Past

Convolution is widely-used and fundamental mathematical operation
in signal processing, statistics, and PDE theory.

Unfortunately the CONV() method in Chebfun for convolving two chebfun 
objects has long been one of the most disappointingly slow features of 
the project. In this talk we will present a new algorithm, which shows 
performance gains on the order of a factor 100.

The key components of the new algorithm are:
* a convolution theorem for Legendre polynomials 
* recurrence relations satisfied by spherical Bessel functions
* recent developments in fast Chebyshev-Legendre transforms [1]

Time-permitting, we shall end with an application from statistics,
using the fact that the probability distribution of the sum of two 
independent random variables is the convolution of their individual 
distributions.

[1] N. Hale and A. Townsend, "A fast, simple, and stable Chebyshev-
Legendre transform using an asymptotic formula”, SISC (to appear).

The accurate and efficient solution of linear systems $Ax=b$ is very important in many engineering and technological applications, and systems of this form also arise as subproblems within other algorithms. In particular, this is true for interior point methods (IPM), where the Newton system must be solved to find the search direction at each iteration. Solving this system is a computational bottleneck of an IPM, and in this talk I will explain how preconditioning and deflation techniques can be used, to lessen this computational burden.  This work is joint with Jacek Gondzio.

The finance literature documents a relation between labor income and

the cross-section of stock returns. One possible explanation for this

is the hedging decisions of investors with relative wealth concerns.

This implies a negative risk premium associated with stock returns

correlated with local undiversifiable wealth, since investors are

willing to pay more for stocks that help their hedging goals. We find

evidence that is consistent with these regularities. In addition, we

show that the effect varies across geographic areas depending on the

size and variability of undiversifiable wealth, proxied by labor income.

This is another opportunity to hear the 2013 LMS Presidential Address:

Abstract: The idea of space is central to the way we think.  It is the technology we have evolved for interpreting our experience of the world.  But space is presumably a human creation, and even inside mathematics it plays a variety of different roles, some modelling our intuition very closely and some seeming almost magical.  I shall point out how the homotopy category in particular breaks away from its own roots.  Then I shall describe how quantum theory leads us beyond the well-established notion of a topological space into the realm of noncommutative geometry.  One might think that noncommutative spaces are not very space-like, and yet it is noncommutativity that makes the world look as it does to us, as a collection of point particles.

In the first part of the talk we are going to build up some intuition about limit-periodic functions and I will explain why they are the 'simplest' class of arithmetic functions appearing in analytic number theory. In the second part, I will give an equivalent description of 'limit-periodicity' by using exponential sums and explain how this property allows us to solve 'twin-prime'-like problems by the circle method.

Liquid Crystals (LC), anisotropic fluids that combine many tensor properties of crystalline solids with the fluidity of liquids, have long been providing major challenges to theorists and molecular modelers. In the classical textbook picture a molecule giving rise to LC phases is represented by a uniaxial rod endowed with repulsive (Onsager) or attractive (Maier-Saupe) interactions or possibly with a combination of the two (van der Waals picture) [1]. While these models have proved able to reproduce at least qualitatively the most common LC phase, the nematic one, and its phase transition to a normal isotropic fluid, they have not been able to deal with quantitative aspects (e.g. the orientational order at the transition) and more seriously, with the variety of novel LC phases and of sophisticated experiments offering increasing detailed observations at the nanoscale. Classical Monte Carlo and molecular dynamics computer simulations that have been successfully used for some time on simple lattice or off-lattice generic models [2-5] have started to offer unprecedented, atomistic level, details of the molecular organization of LC in the bulk and close to surfaces [6,7]. In particular, atomistic simulations are now starting to offer predictive power, opening the possibility of closing the gap between molecular structure and phase organizations. The availability of detailed data from these virtual experiments requires to generalize LC models inserting molecular features like deviation from uniaxiality or rigidity, the inclusion of partial charges etc. Such more detailed descriptions should reflect also in the link between molecular and continuum theories, already developed for the simplest models [8,9], possibly opening the way to a molecular identification of the material and temperature dependent coefficients in Landau-deGennes type free energy functionals.

[1] see, e.g., G. R. Luckhurst and G. W. Gray, eds., The Molecular Physics of Liquid Crystals (Academic Press,, 1979).

[2] P. Pasini and C. Zannoni, eds., Advances in the computer simulations of liquid crystals (Kluwer, 1998)

[3] O. D. Lavrentovich, P. Pasini, C. Zannoni and S. Zumer, eds. Defects in Liquid Crystals: Computer Simulations, Theory and Experiments, (Kluwer, Dordrecht , 2001).

[4] C. Zannoni, Molecular design and computer simulations of novel mesophases, J. Mat. Chem. 11, 2637 (2001).

[5] R.Berardi, L.Muccioli, S.Orlandi, M.Ricci, C.Zannoni, Computer simulations of biaxial nematics, J. Phys. Cond. Matter 20, 1 (2008).

[6] G. Tiberio, L. Muccioli, R. Berardi and C. Zannoni , Towards “in silico” liquid crystals. Realistic Transition temperatures and physical properties for n-cyanobiphenyls via molecular dynamics simulations, ChemPhysChem 10, 125 (2009).

[7] O. Roscioni, L. Muccioli, R. Della Valle, A. Pizzirusso, M. Ricci and C. Zannoni, Predicting the anchoring of liquid crystals at a solid surface: 5-cyanobiphenyl on cristobalite and glassy silica surfaces of increasing roughness, Langmuir 29, 8950 (2013).

[8] 1. J. Katriel, G. F. Kventsel, G. R. Luckhurst and T. J. Sluckin, Free-energies in the Landau and Molecular-field approaches, Liq. Cryst. 1, 337 (1986).

[9] J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a Continuum Theory, Mol. Cryst. Liq. Cryst. 525, 1 (2010).

Computational structures---from simple objects like bits and qubits,

to complex procedures like encryption and quantum teleportation---can

be defined using algebraic structures in a symmetric monoidal

2-category. I will show how this works, and demonstrate how the

representation theory of these structures allows us to recover the

ordinary computational concepts. The structures are topological in

nature, reflecting a close relationship between topology and

computation, and allowing a completely graphical proof style that

makes computations easy to understand. The formalism also gives

insight into contentious issues in the foundations of quantum

computing. No prior knowledge of computer science or category theory

will be required to understand this talk.

Noncommutative projective geometry is the study of quantum versions of projective space and other projective varieties.  Starting with the celebrated work of Artin, Tate and Van den Bergh on noncommutative projective planes, a substantial theory of noncommutative curves and surfaces has been developed, but the classification of noncommutative versions of projective three-space remains unknown.  I will explain how a portion of this classification can be obtained, via deformation quantization, from a corresponding classification of holomorphic foliations due to Cerveau and Lins Neto.  In algebraic terms, the result is an explicit description of the deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.

We show that actin lamellar fragments extracted from cells, lacking

the complex machinery for cell crawling, are spontaneously motile due

solely to actin polymerization forces at the boundary. The motility

mechanism is associated to a morphological instability similar to the

problem of viscous fingering in Hele-Shaw cells, and does not require

the existence of a global polarization of the actin gel, nor the

presence of molecular motors, contrary to previous claims. We base our

study on the formulation of a 2d free-boundary problem and exploit

conformal mapping and center manifold projection techniques to prove

the nonlinear instability of the center of mass, and to find an exact

and simple relation between shape and velocity. A complex subcritical

bifurcation scenario into traveling solutions is unfolded. With the

help of high-precision numerical computation we show that the velocity

is exponentially small close to the bifurcation points, implying a

non-adiabatic mechanism. Examples of traveling solutions and their

stability are studied numerically. Extensions of the approach to more

realistic descriptions of actual biological systems are briefly

discussed.

REF: C. Blanch-Mercader and J. Casademunt, Physical Review Letters

110, 078102 (2013)

The problem is based on real time computation for 4D (3D+time) trajectory planning for unmanned vehicles (UVs). The ability to quickly predict a 4D trajectory/path enables safe, flexible, efficient use of UVs in a collaborative space is a key objective for autonomous mission and task management. 

The problem/topic proposal will consist of 3 challenges: 
1.      A single UV 4D path planning problem.
2.      Multi UV 4D path planning sharing the same space and time.
3.      Assignment of simultaneous tasks for multiple UVs based on the 4D path finding solution.

In joint work with T. Tsankov we study a (yet other) point at which model theory and dynamics intersect. On the one hand, a (metric) aleph_0-categorical structure is determined, up to bi-interpretability, by its automorphism group, while on the other hand, such automorphism groups are exactly the Roelcke precompact ones. One can further identify formulae on the one hand with Roelcke-continuous functions on the other hand, and similarly stable formulae with WAP functions, providing an easy tool for proving that a group is Roelcke precompact and for calculating its Roelcke/WAP compactification. Model-theoretic techniques, transposed in this manner into the topological realm, allow one to prove further that if R(G) = W(G); then G is totally minimal.

We give a short exposition on the zeta determinant for a Laplace - type operator on a closed Manifold as first described by Ray and Singer in their attempt to find an analytic counterpart to R-torsion.

We combine recent breakthroughs in modularity lifting with a
3-5-7 modularity switching argument to deduce modularity of elliptic curves over real
quadratic fields. We
discuss the implications for the Fermat equation. In particular we
show that if d is congruent
to 3 modulo 8, or congruent to 6 or 10 modulo 16, and $K=Q(\sqrt{d})$
then there is an
effectively computable constant B depending on K, such that if p>B is prime,
and $a^p+b^p+c^p=0$ with a,b,c in K, then abc=0.   This is based on joint work with Nuno Freitas (Bayreuth) and Bao Le Hung (Harvard).

Combining classifiers into an ensemble aims at a more accurate and robust classification decision compared to that of a single classifier. For a successful ensemble, the individual classifiers must be as diverse and as accurate as possible. Achieving both simultaneously is impossible, hence compromises have been sought by a variety of ingenious ensemble creating methods. While diversity has been in the focus of the classifier ensemble research for a long time now, the importance of the combination rule has been often marginalised. Indeed, if the ensemble members are diverse, a simple majority (plurality) vote will suffice. However, engineering diversity is not a trivial problem. A bespoke (trainable) combination rule may compensate for the flaws in preparing the individual ensemble members. This talk will introduce classifier ensembles along with some combination rules, and will demonstrate the merit of choosing a suitable combination rule.

An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth.

In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market,

cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to

optimal execution problems and to other market frictions such as proportional and fixed transaction costs.

Direct-search methods are a class of popular derivative-free

algorithms characterized by evaluating the objective function

using a step size and a number of (polling) directions.

When applied to the minimization of smooth functions, the

polling directions are typically taken from positive spanning sets

which in turn must have at least n+1 vectors in an n-dimensional variable space.

In addition, to ensure the global convergence of these algorithms,

the positive spanning sets used throughout the iterations

must be uniformly non-degenerate in the sense of having a positive

(cosine) measure bounded away from zero.

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However, recent numerical results indicated that randomly generating

the polling directions without imposing the positive spanning property

can improve the performance of these methods, especially when the number

of directions is chosen considerably less than n+1.

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In this talk, we analyze direct-search algorithms when the polling

directions are probabilistic descent, meaning that with a certain

probability at least one of them is of descent type. Such a framework

enjoys almost-sure global convergence. More interestingly, we will show

a global decaying rate of $1/\sqrt{k}$ for the gradient size, with

overwhelmingly high probability, matching the corresponding rate for

the deterministic versions of the gradient method or of direct search.

Our analysis helps to understand numerical behavior and the choice of

the number of polling directions.

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This is joint work with Clément Royer, Serge Gratton, and Zaikun Zhang.

In this talk we are concerned with the stability of steady transonic shocks in supersonic flow around a wedge. 2-D and M-D potential stability will be presented.

This talk is based on the joint works with Prof. G.-Q. Chen, and Prof. S.X. Chen.

Many interesting properties of groups are inherited by their subgroups examples of such are finiteness, residual finiteness and being free. People have asked whether hyperbolicity is inherited by subgroups, there are a few counterexamples in this area. I will be detailing the proof of some of these including a construction due to Rips of a finitely generated not finitely presented subgroup of a hyperbolic group and an example of a finitely presented subgroup which is not hyperbolic.

A major project in number theory runs as follows. Suppose some Diophantine equation has infinitely many integer solutions. One can then ask how common solutions are: roughly how many solutions are there in integers $\in [ -B, \, B ] $? And ideally one wants an answer in terms of the geometry of the original equation.

What if we ask the same question about Diophantine inequalities, instead of equations? This is surely a less deep question, but has the advantage that all the geometry we need is over $\mathbb{R}$. This makes the best-understood examples much easier to state and understand.

I will start by introducing Somos sequences, defined by innocent-looking quadratic recursions which, surprisingly, always return integer values. I will then explain how they can be viewed in a much larger context, that of the Laurent phenomenon in the theory of cluster algebras. Some further steps take us to the the quantum cluster positivity conjecture of Berenstein and Zelevinski. I will finally explain how, following Nagao and Efimov, cohomological Donaldson-Thomas theory leads to a proof of this conjecture in some, perhaps all, cases. This is joint work with Davison, Maulik, Schuermann.

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees.

Joint work with Christian Borgs, Jennifer T. Chayes, and Henry Cohn.