex structures $I,J,K$ satisfying the quaternion relations, i.e. $I^2=J^2=K^2=IJK=-1$, and such that $(M,g)$ is Kähler with respect to each of them. I will describe a construction due to Kronheimer which gives such a structure on the cotangent bundle of any complex reductive group.
16:00
The Hasse norm principle for abelian extensions
Abstract
Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^*$ and $K^*$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to J_K$ restricts to the usual field norm $N: L^*\to K^*$ on $L^*$. Thus, if an element of $K^*$ is a norm from $L^*$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^*$ which is a norm from $J_L$ is in fact a norm from $L^*$.
The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.
This is joint work with Christopher Frei and Daniel Loughran.
The spreading of a surfactant-laden drop down an inclined and pre-wetted substrate - Numerics, Asymptotics and Linear Stability Analysis
Abstract
Surfactants are chemicals that adsorb onto the air-liquid interface and lower the surface tension there. Non-uniformities in surfactant concentration result in surface tension gradients leading to a surface shear stress, known as a Marangoni stress. This stress, if sufficiently large, can influence the flow at the interface.
Surfactants are ubiquitous in many aspects of technology and industry to control the wetting properties of liquids due to their ability to modify surface tension. They are used in detergents, crop spraying, coating processes and oil recovery. Surfactants also occur naturally, for example in the mammalian lung. They reduce the surface tension within the liquid lining the airways, which assists in preventing the collapse of the smaller airways. In the lungs of premature infants, the quantity of surfactant produced is insufficient as the lungs are under- developed. This leads to a respiratory distress syndrome which is treated by Surfactant Replacement Therapy.
Motivated by this medical application, we theoretically investigate a model problem involving the spreading of a drop laden with an insoluble surfactant down an inclined and pre-wetted substrate. Our focus is in understanding the mechanisms behind a “fingering” instability observed experimentally during the spreading process. High-resolution numerics reveal a multi-region asymptotic wave-like structure of the spreading droplet. Approximate solutions for each region is then derived using asymptotic analysis. In particular, a quasi-steady similarity solution is obtained for the leading edge of the droplet. A linear stability analysis of this region shows that the base state is linearly unstable to long-wavelength perturbations. The Marangoni effect is shown to be the dominant driving mechanism behind this instability at small wavenumbers. A small wavenumber stability criterion is derived and it's implication on the onset of the fingering instability will be discussed.
CUR Matrix Factorizations: Algorithms, Analysis, Applications
Abstract
12:00
Regularity Theory for Symmetric-Convex Functionals of Linear Growth
Abstract
11:00