Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Noncommutative Geometry, Quantum Fields and Motives

Gauge theory for embedded surfaces, II

Integral formulas in Riemannian geometry

By Lipman Bers London : Chapman and Hall Ltd. Pp. xv + 164. Price 62s. In the preface to this, the third volume in Surveys in Applied Mathematics, F. J. Weyl the Director of the Mathematical Sciences Division of the Office of Naval Research gives broadly speaking two aims as the basis for the series. These are (i) the need to make readily available up-to-date results in selected areas of research and (ii) the need to relate contributions from both sides of the iron curtain in order to produce a balanced appreciation of the present state of knowledge in any such area.

https://iopscience.iop.org/article/10.1088/0031-9112/10/5/018/pdf

Mathematical Aspects of Subsonic and Transonic Gas Dynamics

In this paper, we continue to consider the 2-dimensional (open) Toda system (Toda lattice) for $SU(N+1)$ We give a much more precise bubbling behavior of solutions and study its existence in some critical cases

Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions

For a connection on a principalSU(2) bundle over a base space with a codimension two singular set, a limit holonomy condition is stated. In dimension four, finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy. If there is no holonomy, then a codimension two removable singularity theorem is obtained.

/pdf/classification-of-singular-sobolev-connections-by-their-2vh2p2hqym.pdf

Classification of singular Sobolev connections by their holonomy

An Abelian gauge field theory framed on a complex line bundle L over a compact Riemann surface M is developed which allows the coexistence, simultaneously in the same model, of magnetic vortices and antivortices represented by the N zeros and P poles of a section of L. The quantized minimum energy E is given in terms of the first Chern class c 1 (L) and by a certain intersection number obtained from the multivortices. We show that E = 2π(N + P). To realize such topological invariants as minimum energies, an existence and uniqueness theorem is established under the necessary and sufficient condition that |c 1 (L)| = |N − P|

Abelian gauge theory on Riemann surfaces and new topological invariants

Motivated by Born–Infeld geometric electromagnetic theory, we consider a series of nonlinear equations which extend the minimal surface equations and the related, generalized, Bernstein problems. We study the relation of these equations and the conditions which lead to the triviality of the solutions. We also study a non-Abelian extension of these equations and establish a gap theorem for the Yang–Mills–Born–Infeld fields. We then couple the Born–Infeld electromagnetism with a Higgs scalar field and obtain an existence theorem for the self-dual multiple cosmic string solutions on a closed surface characterized jointly by the first Chern class and the Thom class formulated over the hosting complex line bundle.

https://iopscience.iop.org/article/10.1088/0951-7715/20/5/008/pdf

Generalized Bernstein property and gravitational strings in Born?Infeld theory

It has been shown in the work of Chakrabarti, Sherry and Tchrakian that the chiral SO±(4 p) Yang–Mills theory in the Euclidean 4 p (p≥ 2) dimensions allows an axially symmetric self-dual system of equations similar to Witten's instanton equations in the classical 4-dimensional SU(2)∼SO±(4) theory and the solutions represent a new class of instantons. However the rigorous existence of these higher-dimensional instanton solutions has remained open except for the solution of unit charge representing a single instanton. In this paper we establish an existence and uniqueness theorem for multi-instantons of arbitrary charges in the case p≥ 2. These solutions are the first known instantons, with the Chern–Pontryagin index greater than one, of the Yang–Mills model in higher dimensions. Our approach is a study of a nonlinear variational equation defined on the Poincare half plane.

Multiple Instantons Representing Higher-Order Chern-Pontryagin Classes, II

We consider a certain invariantly defined nonlinear system of partial differential equations on a Riemannian manifold. Since a special case describes a steady, irrotional, compressible flow on the manifold, it is natural to refer to the (square of) the pointwise norm of the solution as the speed of the flow and to the density of the flow. Under appropriate restrictions on the density, the system is elliptic and we obtain a sub-elliptic estimate and a maximum principle for the speed of the flow in terms of the curvature of the manifold.

/pdf/a-sub-elliptic-estimate-for-a-class-of-invariantly-defined-3trjc202mw.pdf

Robert Sibner

Papers

Classification of singular Sobolev connections by their holonomy

Abelian gauge theory on Riemann surfaces and new topological invariants

Generalized Bernstein property and gravitational strings in Born?Infeld theory

Multiple Instantons Representing Higher-Order Chern-Pontryagin Classes, II

A sub-elliptic estimate for a class of invariantly defined elliptic systems