Issue 2, Volume 40, June 2015

2015

1. Stein Meets Malliavin in Normal Approximation Louis H. Y. Chen PDF

   ▸ Abstract

   Stein’s method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation, the functional equation is a first-order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (Probab. Theory Relat. Fields 145(1–2), 75–118, 2009) established a fundamental connection between Stein’s method for normal approximation and Malliavin calculus through integration by parts. This connection is exploited to obtain error bounds in total variation in central limit theorems for functionals of general Gaussian processes. Of particular interest is the fourth moment theorem which provides error bounds of the order $\sqrt {\mathbb {E}({F_{n}^{4}})-3}$ in the central limit theorem for elements $\{F_n \}_{n\geq 1}$ of Wiener chaos of any fixed order such that $\mathbb {E}({F_{n}^{2}}) = 1$. This paper is an exposition of the work of Nourdin and Peccati with a brief introduction to Stein’s method and Malliavin calculus. It is based on a lecture delivered at the Annual Meeting of the Vietnam Institute for Advanced Study in Mathematics in July 2014.

   205-230
2. Around the Poincaré Lemma, After Beilinson Luc Illusie PDF

   ▸ Abstract

   The following notes are an expanded version of talks on Beilinson’s paper as reported by Beilinson (J. AMS 25, 715–738, 2012) given at the department of mathematics of the university of Padova on October 25 and 26, 2012, at the Séminaire RÉGA (Réseau d’Étudiants en Géométrie Algébrique, Institut Henri Poincaré, Paris) on December 12, 2012, lectures at the KIAS (Seoul) on January 3 and 4, 2013, a course at the Morningside Center of Mathematics (Beijing) on February 18, 25, and March 4, 11, 2013, and a course at the VIASM (Hanoi) on September 4, 5, 9, 10, 11, 2014. I do not discuss Beilinson’s next paper as reported by Beilinson (Camb. J. of Math. 1, 1–51, 2013), which deals with refinements of the comparison theorem between p-adic étale cohomology and de Rham cohomology.

   231-253
3. Feynman-Kac Representation of Fully Nonlinear PDEs and Applications Huyên Pham PDF

   ▸ Abstract

   The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present the implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance.

   255-269
4. Balloons and Hoops and their Universal Finite-Type Invariant, BF Theory, and an Ultimate Alexander Invariant Dror Bar-Natan PDF

   ▸ Abstract

   Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second homotopy groups of a topological space --- hoops can be composed as in $\pi_1$, balloons as in $\pi_2$, and hoops ``act'' on balloons as $\pi_1$ acts on $\pi_2$. We observe that ordinary knots and tangles in 3-space map into KBH in 4-space and become amalgams of both balloons and hoops. We give an ansatz for a tree and wheel (that is, free-Lie and cyclic word) -valued invariant $\zeta$ of (ribbon) KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that $\zeta$ is a complete evaluation of the BF topological quantum field theory in 4D. We show that a certain ``reduction and repackaging'' of $\zeta$ is an ``ultimate Alexander invariant'' that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that should be a wonderful playground.

   271-329
5. The Decomposition of $\mathrm{Hom}_k^{}(S,k)$ into Indecomposable Injectives Amnon Neeman PDF

   ▸ Abstract

   Let $S$ be an algebra essentially of finite type over a field $k$. Then $\mathrm{Hom}_k^{}(S,k)$ is an injective $S$--module, and the Matlis structure theorem (Matlis, E.: Pacific J. Math. 8, 511–528 1958) tells us that it can be written as a direct sum of indecomposable injectives. We compute the multiplicities of these injectives. Let $\mathfrak{p}$ be a prime ideal in $S$, and let $I(\mathfrak{p})$ be the injective hull of $S/\mathfrak{p}$. If the residue field $k(\mathfrak{p})$ is algebraic over $k$ then the multiplicity of $I(\mathfrak{p})$ is $\mu(\mathfrak{p})=1$. If the transcendence degree of $k(\mathfrak{p})$ over $k$ is $\geq1$ then $\mu(\mathfrak{p})\geq{|\#k|}^{\aleph_0}$, that is the multiplicity is no less than the cardinality of the field $k$ raised to the power $\aleph_0$. If $S$ is finitely generated over $k$ then equality holds, that is $\mu(\mathfrak{p})={|\#k|}^{\aleph_0}$. For $k(\mathfrak{p})$ of transcendence degree $\leq1$ the result is not surprising, but for $k(\mathfrak{p})$ of transcendence degree $\geq2$ it is not clear that $\mu(\mathfrak{p})\neq 0$. We prove the result by induction on the transcendence degree, and the key is that we produce an injective map, from a space whose dimension we know by induction and into the space whose dimension we want to estimate. The interest in the result comes from the fact that the size of $\mu(\mathfrak{p})$ measures the failure of a natural map $\psi(f):f^\times\longrightarrow f^!$ to be an isomorphism. Here $f^\times$ and $f^!$ are the twisted inverse image functors of Grothendieck duality.

   331-338
6. Alexander Representation of Tangles Stephen Bigelow , Alessia Cattabriga , Vincent Florens PDF

   ▸ Abstract

   A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of ${\mathbb {Z}}[t,t^{-1}]$-modules. For (1,1)-tangles (i.e., tangles with one endpoint on each disk), this invariant coincides with the Alexander polynomial of the link obtained by taking the closure of the tangle. We use the notion of plat position of a tangle to give a constructive proof of invariance in this case.

   339-352