Large-N Limits of Spaces and Structures by Irfan Alam and Ambar N. Sengupta. Spaces and structures varying with a numerical parameter appear in a variety of contexts. In this paper we explore a corresponding limiting structure using the method of ultraproducts. In the case of a sequence of compact groups we construct an invariant measure on the limiting group 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 acting on the ring of polynomials , where 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 , where 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 .

*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 should be .

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).