MAA Press: An Imprint of the American Mathematical Society
The Carus Mathematical Monographs. Volume: 37
2021; 181 pp; Softcover
Print ISBN: 978-1-4704-6031-0
Erd?s asked how many distinct distances must there be in a set of n points in the plane. Falconer asked a continuous analogue, essentially asking what is the minimal Hausdorff dimension required of a compact set in order to guarantee that the set of distinct distances has positive Lebesgue measure in R. The finite field distance problem poses the analogous question in a vector space over a finite field. The problem is relatively new but remains tantalizingly out of reach. This book provides an accessible, exciting summary of known results. The tools used range over combinatorics, number theory, analysis, and algebra. The intended audience is graduate students and advanced undergraduates interested in investigating the unknown dimensions of the problem. Results available until now only in the research literature are clearly explained and beautifully motivated. A concluding chapter opens up connections to related topics in combinatorics and number theory: incidence theory, sum-product phenomena, Waring's problem, and the Kakeya conjecture.
Undergraduate and graduate students and researchers interested in combinatorics.
Mathematical Surveys and Monographs,Volume: 255
2021; 192 pp; Softcover
Print ISBN: 978-1-4704-6436-3
This book is dedicated to the qualitative theory of the stochastic one-dimensional Burgers equation with small viscosity under periodic boundary conditions and to interpreting the obtained results in terms of one-dimensional turbulence in a fictitious one-dimensional fluid described by the Burgers equation. The properties of one-dimensional turbulence which we rigorously derive are then compared with the heuristic Kolmogorov theory of hydrodynamical turbulence, known as the K41 theory. It is shown, in particular, that these properties imply natural one-dimensional analogues of three principal laws of the K41 theory: the size of the Kolmogorov inner scale, the 2/3-law, and the Kolmogorov?Obukhov law.
The first part of the book deals with the stochastic Burgers equation, including the inviscid limit for the equation, its asymptotic in time behavior, and a theory of generalised L1-solutions. This section makes a self-consistent introduction to stochastic PDEs. The relative simplicity of the model allows us to present in a light form many of the main ideas from the general theory of this field. The second part, dedicated to the relation of one-dimensional turbulence with the K41 theory, could serve for a mathematical reader as a rigorous introduction to the literature on hydrodynamical turbulence, all of which is written on a physical level of rigor.
Graduate students and researchers interested in stochastic differential equations and turbulence.
Mathematical Surveys and Monographs, Volume: 256
2021; 192 pp; Softcover
Print ISBN: 978-1-4704-6552-0
Integrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.
Going through the book, readers will find themselves at the forefront of this rapidly developing research field.
Undergraduate students and researchers interested in mathematical aspects of integrable models in quantum field theory.
Graduate Studies in Mathematics, Volume: 213
2021; 322 pp; Hardcover
Print ISBN: 978-1-4704-6511-7
This book gives an extensive survey of many important topics in the theory of Hamilton?Jacobi equations with particular emphasis on modern approaches and viewpoints. Firstly, the basic well-posedness theory of viscosity solutions for first-order Hamilton?Jacobi equations is covered. Then, the homogenization theory, a very active research topic since the late 1980s but not covered in any standard textbook, is discussed in depth. Afterwards, dynamical properties of solutions, the Aubry?Mather theory, and weak Kolmogorov?Arnold?Moser (KAM) theory are studied. Both dynamical and PDE approaches are introduced to investigate these theories. Connections between homogenization, dynamical aspects, and the optimal rate of convergence in homogenization theory are given as well.
The book is self-contained and is useful for a course or for references. It can also serve as a gentle introductory reference to the homogenization theory.
Graduate students and researchers interested in Hamilton?Jacobi equations and viscosity solutions.
Graduate Studies in Mathematics, Volume: 214
2021; 309 pp; Hardcover
Print ISBN: 978-1-4704-6014-3
This book develops a mathematical theory for finance, based on a simple and intuitive absence-of-arbitrage principle. This posits that it should not be possible to fund a non-trivial liability, starting with initial capital arbitrarily near zero. The principle is easy-to-test in specific models, as it is described in terms of the underlying market characteristics; it is shown to be equivalent to the existence of the so-called “Kelly” or growth-optimal portfolio, of the log-optimal portfolio, and of appropriate local martingale deflators. The resulting theory is powerful enough to treat in great generality the fundamental questions of hedging, valuation, and portfolio optimization.
The book contains a considerable amount of new research and results, as well as a significant number of exercises. It can be used as a basic text for graduate courses in Probability and Stochastic Analysis, and in Mathematical Finance. No prior familiarity with finance is required, but it is assumed that readers have a good working knowledge of real analysis, measure theory, and of basic probability theory. Familiarity with stochastic analysis is also assumed, as is integration with respect to continuous semimartingales.
Graduate students and researchers interested in math finance.
Mathematical Surveys and Monographs,Volume: 257
2021; 354 pp; Softcover
MSC: Primary 42; 46; Secondary 26; 28; 31; 35
Print ISBN: 978-1-4704-6575-9
This book discusses advances in maximal function methods related to Poincare and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy's inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are self-improving, such as Hardy's inequalities, capacity density conditions, and reverse Holder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy's inequalities. At the end of the book, the theory of weak solutions to the p-Laplace equation and the use of maximal function techniques is this context are discussed.
The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.
Graduate students and researchers interested in functional analysis.