Past

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust and purely deterministic translation operator in L^2-directions between models. In the concrete context of the generalized parabolic Anderson model in 2D -one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.

Work of Schoen--Yau in the 70's/80's shows that area-minimizing (actually stable) two-sided surfaces in three-manifolds of non-negative scalar curvature are of a special topological type: a sphere, torus, plane or cylinder. The torus and cylinder cases are "borderline" for this estimate. It was shown by Cai--Galloway in the late 80's that the torus can only occur in a very special ambient three manifold. We complete the story by showing that a similar result holds for the cylinder. The talk should be accessible to those with a basic knowledge of curvature in Riemannian geometry.

In this talk, I will present recent results that give the necessary mathematical foundation for the study of rough path driven PDEs in the framework of weak solutions. The main tool is a new rough Gronwall Lemma argument whose application is rather wide: among others, it allows to derive the basic energy estimates leading to the proof of existence for e.g. parabolic RPDEs. The talk is based on a joint work with Aurelien Deya, Massimiliano Gubinelli and Samy Tindel.

It was discovered in the 1970s that black holes are thermodynamic objects carrying entropy, thus suggesting that they are really an ensemble of microscopic states. This idea has been realized in a remarkable manner in string theory, wherein one can describe these ensembles in a class of models. These ensembles are known, however, to contain configurations other than isolated black holes, and it remains an outstanding problem to precisely isolate a black hole in the microscopic ensemble. I will describe how this problem can be solved completely in N=4 string theory. The solution involves surprising relations to mock modular forms -- a class of functions first discovered by S. Ramanujan about 95 years ago. 

Who are you?  What motivates you?  What's important to you?  How do you react to challenges and adversities?  In this session we will explore the power of self-awareness (understanding our own characters, values and motivations) and introduce assertiveness skills in the context of building positive and productive relationships (with colleagues, collaborators, students and others).

Accurate methods for the first-order advection equation, used for example in tracking contaminants in fluids, usually exploit the theory of characteristics. Such methods are described and contrasted with methods that do not make use of characteristics.

Then the second-order wave equation, in the form of a first-order system, is considered. A review of the one-dimensional theory using solutions of various Riemann problems will be provided. In the special case that the medium has the ‘Goupillaud’ property, that waves take the same time to travel through each layer, one can derive exact solutions even when the medium is spatially heterogeneous. The extension of this method to two-dimensional problems will then be discussed. In two-dimensions it is not apparent that exact solutions can be found, however by exploiting a generalised Goupillaud property, it is possible to calculate approximate solutions of high accuracy, perhaps sufficient to be of benchmark quality. Some two-dimensional simulations, using exact one-dimensional solutions and operator splitting, will be described and a numerical evaluation of accuracy will be given.

The possibility of a landscape of metastable vacua raises the question of what fraction of vacua are truly long lived. Naively any would-be vacuum state has many nearby decay paths, and all possible decays must be suppressed. An interesting model of this phenomena consists of N scalars with a random potential of fourth order. We show that the scaling of the typical minimal bounce action with N is readily understood. We discuss the extension to more realistic landscape models as well as the effects of gravity. 

When estimated volatilities are not in perfect agreement with reality, delta hedged option portfolios will incur a non-zero profit-and-loss over time. There is, however, a surprisingly simple formula for the resulting hedge error, which has been known since the late 90s. We call this The Fundamental Theorem of Derivative Trading. This is a survey with twists of that result. We prove a more general version and discuss various extensions (including jumps) and applications (including deriving the Dupire-Gyo ̈ngy-Derman-Kani formula). We also consider its practical consequences both in simulation experiments and on empirical data thus demonstrating the benefits of hedging with implied volatility.

In the year 2003 Yuri Ershov gave a talk at a conference in Teheran on
his notion of ``extremal valued fields''. He proved that algebraically
complete discretely valued fields are extremal. However, the proof
contained a mistake, and it turned out in 2009 through an observation by
Sergej Starchenko that Ershov's original definition leads to all
extremal fields being algebraically closed. In joint work with Salih
Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate
definition and then characterized extremal valued fields in several
important cases.

We call a valued field (K,v) extremal if for all natural numbers n and
all polynomials f in K[X_1,...,X_n], the set of values {vf(a_1,...,a_n)
| a_1,...,a_n in the valuation ring} has a maximum (which is allowed to
be infinity, attained if f has a zero in the valuation ring). This is
such a natural property of valued fields that it is in fact surprising
that it has apparently not been studied much earlier. It is also an
important property because Ershov's original statement is true under the
revised definition, which implies that in particular all Laurent Series
Fields over finite fields are extremal. As it is a deep open problem
whether these fields have a decidable elementary theory and as we are
therefore looking for complete recursive axiomatizations, it is
important to know the elementary properties of them well. That these
fields are extremal could be an important ingredient in the
determination of their structure theory, which in turn is an essential
tool in the proof of model theoretic properties.

The notion of "tame valued field" and their model theoretic properties
play a crucial role in the characterization of extremal fields. A valued
field K with separable-algebraic closure K^sep is tame if it is
henselian and the ramification field of the extension K^sep|K coincides
with the algebraic closure. Open problems in the classification of
extremal fields have recently led to new insights about elementary
equivalence of tame fields in the unequal characteristic case. This led
to a follow-up paper. Major suggestions from the referee were worked out
jointly with Sylvy Anscombe and led to stunning insights about the role
of extremal fields as ``atoms'' from which all aleph_1-saturated valued
fields are pieced together.

I will collect some results about the study of topological and algebraic invariants of this moduli space by using non-abelian Hodge theory. Some keywords are: Higgs bundles, Mixed Hodge structures.

I will discuss the notion of badly approximable points and recent progress and problems in this area, including Schmidt's conjecture, badly approximable points on manifolds and real numbers badly approximable by algebraic numbers.

The problem of optimal stopping with finite horizon in discrete time
is considered in view of maximizing the expected gain. The algorithm
presented in this talk is completely nonparametric in the sense that it
uses observed data from the past of the process up to time -n+1 (n being
a natural number), not relying on any specific model assumption. Kernel
regression estimation of conditional expectations and prediction theory
of individual sequences are used as tools.
The main result is that the algorithm is universally consistent: the
achieved expected gain converges to the optimal value for n tending to
infinity, whenever the underlying process is stationary and ergodic.
An application to exercising American options is given.

Honey poured from a sufficient height onto toast undergoes the well-known `liquid rope coiling’ instability.

We have studied this instability using a combination of laboratory experiments, theory, and numerics, with the aim of determining phase diagrams and scaling laws for the different coiling modes. Finite-amplitude coiling has four distinct modes - viscous, gravitational, inertio-gravitational, and inertial - depending on how the viscous forces that resist deformation of the rope are balanced. The inertio-gravitational mode is particularly interesting as it involves resonance between the coiling portion of the rope and its long trailing `tail’. Further experiments using less viscous fluids reveal that the rope can exhibit five different morphologies, of which steady coiling is only one. We determine the detailed phase diagram of these morphologies, which includes a novel `liquid supercoiling’

state in which the coiled cylinder formed by the primary coiling instability undergoes in turn its own complex buckling instability.  We show that the onset of these different patterns is determined by a non-penetrability condition which takes different forms in the viscous, gravitational and inertial limits. To close, we will briefly evoke two additional related phenomena: spiral waves of bubbles generated by coiling, and the `fluid mechanical sewing machine’ in which the fluid falls onto a moving belt.

The aim of this talk is to design an efficient multigrid method for constrained convex optimization problems arising from discretization  of  some  underlying  infinite  dimensional  problems. Due  to problem  dependency  of this approach, we only consider bound constraints with (possibly) a linear equality constraint. As our aim is to target large-scale problems, we want to avoid computation of second 
derivatives of the objective function, thus excluding Newton like methods. We propose a smoothing operator that only uses first-order information and study the computational efficiency of the resulting method. In the second part, we consider application of multigrid techniques to more general optimization problems, in particular, the topology design problem.

I will talk about the boundaries of CAT(0) groups giving definitions, some examples and will state some theorems. I may even prove something if there is time. 

I will talk about a remarkable theorem by Paulin, which says
that if a one-ended hyperbolic group has infinite outer automorphism
group, then it splits over a two-ended subgroup. In particular, this
gives a condition which ensures a hyperbolic group doesn't have property
(T).

We will discuss a result to the effect that the moduli space of log stable maps to a toric variety X is "the same" as the Morse-theoretic moduli space of broken gradient flow lines in the "differentiable realization" Y of the fan for X.  This is joint work with Sam Molcho.

 I will talk about Jochen’s theorem about the existence of some non-trivial Henselian valuation given by investigating the absolute Galois group.