Contemporary Mathematics

In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation procedure can then be evaluated using the bottleneck distance between the estimated persistent homology and the true persistent homology. The connection to statistics comes from the fact that when viewed as a nonparametric regression problem, the bottleneck distance is bounded by the sup-norm loss. Consequently, a sharp asymptotic minimax bound is determined under the sup-norm risk over Hölder classes of functions for the nonparametric regression problem on manifolds. This provides good convergence properties for the persistent homology estimator in terms of the expected bottleneck distance.

In this paper we observe that the existence of an sl(2, R) can help us in organizing the analysis and computation of time-dependent symmetries of nonlinear evolution equations. We apply this idea to the Burgers and Ibragimov-Shabat equations. We then use sl(2, R) to compute their Bäcklund transformations.

In 1961 George Mackey introduced the concept of a virtual group as an equivalence class under similarity of ergodic measured groupoids, and he developed this circle of ideas in subsequent papers in 1963 and 1966. The goal here is first to explain these ideas, place them in a larger context, and then to show how they have influenced and helped to shape developments in four different but related areas of research over the following 45 years. These areas include first the general area and the connections between ergodic group actions, von Neumann algebras, measurable group theory and rigidity theorems. Then we turn to the second area concerning topological groupoids, C *-algebras, K-theory and cyclic homology, or as it is now termed non-commutative geometry. We briefly discuss some aspects of Lie groupoids, and finally we shall turn attention to the fourth area of Borel equivalence relations seen as a part of descriptive set theory. In each case we trace the influence that Mackey's ideas have had in shaping each of these four areas of research.

We study geometric quantization of moduli spaces of vector bundles on an algebraic curve X, and its relation to theta functions. In limits when the complex structure of X degenerates, we describe vector spaces of distributions with Verlinde dimension, associated to the quantization of some of these moduli spaces in real polarizations. We show that special bases on these spaces, defined using a trinion decomposition of X, are related by modular transformation matrices for characters of affine Lie algebras, which appear naturally in the Blattner-Kostant-Sternberg (BKS) pairing for different quantizations of the moduli space.

Let C be a clutter and let I be its edge ideal. We present a combinatorial description of the minimal generators of the symbolic Rees algebra Rs(I) of I. It is shown that the minimal generators of Rs(I) are in one to one correspondence with the irreducible parallelizations of C. From our description some major results on symbolic Rees algebras of perfect graphs and clutters will follow. As a byproduct, we give a method, using Hilbert bases, to compute all irreducible parallelizations of C along with all the corresponding vertex covering numbers.

In this paper we begin to explore the mathematical connection between equilibrium shapes of crystalline materials (Wul shapes) and shock wave structures in compressible gas dynamics (Riemann problems). These are radically di erent physical phenomena, but the similar nature of their discontinuous solutions suggests a connection.

We give an overview of Lie's sphere geometry, and discuss Lie's discovery of the line-sphere correspondence. The article's aims are twofold. First we discuss the role played by this result in Lie's mathematical evolution. Second, we indicate how the line-sphere correspondence can be used to describe a class of surfaces with integrable asymptotic lines.

We present major open problems in algebraic coding theory. Some of these problems are classified as Hilbert problems in that they are foundational questions whose solutions would lead to further study. The remainder are classified as Fermat problems in that they are difficult problems in coding theory that have defied solution for a significant period of time.

We present a new proof of a theorem of Schur's from 1905 determining the least common multiple of the orders of all finite groups of complex n × n-matrices whose elements have traces in the field É of rational numbers. The basic method of proof goes back to Minkowski and proceeds by reduction to the case of finite fields.

This paper describes the investment universe of hedge funds from a perspective which is at the same time mathematical in nature and practical in its objectives. It addresses the investment opportunities that hedge funds pursue, the mathematical techniques to understand their performance, the portfolio construction tools needed to construct investments, and risk methodologies useful in assessing performance.

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This article is an introduction to our recent work in harmonic analysis associated with semigroups of operators, in the effort of finding a noncommutative Calderón-Zygmund theory for von Neumann algebras. The classical CZ theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of such metrics -or with very little information on the metric-Markov semigroups of operators appear to be the right substitutes of classical metric/geometric tools in harmonic analysis. Our approach is particularly useful in the noncommutative setting but it is also valid in classical/commutative frameworks.

This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects.

A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. The operational rules are described and the method is illustrated with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.

After a brief history of 'cohomological physics', the Batalin-Vilkovisky complex is given a revisionist presentation as homological algebra, in part classical, in part novel. Interpretation of the higher order terms in the extended Lagrangian is given as higher homotopy Lie algebra and via deformation theory. Examples are given for higher spin particles and closed string field theory.

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

In this paper, we present a new version of cochains in Algebraic Topology, starting with "quantum differential forms". This version provides many examples of modules over the braid group, together with control of the non commutativity of cup-products on the cochain level. If the quantum parameter q is equal to 1, we recover essentially the commutative differential graded algebra of de Rham-Sullivan forms on a simplicial set [1] . For topological applications, we may take either q = 1 if we are dealing with rational coefficients or q = 0 in the general case. In both cases, the quantum formulas are simpler (if q = 0 for instance, the quantum exponential e q (x) is just the function 1/1-x).

In 1986 J. Nuttall published in Constructive Approximation the paper , where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Padé approximants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the socalled Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this paper features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters.

There are four values of s for which the hypergeometric function 2 F 1 ( 1 2 − s, 1 2 + s; 1; ·) can be parametrized in terms of modular forms; namely, s = 0, 1 3 , 1 4 , 1 6 . For the classical s = 0 case, the parametrization is in terms of the Jacobian theta functions θ 3 (q), θ 4 (q) and is related to the arithmetic-geometric mean iteration of Gauss and Legendre. Analogues of the arithmetic-geometric mean are given for the remaining cases. The case s = 1 6 and its relationship to the work of Ramanujan is highlighted. The work presented includes various pieces of joint work with combinations of the following:

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree.

Applying a modification of MacPherson's graph construction to the case of periodic complexes, we give an algebraic construction of Witten's "top Chern class" on the moduli space of algebraic curves with higher spin structures. We show that it satisfies most of the axioms for the spin virtual class and also the so-called descent axiom.

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

A. In this paper we will prove some results about integrability of the weak invariant bundles for partially hyperbolic diffeomorphisms in dimension 3. We deal with the problems of existence and uniqueness in case we have transitivity and denseness of periodic orbits. We prove that if we have two crossing central curves contained in a weak-unstable disk then E cu is uniquely integrable. We also obtain the integrability of the center bundle if the compact periodic center curves are dense.

Ever since Werner Heisenberg's 1927 paper on uncertainty, there has been considerable hesitancy in simultaneously considering positions and momenta in quantum contexts, since these are incompatible observables. But this persistent discomfort with addressing positions and momenta jointly in the quantum world is not really warranted, as was first fully appreciated by Hilbrand Groenewold and José Moyal in the 1940s. While the formalism for quantum mechanics in phase space was wholly cast at that time, it was not completely understood nor widely known-much less generally accepted-until the late 20th century. R. Feynman When Feynman first unlocked the secrets of the path integral formalism and presented them to the world, he was publicly rebuked [8]. "It was obvious," Bohr said, "that such trajectories violated the uncertainty principle." However, in this case, 1 Bohr was wrong. Today path integrals are universally recognized and widely used as an alternative framework to describe quantum behavior, equivalent to-although conceptually distinct from-the usual Hilbert space framework, and therefore completely in accord with Heisenberg's uncertainty principle. The different points of view offered by the Hilbert space and path integral frameworks combine to provide greater insight and depth of understanding. N. Bohr Similarly, many physicists hold the conviction that classical-valued position and momentum variables should not be simultaneously employed in any meaningful formula expressing quantum behavior, simply because this would also seem to violate the uncertainty principle (see Appendix Dirac). However, they are also wrong. Quantum mechanics (QM) can be consistently and autonomously formulated in phase space,

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society,

We give a brief account of some of the traditional ways that genetic algorithms have been applied, and explain how our approach to the use of genetic algorithms for solving problems in combinatorial group theory differs. We find that, in our situation, there seems to be a correlation between successful genetic algorithms and the existence of good non-genetic, sometimes deterministic, algorithms. We use a class of equations in free groups as a test bench. In particular, it allows us to trace the convergence of co-evolution of the population of fitness functions to a deterministic solution.