Past

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.

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

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. 
 

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.

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. 
 

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

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.

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.

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?

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.

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.
 

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.

CANCELLED

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.

I will describe a method to find negatively curved structures on some groups, by manipulating metrics on piecewise hyperbolic complexes. As an example, I will prove that hyperbolic limit groups are CAT(-1).

Torsion in homology are invariants that have received increasing attention over the last twenty years, by the work of Lück, Bergeron, Venkatesh and others. While there are various vanishing results, no one has found a finitely presented group where the torsion in the first homology is exponential over a normal chain with trivial intersection. On the other hand, conjecturally, every 3-manifold group should be an example.

A group is right angled if it can be generated by a list of infinite order elements, such that every element commutes with its neighbors. Many lattices in higher rank Lie groups (like SL(n,Z), n>2) are right angled. We prove that for a right angled group, the torsion in the first homology has subexponential growth for any Farber sequence of subgroups, in particular, any chain of normal subgroups with trivial intersection. We also exhibit right angled cocompact lattices in SL(n,R) (n>2), for which the Congruence Subgroup Property is not known. This is joint work with Nik Nikolov and Tsachik Gelander.

Efficient factorization or efficient computation of class 
numbers would both suffice to break RSA.  However the talk lies more in 
computational number theory rather than in cryptography proper. We will 
address two questions: (1) How quickly can one construct a factor table 
for the numbers up to x?, and (2) How quickly can one do the same for the 
class numbers (of imaginary quadratic fields)? Somewhat surprisingly, the 
approach we describe for the second problem is motivated by the classical 
Hardy-Littlewood method.