# Papers: a selection

Large-N Limits of Spaces and Structures by Irfan Alam and Ambar N. Sengupta. Spaces and structures $X_N$ varying with a numerical parameter $N$ appear in a variety of contexts. In this paper we explore a corresponding limiting structure $X_\infty$ using the method of ultraproducts. In the case of a sequence of compact groups $G_N$ we construct an invariant measure on the limiting group $G_\infty$ and obtain a topological structure on this group using the method of Weil and Kodaira.

Rotational Symmetries in Polynomial Rings Keith Conrad and Ambar N. Sengupta. In this paper we study the action of the rotation generators $M_{jk}=X_j\partial_{X_k}-X_k\partial_{X_j}$ acting on the ring of polynomials $R[X_1,\ldots, X_N]$, where $R$ is commutative ring. Withing this purely algebraic setting we establish numerous results, many of which have classical analytic counterparts. For example, we show that every polynomial is a sum of terms of the form $(X_1^2+\cdots+X_N^2)^sp_s(X)$, where $p_s(X)$ is a harmonic polynomial. We define a purely algebraic counterpart to integration of polynomials over spheres and establish a formula connecting integration of a polynomial over a sphere and iterated powers of the Laplacian operator applied to the polynomial. We establish results such as an algebraic counterpart of the mean-value property of harmonic polynomials. We also include results on zonal harmonics. We determine all simultaneous eigenvectors of the commuting operators $M_{12}, M_{34}, \ldots$.

Categorical geometry and related works

Gauge Transformations for Categorical Bundles Saikat Chatterjee, Amitabha Lahiri and Ambar N. Sengupta, Journal of Geometry and Physics, 133C (2018) pp. 219-241.

Construction of categorical bundles from local data Saikat Chatterjee, Amitabha Lahiri and Ambar N. Sengupta, Theory and Applications of Categories, Vol. 31, 2016, No. 14, pp 388-417.

Twisted-Product Categorical Bundles  Saikat Chatterjee, Amitabha Lahiri and Ambar N. Sengupta, Journal of Geometry and Physics, Volume 98, December 2015, Pages 128-149.

Connections on decorated path space bundle Saikat Chatterjee, Amitabha Lahiri and Ambar N. Sengupta, Journal of Geometry and Physics, Volume 112, February 2017, Pages 147-174.

Twisted Actions of Categorical Groups Saikat Chatterjee, Amitabha Lahiri and Ambar N. Sengupta, Theory and Applications of Categories, Vol. 29, No. 8, 2014, pp. 215-255

Pathspace Connections and Categorical Geometry Saikat Chatterjee, Amitabha Lahiri and Ambar N. Sengupta, Journal of Geometry and Physics, Volume 75, January 2014

A Morphism Double Category and Monoidal Structure Algebra, Volume 2013 (2013), Article ID 460582

Parallel Transport over Pathspaces Saikat Chatterjee, Amitabha Lahiri, Ambar N. Sengupta, Reviews in Mathematical Physics 9 (2010) 1033-1059.

Negative Forms and Pathspace Forms Saikat Chatterjee, Amitabha Lahiri, Ambar N. Sengupta,   International Journal of Geometric Methods in Modern Physics, Vol. 5, No. 4 (June 2008) 573-586.

Infinite-dimensional geometry and probability

Polynomials and High Dimensional Spheres Amy Peterson and Ambar N. Sengupta. Nonlinear Analysis, Volume 187, (2019), Pages 18-48

Limiting Means of Spherical Slices  Amy Peterson and Ambar N. Sengupta. Communications on Stochastic Analysis, Volume 12, Number 3, (2018), Pages 271-281.

The Gaussian Limit for High-Dimensional Spherical Means  Amy Peterson and Ambar N. Sengupta. Journal of Functional Analysis, Volume 276, Issue 3 (2019). Pages 815-856.

The Gaussian Radon as a Limit of Spherical Transforms,  Ambar N. Sengupta, Journal of Functional Analysis, Volume 271, Issue 11 (2016) 3242-3268. Correction: The equation for the subspace $L_N$ should be $\frac{p}{||u_{(N)}||}u_{(N)}+u_{(N)}^\perp$.

The Gaussian Radon Transform in Classical Wiener Space Irina Holmes and Ambar N. Sengupta, Communications on Stochastic Analysis Vol 8, No. 2 (2014) 247-268.

The Gaussian Radon Transform and Machine Learning Irina Holmes and Ambar N. Sengupta, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 18, No. 03, 1550019 (2015).

A Gaussian Radon Transform for Banach Spaces Irina Holmes and Ambar N. Sengupta, Journal of Functional Analysis, Volume 263, Issue 11, 1 December 2012, Pages 3689-3706

A Support Theorem for a Gaussian Radon Transform in Infinite Dimensions Jeremy J. Becnel and Ambar N. Sengupta, Transactions of the American Mathematical Society, 364 (2012), 1281-1291.

The Radon-Gauss Transform Vochita Mihai and Ambar N. Sengupta, Soochow Journal of Mathematics, Volume 33, 415-434 (2007). The Radon-Gauss Transform

Finance and Mathematics

Identities and Inequalities for CDO Tranche Sensitivities Claas Becker and Ambar N. Sengupta, Communications on Stochastic Analysis, vol. 7, no. 3 (2013).

Temporal Correlation of Defaults in Subprime Securitization Eric Hillebrand, Ambar N. Sengupta, Junyue Xu, Communications on Stochastic Analysis, Vol. 6, Number 3 (2012) 487-511

Quantum Physics

Complex Phase Space and Weyl’s Commutation Relations  Sergio Albeverio and Ambar N. Sengupta. (updated November, 2015)

Finite Geometries with Qubit Operators Ambar N. Sengupta, Quantum Probability, and Related Topics, Volume: 12, Issue: 2 (2009) pp. 359-366.

Quantum Yang-Mills in the large-N limit

Quantum Free Yang-Mills on the Plane Michael Anshelevich and Ambar N. Sengupta, Journal of Geometry and Physics, Volume 62, Issue 2, February 2012, Pages 330343

Traces in two-dimensional QCD: The large-N limit Ambar N. Sengupta, pages 193-212 in ‘Traces in Geometry, Number Theory and Quantum Fields’, edited by Sergio Albeverio, Matilde Marcolli, Sylvie Paycha, and Jorge Plazas, published by Vieweg (2008).

Chern-Simons Theory

A Mathematical Construction of the Non-Abelian Chern-Simons Functional Integral Sergio Albeverio and Ambar Sengupta, Commun. Math. Phys. 186, 563-579 (1997).

Chern-Simons Theory, Hida Distributions, and State Models Sergio Albeverio, Atle Hahn, Ambar N. Sengupta, Infinite Dimensional Analysis, Quantum Probability and Related Topics 6(Special Issue on Probability and Geometry) (2003) 65-81.

Quantum Yang-Mills for Surfaces

Gauge Theory in Two Dimensions: Topological, Geometric and Probabilistic Aspects Ambar N. Sengupta, pages 109-129 in ‘Stochastic Analysis in Mathematical Physics’ edited by Gerard Ben Arous, Ana Bela Cruzeiro, Yves Le Jan, and Jean-Claude Zambrini, published by World Scientific (2008)

The Volume Measure for Flat Connections as Limit of the Yang–Mills Measure Ambar N. Sengupta, Journal of Geometry and Physics 47 398-426 (2003).

Sewing Yang-Mills Measures for non-trivial Bundles Ambar N. Sengupta, Infinite Dimensional Analysis, Quantum Probability and Related Topics 6 (Special Issue on Probability and Geometry) (2003) 39-52.

The Moduli Space of Flat Connections on Oriented Surfaces with Boundary Ambar N. Sengupta, Journal of Functional Analysis 190, 179-232 (2002) : Special Issue dedicated to the memory of I. E. Segal.

Sewing Symplectic Volumes for Flat Connections over Compact Surfaces Ambar N. Sengupta, Journal of Geometry and Physics, 32 (2000) 269-292.  Over the years I have found the  determinant and disintegration formulas in sections 2 and 3 of this paper to be very useful in other contexts as well.

The moduli space of flat SU(2) and SO(3) connections over surfaces. J. Geom. Phys. 28 (1998), no. 3-4, 209–254.

Yang-Mills on Surfaces with Boundary : Quantum Theory and
Symplectic Limit
, Ambar Sengupta, Communications in Mathematical Physics 183, 661-706 (1997).

The Moduli Space of Yang-Mills Connections over a Compact Surface Ambar Sengupta, Reviews in Mathematical Physics 9, 77-121 (1997).

The Segal-Bargmann transform

An Infinite dimensional identity for the Segal-Bargmann Transform Jeremy Becnel and Ambar N. Sengupta, Proceedings of the American Mathematical Society 135 (2007), 2995-3003.

Holomorphic Fock spaces for Positive Linear Transformations Ray Fabec, Gestur Olafsson, Ambar N. Sengupta, Mathematica Scandinavica, 98, 262-282 (2006).

The Segal-Bargmann Transform for Two Dimensional Yang-Mills, Sergio Albeverio, Brian C. Hall and Ambar N. Sengupta, Infinite Dimensional Analysis, Quantum Probability and Related Topics 2 (1999) 27-49.

The Segal-Bargmann transform for path spaces in groups, Brian C. Hall and Ambar N. Sengupta, Journal of Functional Analysis 152 (1998).