Hardcover ISBN: 978-1-4704-7431-7
Expected availability date: June 12, 2024
Graduate Studies in Mathematics Volume: 241;
2024; 293 pp
MSC: Primary 14; 90; 68; 12; 52
This book provides a comprehensive and user-friendly exploration of the tremendous recent developments that reveal the connections between real algebraic geometry and optimization, two subjects that were usually taught separately until the beginning of the 21st century. Real algebraic geometry studies the solutions of polynomial equations and polynomial inequalities over the real numbers. Real algebraic problems arise in many applications, including science and engineering, computer vision, robotics, and game theory. Optimization is concerned with minimizing or maximizing a given objective function over a feasible set. Presenting key ideas from classical and modern concepts in real algebraic geometry, this book develops related convex optimization techniques for polynomial optimization. The connection to optimization invites a computational view on real algebraic geometry and opens doors to applications.
Intended as an introduction for students of mathematics or related fields at an advanced undergraduate or graduate level, this book serves as a valuable resource for researchers and practitioners. Each chapter is complemented by a collection of beneficial exercises, notes on references, and further reading. As a prerequisite, only some undergraduate algebra is required.
Undergraduate and graduate students interested in real algebraic geometry and polynomial and semidefinite optimization.
Foundations
Univariate real polynomials
From polyhedra to semialgebraic sets
The Tarski-Sidenberg principle and elimination of quantifiers
Cylindrical algebraic decomposition
Linear, semidefinite, and conic optimization
Positive polynomials, sums of suares and convexity
Positive polynomials
Polynomial optimization
Spectrahedra
Outlook
Stable and hyperbolic polynomials
Relative entropy methods in semialgebraic optimzation
Background material
Notation
Bibliography
Index
Hardcover ISBN: 978-1-4704-7655-7
Product Code: GSM/242
Softcover ISBN: 978-1-4704-7677-9
Product Code: GSM/242.S
Expected availability date: July 06, 2024
Graduate Studies in Mathematics Volume: 242;
2024; 179 pp
MSC: Primary 32; 30; 37;
This textbook offers an accessible introduction to translation surfaces. Building on modest prerequisites, the authors focus on the fundamentals behind big ideas in the field: ergodic properties of translation flows, counting problems for saddle connections, and associated renormalization techniques. Proofs that go beyond the introductory nature of the book are deftly omitted, allowing readers to develop essential tools and motivation before delving into the literature.
Beginning with the fundamental example of the flat torus, the book goes on to establish the three equivalent definitions of translation surface. An introduction to the moduli space of translation surfaces follows, leading into a study of the dynamics and ergodic theory associated to a translation surface. Counting problems and group actions come to the fore in the latter chapters, giving a broad overview of progress in the 40 years since the ergodicity of the Teichmuller geodesic flow was proven. Exercises are included throughout, inviting readers to actively explore and extend the theory along the way.
Translation Surfaces invites readers into this exciting area, providing an accessible entry point from the perspectives of dynamics, ergodicity, and measure theory. Suitable for a one- or two-semester graduate course, it assumes a background in complex analysis, measure theory, and manifolds, while some familiarity with Riemann surfaces and ergodic theory would be beneficial.
Graduate students and researchers interested in research questions related to translation surfaces.
Introduction
Three definitions
Moduli spaces of translation surfaces
Dynamical systems and ergodic theory
Renormalization
Counting and equidistribution
Lattice surfaces
Conclusion
Bibliography
Index
A C * -algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's KK-theory to a commutative C
? -algebra. This paper is motivated by the problem of establishing the range of validity of the UCT, and in particular, whether the UCT holds for all nuclear C
? -algebras.
We introduce the idea of a C ?
-algebra that "decomposes" over a class C of C ? -algebras. Roughly, this means that locally there are approximately central elements that approximately cut the C
?
-algebra into two C ?
-subalgebras from C that have well-behaved intersection. We show that if a C ?
-algebra decomposes over the class of nuclear, UCT C ?
-algebras, then it satisfies the UCT. The argument is based on a Mayerr?Vietoris principle in the framework of controlled KK-theory; the latter was introduced by the authors in an earlier work. Nuclearity is used via Kasparov's Hilbert module version of Voiculescu's theorem, and Haagerup's theorem that nuclear C ? -algebras are amenable.
We say that a C ? -algebra has finite complexity if it is in the smallest class of C ? -algebras containing the finite-dimensional C? -algebras, and closed under decomposability; our main result implies hat all C
? -algebras in this class satisfy the UCT. The class of C ? -algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a C
? -algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear C
? -algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear C ? -algebras.
ISBN print 978-3-98547-066-2
This is the first in a series of papers on projective positive energy representations of gauge groups. Let Ξ→M be a principal fiber bundle, and let Γ
c
?
(M,Ad(Ξ)) be the group of compactly supported (local) gauge transformations. If P is a group of “space?time symmetries” acting on Ξ→M, then a projective unitary representation of Γ
c
?
(M,Ad(Ξ))?P is of \textit{positive energy} if every “timelike generator” p
0
?
∈p gives rise to a Hamiltonian H(p
0
?
) whose spectrum is bounded from below. Our main result shows that in the absence of fixed points for the cone of timelike generators, the projective positive energy representations of the connected component Γ
c
?
(M,Ad(Ξ))
0
?
come from 1-dimensional P-orbits. For compact M this yields a complete classification of the projective positive energy representations in terms of lowest weight representations of affine Kac?Moody algebras. For noncompact M, it yields a classification under further restrictions on the space of ground states.
In the second part of this series we consider larger groups of gauge transformations, which contain also global transformations. The present results are used to localize the positive energy representations at (conformal) infinity.
ISBN print
978-3-98547-067-9
We investigate the problem of the Levy flight foraging hypothesis in an ecological niche described by a bounded region of space, with either absorbing or reflecting boundary conditions.
To this end, we consider a forager diffusing according to a fractional heat equation in a bounded domain and we define several efficiency functionals whose optimality is discussed in relation to the fractional exponent s∈(0,1) of the diffusive equation.
Such an equation is taken to be the spectral fractional heat equation (with Dirichlet or Neumann boundary conditions).
We analyze the biological scenarios in which a target is close to the forager or far from it. In particular, for all the efficiency functionals considered here, we show that if the target is close enough to the forager, then the most rewarding search strategy will be in a small neighborhood of s=0.
Interestingly, we show that s=0 is a global pessimizer for some of the efficiency functionals. From this, together with the aforementioned optimality results, we deduce that the most rewarding strategy can be unsafe or unreliable in practice, given its proximity with the pessimizing exponent, thus the forager may opt for a less performant, but safer, hunting method.
The biological literature has collected several pieces of evidence of foragers diffusing with very low Levy exponents, often in relation with a high energetic content of the prey. It is thereby suggestive to relate these patterns, which are induced by distributions with a very fat tail, with a high-risk/high-gain strategy, in which the forager adopts a potentially very profitable, but also potentially completely unrewarding, strategy due to the high value of the possible outcome.