Past

Compressed sensing and matrix completion are techniques by which simplicity in data can be exploited for more efficient data acquisition. For instance, if a matrix is known to be (approximately) low rank then it can be recovered from few of its entries. The design and analysis of computationally efficient algorithms for these problems has been extensively studies over the last 8 years. In this talk we present a new algorithm that balances low per iteration complexity with fast asymptotic convergence. This algorithm has been shown to have faster recovery time than any other known algorithm in the area, both for small scale problems and massively parallel GPU implementations. The new algorithm adapts the classical nonlinear conjugate gradient algorithm and shows the efficacy of a linear algebra perspective to compressed sensing and matrix completion.

Several objects can be associated to a list of vectors with integer coordinates: among others, a family of tori called toric arrangement, a convex polytope called zonotope, a function called vector partition function; these objects have been described in a recent book by De Concini and Procesi. The linear algebra of the list of vectors is axiomatized by the combinatorial notion of a matroid; but several properties of the objects above depend also on the arithmetics of the list. This can be encoded by the notion of a "matroid over Z". Similarly, applications to tropical geometry suggest the introduction of matroids over a discrete valuation ring.Motivated by the examples above, we introduce the more general notion of a "matroid over a commutative ring R". Such a matroid arises for example from a list of elements in a R-module. When R is a Dedekind domain, we can extend the usual properties and operations holding for matroids (e.g., duality). We can also compute the Tutte-Grothendieck ring of matroids over R; the class of a matroid in such a ring specializes to several invariants, such as the Tutte polynomial and the Tutte quasipolynomial. We will also outline other possible applications and open problems. (Joint work with Alex Fink).

In 1992 (or thereabouts) Bloch and Kriz gave the first explicit definition of the category of mixed Tate motives (MTM). Their definition relies heavily on the theory of algebraic cycles. Unfortunately, traditional methods of representing algebraic cycles (such as in terms of formal linear combinations of systems of polynomial equations) are notoriously difficult to work with, so progress in capitalizing on this description of the category to illuminate outstanding conjectures in the field has been slow. More recently, Gangl, Goncharov, and Levin suggested a simpler way to understand this category (and by extension, algebraic cycles more generally) by relating specific algebraic cycles to rooted, decorated, planar trees. In our talks, describing work in progress, we generalize this correspondence and attempt to systematize the connection between algebraic cycles and graphs. We will construct a Lie coalgebra L from a certain algebra of admissible graphs, discuss various properties that it satisfies (such as a well defined and simply described realization functor to the category of mixed Hodge structures), and relate the category of co-representations of L to the category MTM. One promising consequence of our investigations is the appearance of alternative bases of rational motives that have not previously appeared in the literature, suggesting a richer rational structure than had been previously suspected. In addition, our results give the first bounds on the complexity of computing admissibility of algebraic cycles, a previously unexplored topic.

When applied to an inequality constrained optimization problem, interior point methods generate iterates that belong to the interior of the set determined by the constraints, thus avoiding/ignoring the combinatorial aspect of the solution. This comes at the cost of difficulty in predicting the optimal active constraints that would enable termination.  We propose the use of controlled perturbations to address this challenge. Namely, in the context of linear programming with only nonnegativity constraints as the inequality constraints, we consider perturbing the nonnegativity constraints so as to enlarge the feasible set. Theoretically, we show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close to the central path of the perturbed problem and that a primal-dual path-following algorithm applied to the perturbed problem is able to predict the optimal active-set of the original problem when the duality gap (for the perturbed problem) is not too small. Encouraging preliminary numerical experience is obtained when comparing the perturbed and unperturbed interior point algorithms' active-set predictions for the purpose of cross-over to simplex.

A non-parametric test for dependence between sets of random variables based on the entropy rate is proposed. The test has correct size, unit asymptotic power, and can be applied to test setwise cross sectional and serial dependence. Using Monte Carlo experiments, we show that the test has favourable small-sample properties when compared to other tests for dependence. The ‘trick’ of the test relies on using universal codes to estimate the entropy rate of the stochastic process generating the data, and simulating the null distribution of the estimator through subsampling. This approach avoids having to estimate joint densities and therefore allows for large classes of dependence relationships to be tested. Potential economic applications include model specification, variable and lag selection, data mining, goodness-of-fit testing and measuring predictability.

An old conjecture by A. Zygmund proposes

a Lebesgue Differentiation theorem along a

Lipschitz vector field in the plane. E. Stein

formulated a corresponding conjecture about

the Hilbert transform along the vector field.

If the vector field is constant along

vertical lines, the Hilbert transform along

the vector field is closely related to Carleson's

operator. We discuss some progress in the area

by and with Michael Bateman and by my student

Shaoming Guo.

In this talk I will explain the basic motivation behind the trace formula and give some simple examples. I will then discuss how it can be used to prove things about automorphic representations on general reductive groups.

We will present a somewhat different proof of Agol's theorem that

3-manifolds 

with RFRS fundamental group admit a finite cover which fibers over S^1.

This is joint work with Takahiro Kitayama.

The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.

Despite the ability of the stochastic volatility models along with their multivariate and multi-factor extension to describe the dynamics of the asset returns, these
models are very difficult to calibrate to market information. The recent financial crises, however, highlight that we can not use simplified models to describe the fincancial returns. Therefore, our statistical methodologies have to be improved. We propose a non parametricmethodology based on the use of the Fourier transform and the high frequency data which allows to estimate the diffusion and the leverage components of a general stochastic volatility model driven by continuous Brownian semimartingales. Our estimation procedure is based only on a pre-estimation of the Fourier coefficients of the volatility process and on the use of the Bohr convolution product as in Malliavin and Mancino 2009. This approach constitutes a novelty in comparison with the non-parametric methodologies proposed in the literature generally based on a pre-estimation of the spot volatility and in virtue of its definition it can be directly applied in the case of irregular tradingobservations of the price path an microstructure noise contaminations.

 We will introduce 2 types of topological games (Menger and
> Telgársky) and show how the existence or non-existence of winning
> strategies implies certain properties of the underlying topological
> space. We will then show how these, and related properties, interact
> D-spaces.

Hazardous geophysical mass flows, such as snow avalanches, debris-flows and pyroclastic flows, often spontaneously develop large particle rich levees that channelize the flow and enhance their run-out. Measurements of the surface velocity near an advancing flow front have been made at the United States Geological Survey (USGS) debris-flow flume, where 10m^3 of water saturated sand and gravel are allowed to flow down an 80m chute onto a run-out pad. In the run-out phase the flow front is approximately invariant in shape and advances at almost constant speed. By tracking the motion of surface tracers and using a simple kinematic model, it was possible to infer bulk motion as incoming material is sheared towards the front, over-run and shouldered to the side. At the heart of the levee formation process is a subtle segregation-mobility feedback effect. Simple models for particle segregation and the depth-averaged motion of granular avalanches are described and one of the first attempts is made to couple these two types of models together. This process proves to be non-trivial, yielding considerable complexity as well as pathologies that require additional physics to be included.

tba

It is well known that infinite perfect information two person games at low levels in the arithmetic hierarchy of sets have winning strategies for one of the players, and moreover this fact can be proven in analysis alone. This has led people to consider reverse mathematical analyses of precisely which subsystems of second order arithmetic are needed. We go over the history of these results. Recently Montalban and Shore gave a precise delineation of the amount of determinacy provable in analysis. Their arguments use concretely given levels of the Gödel constructible hierarchy. It should be possible to lift those arguments to the amount of determinacy, properly including analytic determinacy, provable in stronger theories than the standard ZFC set theory. We summarise some recent joint work with Chris Le Sueur.

I will explain how, given a crepant morphism with one-dimensional fibres between 3-folds, it is possible to use noncommutative deformations to run the MMP in a satisfyingly algorithmic fashion.  As part of this, a flop is viewed homologically as the solution to a universal property, and so is constructed not by changing GIT, but instead by changing the algebra. Carrying this extra information of the new algebra allows us to continue to flop, and thus continue the MMP, without having to calculate everything from scratch. Proving things in this manner does in fact have other consequences too, and I will explain some them, both theoretical and computational.

A distinct covering system of congruences is a collection

\[

(a_i \bmod m_i), \qquad 1\ \textless\ m_1\ \textless\ m_2\ \textless\ \ldots\ \textless\ m_k

\]

whose union is the integers. Erd\"os asked whether there are covering systems for which $m_1$ is arbitrarily large. I will describe my negative answer to this problem, which involves the Lov\'{a}sz Local Lemma and the theory of smooth numbers.

Strongly nonlinear problems, written abstractly in the form N[u]=0, are typically difficult to analyze unless they possess special properties. However, if we are able to find a quasi-solution u_0 in the sense that the residual N[u_0] := R is small, then it is possible to analyze a strongly nonlinear problem with weakly nonlinear analysis in the following manner: We decompose u=u_0 + E; then E satisfies L E = -N_1 [E] - R, where L is the Fre'chet derivative of the operator N and N_1 [E] := N[u_0+E]-N[u_0]-L E contains all the nonlinearity. If L has a suitable inversion property and the nonlinearity N_1 is sufficiently regular in E, then weakly nonlinear analysis of the error E through contraction mapping theorem gives rise to control of the error E. What is described above is quite routine. The only new element is to determine a quasi-solution u_0, which is typically found through a combination of classic orthogonal polynomial representation and exponential asymptotics.

This method has been used in a number of nonlinear ODEs arising from reduction of PDEs. We also show how it can be extended to integro-differential equations that arise in study of deep water waves of permanent form. The method is quite general and can in principle be applied to nonlinear PDEs as well.

NB. Much of this is joint work with O. Costin and other collaborators.

We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zero-interest rate setting, starting with an initial wealth of 1 and using only positive portfolios, to superreplicate a constant c>1. The optimal arbitrage strategy is the strategy for which this constant has the highest possible value. Our definition of optimal arbitrage is similar to the one in Fenrholz and Karatzas (2010), where optimal relative arbitrage with respect to the market portfolio is studied. In this work we present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly. We then develop several new examples of market models with arbitrage, which are based on economic agents' views concerning the impossibility of certain events rather than ad hoc constructions. We also explore the concept of fragility of arbitrage introduced in Guasoni and Rasonyi (2012), and provide new examples of arbitrage models which are not fragile in this sense.

References:

Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. The Annals of Applied Probability, 20(4):1179–1204.

Guasoni, P. and Rasonyi, M. (2012). Fragility of arbitrage and bubbles in diffusion models. preprint.