The Outer Model Programme investigates L-like forcing extensions of the universe, where we say that a model of Set Theory is L-like if it satisfies properties of Goedel's constructible universe of sets L. I will introduce the Outer Model Programme, talk about its history, motivations, recent results and applications. I will be presenting joint work with Sy Friedman and Philipp Luecke.
I discuss models for the planar spreading of a viscous fluid between an elastic lid and an underlying rigid plane. These have application to the growth of magmatic intrusions, as well as to other industrial and biological processes; simple experiments of an inflated blister will be used for motivation. The height of the fluid layer is described by a sixth order non-linear diffusion equation, analogous to the fourth order equation that describes surface tension driven spreading. The dynamics depend sensitively on the conditions at the contact line, where the sheet is lifted from the substrate and where some form of regularization must be applied to the model. I will explore solutions with a pre-wetted film or a constant-pressure fluid lag, for flat and inclined planes, and compare with the analogous surface tension problems.
Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!
This talk is a basic introduction to the wonderful world of Higgs bundles on a Riemann Surface, and their moduli space. We will only survey the basics of the theory focusing on the rich geometry of the moduli space of Higgs bundles, and the relation to moduli space of vector bundles. In the end we consider small applications of Higgs bundles. As this talk will be very basic we won't go into any new developments of the theory, but just mention the areas in which Higgs bundles are used today.
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And applications to problems involving pointwise constraints on the gradient of the state on non smooth polygonal domains
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In this talk we are concerned with an analysis of Moreau-Yosida regularization of pointwise state constrained optimal control problems. As recent analysis has already revealed the convergence of the primal variables is dominated by the reduction of the feasibility violation in the maximum norm.
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We will use a new method to derive convergence of the feasibility violation in the maximum norm giving improved the known convergence rates.
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Finally we will employ these techniques to analyze optimal control problems governed by elliptic PDEs with pointwise constraints on the gradient of the state on non smooth polygonal domains. For these problems, standard analysis, fails because the control to state mapping does not yield sufficient regularity for the states to be continuously differentiable on the closure of the domain. Nonetheless, these problems are well posed. In particular, the results of the first part will be used to derive convergence rates for the primal variables of the regularized problem.
We consider a problem of maximising lifetime utility of consumption subject to a drawdown constraint on undiscounted wealth
process. This problem was solved by Elie and Touzi in the case of zero interest rate. We apply methodology of Azema-Yor processes to connect
constrained and unconstrained wealth processes, which allows us to get the results for non-zero interest rate.
The theory of equations
over groups goes back to the very beginning of group theory and is
linked to many deep problems in mathematics, such as the Diophantine
problem over rationals. In this talk, we shall survey some of the key
results on equations over groups,
give an outline of the Makanin-Razborov process (an algorithm for
solving equations over free groups) and its connections to other results
in group theory and low-dimensional topology.
We will present a regularity result for degenerate elliptic equations in nondivergence form.
In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity-- provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity.
In the Greek mythology the hydra is a many-headed poisonous beast. When cutting one of its heads off, it will grow two more. Inspired by how Hercules defeated the hydra, Dison and Riley constructed a family of groups defined by two generators and one relator, which is an Engel word: the hydra groups. I will talk about its remarkably wild subgroup distortion and its hyperbolic cousin. Very recent discussions of Baumslag and Mikhailov show that those groups are residually torsion-free nilpotent and they introduce generalised hydra groups.
When two drops come into contact they will rapidly merge and form a single drop. Here we address the coalescence of drops on a substrate, focussing on the initial dynamics just after contact. For very viscous drops we present similarity solutions for the bridge that connects the two drops, the size of which grows linearly with time. Both the dynamics and the self-similar bridge profiles are verified quantitatively by experiments. We then consider the coalescence of water drops, for which viscosity can be neglected and liquid inertia takes over. Once again, we find that experiments display a self-similar dynamics, but now the bridge size grows with a power-law $t^{2/3}$. We provide a scaling theory for this behavior, based on geometric arguments. The main result for both viscous and inertial drops is that the contact angle is important as it determines the geometry of coalescence -- yet, the contact line dynamics appears irrelevant for the early stages of coalescence.
A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover
admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I'll explain
how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in
the context of the Kobayashi-Hitchin correspondence.
This talk will give an introduction to the recent paper by Arkani Hamed et. al. arxiv:1212:5605.
Results of Bourgain and Kindler-Safra state that if $f$ is a Boolean function on $\{0,1\}^n$, and
the Fourier transform of $f$ is highly concentrated on low frequencies, then $f$ must be close
to a ‘junta’ (a function depending upon a small number of coordinates). This phenomenon is
known as ‘Fourier stability’, and has several interesting consequences in combinatorics,
theoretical computer science and social choice theory. We will describe some of these,
before turning to the analogous question for Boolean functions on the symmetric group. Here,
genuine stability does not occur; it is replaced by a weaker phenomenon, which we call
‘quasi-stability’. We use our 'quasi-stability' result to prove an isoperimetric inequality
for $S_n$ which is sharp for sets of size $(n-t)!$, when $n$ is large. Several open questions
remain. Joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Weizmann
Institute).
Let ${T_1,...,T_l}$ be a collection of differential operators
with constant coefficients on the torus $\mathbb{T}^n$. Consider the
Banach space $X$ of functions $f$ on the torus for which all functions
$T_j f$, $j=1,...,l$, are continuous. The embeddability of $X$ into some
space $C(K)$ as a complemented subspace will be discussed. The main result
is as follows. Fix some pattern of mixed homogeneity and extract the
senior homogeneous parts (relative to the pattern chosen)
${\tau_1,...,\tau_l}$ from the initial operators ${T_1,...,T_l}$. If there
are two nonproportional operators among the $\tau_j$ (for at least one
homogeneity pattern), then $X$ is not isomorphic to a complemented
subspace of $C(K)$ for any compact space $K$.
The main ingredient of the proof is a new Sobolev-type embedding
theorem. It generalises the classical embedding of
${\stackrel{\circ}{W}}_1^1(\mathbb{R}^2)$ to $L^2(\mathbb{R}^2)$. The difference is that
now the integrability condition is imposed on certain linear combinations
of derivatives of different order of several functions rather than on the
first order derivatives of one function.
This is a joint work with D. Maksimov and D. Stolyarov.
Abstract: When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.
We study portfolio optimization and contingent claim valuation in markets where illiquidity may affect the transfer of wealth over time and between investment classes. In addition to classical frictionless markets and markets with transaction costs, our model covers nonlinear illiquidity effects that arise in limit order markets. We extend basic results on arbitrage bounds, attainable claims and optimal portfolios to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of portfolio optimization under conditions that extend the no-arbitrage condition in the classical linear market model.
The operation of sub-cellular processes in living organisms often require the transfer of biopolymers across impermeable lipid membranes. The emergence of new experimental techniques for manipulation of single molecules at nanometer scales have made possible in vitro experiments that can directly probe such translocation processes in cells as well as in synthetic systems. Some of these ideas have spawned novel bio-technologies with many more likely to emerge in the near future. In this talk I would review some of these experiments and attempt to provide a quantitative understanding of the data in terms of physical laws, primarily mechanics and electrostatics.