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Cancellation for Surfaces Revisited
The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism
If the cancellation does not hold then
eBook
Algebraic geometry over C∞-rings
If X is a manifold then the \\mathbb R-algebra C^\\infty (X) of smooth functions c:X\\rightarrow \\mathbb R is a C^\\infty -ring. That is, for each smooth function f:\\mathbb R^n\\rightarrow \\mathbb R there is an n-fold operation \\Phi _f:C^\\infty (X)^n\\rightarrow C^\\infty (X) acting by \\Phi _f:(c_1,\\ldots ,c_n)\\mapsto f(c_1,\\ldots ,c_n), and these operations \\Phi _f satisfy many natural identities. Thus, C^\\infty (X) actually has a far richer structure than the obvious \\mathbb R-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by C^\\infty -rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C^\\infty -schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on C^\\infty -schemes, and C^\\infty -stacks, in particular Deligne-Mumford C^\\infty-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C^\\infty-rings and C^\\infty -schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, \"derived\" versions of manifolds and orbifolds related to Spivak's \"derived manifolds\".
eBook
Modular forms
This book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat's last theorem and the approximation of pi. It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it.
eBook
Topological Complexity and Related Topics
This volume contains the proceedings of the mini-workshop on Topological Complexity and Related Topics, held from February 28-March 5, 2016, at the Mathematisches Forschungsinstitut Oberwolfach. Topological complexity is a numerical homotopy invariant, defined by Farber in the early twenty-first century as part of a topological approach to the motion planning problem in robotics. It continues to be the subject of intensive research by homotopy theorists, partly due to its potential applicability, and partly due to its close relationship to more classical invariants, such as the Lusternik-Schnirelmann category and the Schwarz genus. This volume contains survey articles and original research papers on topological complexity and its many generalizations and variants, to give a snapshot of contemporary research on this exciting topic at the interface of pure mathematics and engineering.
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Topology, Geometry, and Dynamics
Vladimir Abramovich Rokhlin (8/23/1919-12/03/1984) was one of the leading Russian mathematicians of the second part of the twentieth century. His main achievements were in algebraic topology, real algebraic geometry, and ergodic theory.The volume contains the proceedings of the Conference on Topology, Geometry, and Dynamics: V. A. Rokhlin-100, held from August 19-23, 2019, at The Euler International Mathematics Institute and the Steklov Institute of Mathematics, St. Petersburg, Russia.The articles deal with topology of manifolds, theory of cobordisms, knot theory, geometry of real algebraic manifolds and dynamical systems and related topics. The book also contains Rokhlin's biography supplemented with copies of actual very interesting documents.
eBook
Representations of Algebras, Geometry and Physics
This volume contains selected expository lectures delivered at the 2018 Maurice Auslander Distinguished Lectures and International Conference, held April 25-30, 2018, at the Woods Hole Oceanographic Institute, Woods Hole, MA.Reflecting recent developments in modern representation theory of algebras, the selected topics include an introduction to a new class of quiver algebras on surfaces, called \"geodesic ghor algebras\", a detailed presentation of Feynman categories from a representation-theoretic viewpoint, connections between representations of quivers and the structure theory of Coxeter groups, powerful new applications of approximable triangulated categories, new results on the heart of a t-structure, and an introduction to methods of constructive category theory.
eBook