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A group is an abstract mathematical object, a set with elements and an operation satisfying certain axioms. A representation of a group realizes the elements of the group concretely as geometric symmetries.

The simplest meaningful object we can construct out of a field F and a group G is a vector space over F, with the elements of Gas basis.

We are now ready to plunge into a fuller exploration of the group algebra F[G], where G is a group and F is a field. Our focus is, of course, on the case where Gis finite.

Having survived the long exploration of semisimple structure, we may think that midway on our journey we find ourselves in deep woods, the right path lost [16]. But this is not the time to abandon hope; instead, we plunge right into untangling the structure of representations of an important family of groups, the permutation groups S n . This will be the only important class of finite groups to which we will apply all the machinery we have manufactured. A natural pathway beyond this is the study of representations of reflection groups.

The unitary group U(N) consists of all N ×N complex matrices U that satisfy the unitarity condition $${U}^{{_\ast}}U = I.$$ It is a group under matrix multiplication, and, being a subset of the linear space of all N ×N complex matrices, it is a topological space as well. Multiplication of matrices is, clearly, continuous.

We construct a free-probability quantum Yang-Mills theory on the two dimensional plane, determine the Wilson loop expectation values, and show that this theory is the $N=\infty$ limit of U(N) quantum Yang-Mills theory on the plane., 24 pages, tikz figures

The Yang-Mills measure for gauge fields over the two-sphere is constructed using a conditioned white-noise process. Stochastic differential equations are used to study parallel translation under this measure. Expectation values of a broad class of Wilson loop configurations are computed explicitly and it is shown that these values are invariant under area-preserving diffeomorphisms of the sphere.

We develop a differential geometric framework for parallel transport over path spaces and a corresponding discrete theory, an integrated version of the continuum theory, using a category-theoretic framework., 27pp 3fig pdflatex only; v2: rewritten with several clarifications; v3: minor changes, added references. Version to be published, 30pp

We demonstrate how pricing functionals give rise to pricing mea- sures, using a time-independent framework. For infinite market state spaces, the Gel'fand spectral theory for C ⁄ -algebras is used to obtain the pricing measure. Pricing functionals with additional market information are shown, within this model, to be given by conditional expectations. Our approach is time-independent, and the usual martingale property of prices appears as a special case in our method.

Complex White Noise and Infinite Dimensional Unitary Group (T Hida) A New Explicit Formula for the Solution of the Black-Merton-Scholes Equation (J A Goldstein et al.) An Infinite Dimensional Fourier-Mehler Transform and the Levy Laplacian (K Saito) Quantum Stochastic Dilation of Symmetric Covariant Completely Positive Semigroups with Unbounded Generator (D Goswami & K B Sinha) Donsker's Functional Calculus and Related Questions (P-L Chow & J Potthoff) Stochastic Analysis of Tidal Dynamics Equation (U Manna et al.) Spaces of Test and Generalized Functions of Arcsine White Noise (A Riahi & H Ouerdiane) Probability Measures with Sub-additive Principal Szego-Jacobi Parameters (A I Stan) Adapted Solutions to the 3D Backward Stochastic Navier-Stokes Equations (P Sundar & H Yin) White Noise Analysis in the Theory of 3-manifold Quantum Invariants (A Hahn) Complex Ito Formulas (M Redfern) Volatility Models of the Yield Curve (V Goodman).

The large‐N limit of the quantum Yang‐Mills field over the plane R2, for the gauge group U(N), is shown to lead to a white noise field in the sense of free probability.

We present a description of two dimensional Yang-Mills gauge theory on the plane and on compact surfaces, examining the topological, geometric and probabilistic aspects., 16 pages, presented at a conference titled Stochastic Analysis and Mathematical Physics 2006, organized by Ana Bela Cruzeiro and Jean-Claude Zambrini

We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces., Comment: 12 pp.; the Introduction has been rewritten and mention of cohomology dropped in Proposition 3.2; material slightly reorganized

There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for the ridge regression problem within the framework of abstract Wiener spaces and show how the support vector machine solution to such problems can be interpreted in terms of the Gaussian Radon transform.

Though not obvious at first glance, these ideas are all connected with each other. In the following sections we shall give an account of some of these ideas and indicate how everything ties together. The relation between the algebraic problem of counting solutions of equations in a group with two-dimensional Yang–Mills theory has been explored in several works by Mulase et al. [6, 7]. (We shall not describe relations with Chern–Simons theory.) Ideas from quantum Yang–Mills may be useful in understanding representation varieties of finitely generated groups. For the case of surface groups, it still remains to carry out a complete study of the topological and symplectic structure of the various strata of the moduli space of flat connections from the quantum Yang–Mills point of view, though much progress has been made (initiated by Witten’s works [11,12]).

It has been proposed that a constellation of three TP53 polymorphisms (intron 3 16 bp duplication, codon 72 BstUI, and intron 6 Nci I RFLP at nt 13494) constitute a haplotype predictive of increased cancer risk. We have estimated the allele frequency of these polymorphisms in three endogamous Indian ethnic populations from three different geographic locations (viz. Iyer from south India, Brahmin from central India and Mahishya from eastern India), as well as in head and neck squamous cell carcinoma (HNSCC) patients, and in ethnically matched normal individuals from the eastern region of India. The genotype distributions and allele frequencies of the three polymorphisms in all but one population, as well as in patients, showed a good fit to Hardy-Weinberg equilibrium. Strong linkage disequilibria were observed between all loci in every population examined, except for the 16bp-Nci I haplotype in the Mahishya population. The Mahishya population differed significantly from the other two populations with respect to differences in allele frequency and haplotype frequency. Although there were no significant differences in genotypic frequency at any of the loci between HNSCC patients and the matched control population, the minor allele frequency of codon 72 and intron 3 16 bp polymorphisms showed significant variation. Variation in overall haplotype frequency between patients and normal individuals was significant (p = 0.036) when two rare haplotypes 2-1-2 and 1-2-1 were combined. The rare haplotype 2-1-2 was found to be modestly over represented in HNSCC patients as compared to normal individuals.

A short overview is presented of: (i) the quantum field measure associated with pure Yang-Mills gauge theory on a compact surface, (ii) an approach to the symplectic structure on the [moduli] space of [flat] connections over compact oriented surfaces (with and without boundary), and (iii) the relation between the classical Yang-Mills measure and the volume measure on the moduli space of flat connections.

We relate the semiclassical limit of the quantum Yang-Mills partition function on a compact oriented surface to the symplectic volume of the moduli space of flat connections, by using an explicit expression for the symplectic form. This gives an independent proof of some recent results of Witten and Forman., 10 pages

We describe the construction of the Yang-Mills measure for the (Euclidean) quantum theory of gauge fields associated to a bundle over a compact surface. This measure is obtained from the product of a Gaussian measure and a Haar measure by means of a conditioning process which encodes the topology of the bundle. We then describe expectation values of important random variables (‘Wilson loops variables’) associated to systems of loops on the surface. We also discuss results relating to the limit of the Yang-Mills measure as an associated parameter goes to zero, and the symplectic nature, in certain situations, of the set of minima of the Yang-Mills functional.

The equation of parallel transport is given meaning in the quantized two dimensional Euclidean Yang-Mills theory by interpreting it as a stochastic differential equation in the complete axial gauge. The expectation of products of Wilson loops is evaluated using the stochastic differential calculus and a method of Bralic. Euclidean invariance of the theory is proven.

Consider a closed surfaceΣobtained by pasting two surfacesΣ1andΣ2along some of their boundary components. A formula is derived expressing quantum Yang–Mills functional integrals forΣin terms of the corresponding integrals for the surfacesΣi. Using this, a formula is derived which expresses the symplectic volume measure on the moduli space of flatSU(2) connections over a closed Riemann surfaceΣin terms of the corresponding volume measures for the surfacesΣ1andΣ2, when the latter have one boundary component each.