* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

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Torus Actions and their Applications in Topology and Combinatorics

We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low energy effective field theory approximations, and thereby produce a general formula for the group of Symmetry Protected Topological (SPT) phases in terms of Thom's bordism spectra; the only input is the dimension and symmetry group. We provide computations for fermionic systems in physically relevant dimensions. Other topics include symmetry in quantum field theories, a relativistic 10-fold way, the homotopy theory of relativistic free fermions, and a topological spin-statistics theorem.

Reflection positivity and invertible topological phases

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area

Toric Topology

The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces

A combinatorial construction is used to analyze the properties of polyhedral products and generalized moment-angle complexes with respect to certain operations on CW pairs including exponentiation. This allows for the construction of infinite families of toric manifolds, associated to a given one, in a way which simplifies the combinatorial input and consequently, the presentation of the cohomology rings. The new input is the interaction of a purely combinatorial construction with natural associated geometric constructions related to polyhedral products and toric manifolds. Applications of the methods and results developed here have appeared in literature.

https://www.intlpress.com/site/pub/files/_fulltext/journals/hha/2015/0017/0002/HHA-2015-0017-0002-a008.pdf

Operations on polyhedral products and a new topological construction of infinite families of toric manifolds

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or {\it partial product spaces} which arise as {\it polyhedral product functors} described below. 
In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in Goresky-MacPherson \cite{goresky.macpherson}, Hochster\cite{hochster}, Baskakov \cite{baskakov}, Panov \cite{panov}, and Buchstaber-Panov \cite{buchstaber.panov}. Since the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter \cite{porter} and Ganea \cite{ganea}. The spirit of the work here follows that of Denham-Suciu in \cite{denham.suciu}.

The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces

This article gives a natural decomposition of the suspension of a generalized moment-angle complex or partial product space which arises as the polyhedral product functor described below. The introduction and application of the smash product moment-angle complex provides a precise identification of the stable homotopy type of the values of the polyhedral product functor. One direct consequence is an analysis of the associated cohomology. For the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in earlier work of others as described below. Because the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. Implied, therefore, is a decomposition for the Stanley–Reisner ring of a finite simplicial complex, and natural generalizations.

https://www.pnas.org/content/pnas/106/30/12241.full.pdf

Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces.

https://msp.org/pjm/1987/128-1/pjm-v128-n1-p01-s.pdf

The eta invariant, ${\rm Pin}^c$ bordism, and equivariant ${\rm Spin}^c$ bordism for cyclic $2$-groups.

We describe the integral equivariant cohomology of a weighted projective space in terms of piecewise polynomials, as well as by generators and relations. Unlike the ordinary integral cohomology, this ring distinguishes among weighted projective spaces. We also prove a Chern class formula for weighted
projective bundles.

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Anthony Bahri

Papers

Operations on polyhedral products and a new topological construction of infinite families of toric manifolds

The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces

Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces.

The eta invariant, ${\rm Pin}^c$ bordism, and equivariant ${\rm Spin}^c$ bordism for cyclic $2$-groups.

The equivariant cohomology of weighted projective spaces