Past

• Stephen Peppin
• Chris Prior
• Mark Flegg

Isostatic mounts are used in applications like telescopes and robotics to move and hold part of a structure in a desired pose relative to the rest, by driving some controls rather than driving the subsystem directly. To achieve this successfully requires an understanding of the coupled space of configurations and controls, and of the singularities of the mapping from the coupled space to the space of controls. It is crucial to avoid such singularities because generically they lead to large constraint forces and internal stresses which can cause distortion. In this paper we outline design principles for isostatic mount systems for dynamic structures, with particular emphasis on robots.

This is based on joint work with Dave Jorgensen. Given a Gorenstein algebra,

one can define Tate-Hochschild cohomology groups. These are defined for all

degrees, non-negative as well as negative, and they agree with the usual

Hochschild cohomology groups for all degrees larger than the injective

dimension of the algebra. We prove certain duality theorems relating the

cohomology groups in positive degree to those in negative degree, in the

case where the algebra is Frobenius (for example symmetric). We explicitly

compute all Tate-Hochschild cohomology groups for certain classes of

Frobenius algebras, namely, certain quantum complete intersections.

We develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an $\epsilon$-accurate solution with probability at least $1-\rho$ in at most $O[(2n/\epsilon)\log(1/\rho)]$ iterations, where $n$ is the dimension of the problem. For strongly convex functions the method converges linearly. This extends recent results of Nesterov [Efficiency of coordinate descent methods on huge-scale optimization problems, CORE Discussion Paper \#2010/2], which cover the smooth case, to composite minimization, while at the same time improving the complexity by the factor of 4 and removing $\epsilon$ from the logarithm. More importantly, in contrast with the aforementioned work in which the authors achieve the results by applying their method to a regularized version of the objective function with an unknown scaling factor, we show that this is not necessary, thus achieving true iteration complexity bounds. In the smooth case we also allow for arbitrary probability vectors and non-Euclidean norms. Time permitting, we will also mention new iteration complexity results for a parallel version of the method.

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In the second part of the talk we demonstrate numerically that the algorithm is able to solve huge-scale $\ell_1$-regularized support vector machine and least squares problems with billion variables. Finally, we present preliminary computational results with a GPU-accelerated parallel version of the method, on truss topology design instances, achieving speedups of up to two orders of magnitude when compared to a single-core implementation in C.

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References:

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P. Richtarik and M. Takac, Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function, 2011; http://arxiv.org/abs/1107.2848

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P. Richtarik and M. Takac, Efficient serial and parallel coordinate descent methods for huge-scale truss topology design, 2011; http://www.optimization-online.org/DB_HTML/2011/08/3118.html

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P. Richtarik and M. Takac, Efficiency of randomized coordinate descent methods on minimization problems with a composite objective function, Proceedings of SPARS11

1pm Kawei Wang

\newline Title: A Model of Behavioral Consumption in Contnuous Time

\newline Abstract: Inspired by Jin and Zhou (2008), we try to construct a model

of consumption within the framework of Prospect Theory and Cumulative

Prospect Theory in continuous time.

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1.20 Rasmus Wissmann

\newline Title: A Principal Component Analysis-based Approach for High-Dimensional PDEs in Derivative Pricing

\newline Abstract: Complex derivatives, such as multi asset and path dependent options,

often lead to high-dimensional problems. These are generally hard to

tackle with numerical PDE methods, because the computational effort

necessary increases exponentially with the number of dimensions. We

investigate a Principal Component Analysis-based approach that aims to

make the high-dimensional problem tractable by splitting it into a

number of low-dimensional ones. This is done via a diagonalization of

the PDE according to the eigenvectors of the covariance matrix and a

subsequent Taylor-like approximation. This idea was first introduced by

Reisinger and Wittum for the basic case of a vanilla option on a basket

of stocks [1]. We aim to extend the approach to more complex derivatives

and markets as well as to develop higher order versions. In this talk we

will present the basic ideas, initial results for the example of a

ratchet cap under the LIBOR Market Model and the current plans for

further research.

[1] C. Reisinger and G. Wittum, Efficient Hierarchical Approximation of

High-Dimensional Option Pricing Problems, SIAM Journal of Scientific

Computing, 2007:29

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1.40 Pedro Vitoria

\newline Title: Infinitesimal Mean-Variance and Forward Utility

\newline Abstract: Mean-Variance, introduced by Markowitz in his seminal paper of 1952, is

a classic criterion in Portfolio Theory that is still predominantly used

today in real investment practice. In the academic literature, a number of

interesting results have been produced in continuous-time version of this

model.

In my talk, I will establish a link between the multi-period

Mean-Variance model and its continuous-time limit. A key feature of the

results is that, under suitable but mild technical conditions, it

captures the results of Forward Utility, thus establishing an important

link between Mean-Variance and forward utility maximisation.

We will present a class of compact and connected homogeneous

spaces such that the Ricci flow of invariant Riemannian metrics develops

type I singularities in finite time. We will describe the singular

behaviours that we can get, as we approach the singular time, and the Ricci

soliton that we obtain by blowing up the solution near the singularity.

Finally, we will investigate the existence of ancient solutions when the

isotropy representation decomposes into two inequivalent irreducible

summands.

In topos-valid point-free topology there is a good analogue of regular measures and associated measure theoretic concepts including integration. It is expressed in terms of valuations, essentially measures restricted to the opens. A valuation $m$ is $0$ on the empty set and Scott continuous, as well as satisfying the modular law $$ m(U \cup V) + m(U \cap V) = m(U) + m(V). $$

Of course, that begs the question of why one would want to work with topos-valid point-free topology, but I'll give some general justification regarding fibrewise topology of bundles and a more specific example from recent topos work on quantum foundations.

The focus of the talk is the valuation locale, an analogue of hyperspaces: if $X$ is a point-free space (locale) then its valuation locale $VX$ is a point-free space whose points are the valuations on $X$. It was developed by Heckmann, by Coquand and Spitters, and by myself out of the probabilistic powerdomain of Jones and Plotkin.

I shall discuss the following results, proved in a draft paper "A monad of valuation locales" available at http://www.cs.bham.ac.uk/~sjv/Riesz.pdf:

• V is a strong monad, analogous to the Giry monad of measure theory.
• There is a Riesz theorem that valuations are equivalent to linear functionals on real-valued maps.
• The monad is commutative: this is a categorical way of saying that product valuations exist and there is a Fubini theorem.

The technical core is an analysis of simple maps to the reals. They can be used to approximate more general maps, and provide a means to reducing the calculations to finitary algebra. In particular the free commutative monoid $M(L)$ over a distributive lattice $L$, subject to certain relations including ones deriving from the modular law, can be got as a tensor product in a semilattice sense of $L$ with the natural numbers. It also satisfies the Principle of Inclusion and Exclusion (in a form presented without subtraction).

The Hanna Neumann Conjecture provides a bound on the rank of the intersection of finitely generated subgroups of a free group. We will follow Mineyev's recent elementary and beautiful proof of this longstanding conjecture.

Cell locomotion is essential for early development, angiogenesis, tissue regeneration, the immune response, and wound healing in multicellular organisms, and plays a very deleterious role in cancer metastasis in humans. Locomotion involves the detection and transduction of extracellular chemical and mechanical signals, integration of the signals into an intracellular signal, and the spatio-temporal control of the intracellular biochemical and mechanical responses that lead to force generation, morphological changes and directed movement. While many single-celled organisms use flagella or cilia to swim, there are two basic modes of movement used by eukaryotic cells that lack such structures -- mesenchymal and amoeboid. The former, which can be characterized as `crawling' in fibroblasts or `gliding' in keratocytes, involves the extension of finger-like filopodia or pseudopodia and/or broad flat lamellipodia, whose protrusion is driven by actin polymerization at the leading edge. This mode dominates in cells such as fibroblasts when moving on a 2D substrate. In the amoeboid mode, which does not rely on strong adhesion, cells are more rounded and employ shape changes to move -- in effect 'jostling through the crowd' or `swimming'. Here force generation relies more heavily on actin bundles and on the control of myosin contractility. Leukocytes use this mode for movement through the extracellular matrix in the absence of adhesion sites, as does Dictyostelium discoideum when cells sort in the slug. However, recent experiments have shown that numerous cell types display enormous plasticity in locomotion in that they sense the mechanical properties of their environment and adjust the balance between the modes accordingly by altering the balance between parallel signal transduction pathways. Thus pure crawling and pure swimming are the extremes on a continuum of locomotion strategies, but many cells can sense their environment and use the most efficient strategy in a given context. We will discuss some of the mathematical and computational challenges that this diversity poses.

I shall discuss recent work in which bounds are obtained, generalising/specialising earlier work for general linear groups

The induced graph removal lemma states that for any fixed graph $H$ on $h$ vertices and any $e\textgreater 0$ there exists $d\textgreater0$ such that any graph $G$ with at most $d n^h$ induced copies of $H$ may be made $H$-free by adding or removing atmost $e n^2$ edges. This fact was originally proven by Alon, Fischer, Krivelevich and Szegedy. In this talk, we discuss a new proof and itsrelation to various regularity lemmas. This is joint work with Jacob Fox.

We discuss conjectures and results concerning small gaps between primes. In particular, we consider the work of Goldston, Pintz and Yildrim which shows that infinitely often there are gaps which have size an arbitrarily small proportion of the average gap.

The Farrell-Jones Conjecture predicts a homological formula for K-and L-theory of group rings. Through surgery theory it is important for the classification of manifolds and in particular the Borel conjecture. In this talk I will give an introduction to this conjecture and give an overview about positive results and open questions.

The vacant set is the set of vertices not visited by a random walk on a graph G before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the `thermodynamic limit'

when the graph grows), and the relation of the problem to the random interlacement percolation.  I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.

Khovanov homology is an invariant of knots in S^3 which categorifies the Jones polynomial. Let C be a singular plane curve. I'll describe some conjectures relating the geometry of the Hilbert scheme of points on C to a variant of Khovanov homology which categorifies the HOMFLY-PT polynomial. These conjectures suggest a relation between HOMFLY-PT homology of torus knots and the representation theory of the rational Cherednik algebra. As a consequence, we get some easily testable predictions about the Khovanov homology of torus knots.