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Handbook of Geometry and Topology of
Singularities VIII

Handbook of Geometry and Topology of Singularities VIII

Synthesis Lectures on Mathematics & Statistics

This book presents in-depth explanations of well-known and recognized behaviors of neural networks in machine learning.&nbsp; In addition, the author provides novel technical analyses of behaviors of discrete-time dynamical systems modeled as difference equations. &nbsp;These analyses and their outcomes are closely related to models of very well-known neural networks such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) neural networks, which are widely used in machine learning and artificial intelligence (AI) applications.&nbsp; The author also discusses difference equations and their relevance to neural networks, machine learning, and AI.
&nbsp;In addition, this book:
<ul>
<li>Includes characterizations of difference equations and technical prospectives of discrete-time systems</li>
<li>Provides new insights into the dynamical behaviors of some of the most popular neural networks used in machine learning</li>
<li>Discusses novel technical analyses of discrete-time dynamical systems modeled as difference equations</li>
</ul>

Difference Equations and Machine Learning

SISSA Springer Series

Topological Degree with Applications

Princeton Mathematical Series

"A natural question that sits at the nexus of algebraic geometry, differential geometry, and geometric group theory is: which groups can be realized as fundamental groups of compact Kèahler manifolds, called "Kèahler groups"? Roughly speaking, the fundamental group of a manifold measures the number of "holes." Many restrictions are known, and many examples are known; but mathematicians are far from having a precise conjecture about which groups are Kèahler. The question serves as a fruitful connection between several major areas of geometry and complex analysis. Py's book is an up-to-date pedagogical survey of the central theorems and methods for the study of Kèahler groups including, where illuminating, detailed proofs. It includes results of Gromov, Schoen, Napier, Ramachandran, Corlette, Simpson, Delzant, Arapura, and Nori. The charm of the subject is that different methods yield information of different flavors, and the challenge is to draw these threads together. This book leans toward geometric group theory, but it gives a coherent account of great value to anyone interested in Kèahler groups - and in Kèahler manifolds more broadly. The emphasis is on unity and cross-fertilization among approaches"--

Lectures on Kähler Groups

Advanced Fractional Dynamic Equations on Time Scales

Boundary Value Problems

Frontiers in Mathematics

This monograph explores the finiteness and structure of Shafarevich-Tate groups of abelian varieties over global fields of any characteristic. Readers will better understand how the methods of Euler systems and Kolyvagin systems can be adapted to Heegner points and CM points. Also offered is a comprehensive overview of the most important classical and recent results on these objects. Shafarevich-Tate Groups will be a valuable resource to those interested in this active area of research.

Shafarevich-Tate Groups

Compact Textbooks in Mathematics

This book serves as an introduction to topology, a branch of mathematics that studies the qualitative properties of geometric objects. It is designed as a bridge between elementary courses in analysis and linear algebra and more advanced classes in algebraic and geometric topology, making it particularly suitable for both undergraduate and graduate mathematics students. Additionally, it can be used for self-study.The authors employ the modern language of category theory to unify and clarify the concepts presented, with definitions supported by numerous examples and illustrations. The book includes over 170 exercises that reinforce and deepen the understanding of the material. Many sections feature brief insights into advanced topics, providing a foundation for study projects or seminar presentations.In addition to set-theoretic topology, the book covers essential concepts such as fundamental groups, covering spaces, bundles, sheaves, and simplicial methods, which are vital in contemporary geometry and topology.

A Basic Course in Topology

Princeton Legacy Library

An authoritative introduction to the essential features of étale cohomology A. Grothendieck’s work on algebraic geometry is one of the most important mathematical achievements of the twentieth century. In the early 1960s, he and M. Artin introduced étale cohomology to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry but also in several different branches of number theory and in the representation theory of finite and p-adic groups. In this classic book, James Milne provides an invaluable introduction to étale cohomology, covering the essential features of the theory. Milne begins with a review of the basic properties of flat and étale morphisms and the algebraic fundamental group. He then turns to the basic theory of étale sheaves and elementary étale cohomology, followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Milne proves the fundamental theorems in étale cohomology—those of base change, purity, Poincaré duality, and the Lefschetz trace formula—and applies these theorems to show the rationality of some very general L-series.

Étale Cohomology

An accessible, motivated introduction to one of the most dynamic areas of mathematicsDecades ago, Mumford wrote that algebraic geometry "seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics." The revolution has now fully come to pass and has fundamentally changed how we think about many fields of mathematics. This book provides a thorough foundation in the powerful ideas that now shape the landscape, with an informal yet rigorous exposition that builds intuition for the formidable machinery. It begins with a discussion of categorical thinking and sheaves and then develops the notion of schemes and varieties as examples of "geometric spaces" before discussing their specific aspects. The book goes on to cover topics such as dimension and smoothness, vector bundles and their natural generalizations, and important cohomological tools and their applications. Important optional topics are included in starred sections. A comprehensive introduction certain to become the standard book on the subjectFeatures a wealth of exercises that enable students to learn by doingRequires few prerequisites, developing the tools students need to succeed, from category theory and sheaf theory to commutative and homological algebraUses an example-driven approach that builds mathematical intuitionA self-contained textbook for graduate students and an essential reference for researchers

Foundations of Algebraic Geometry

The Rising Sea

VON NULL AUF MATHE

Im Bogen zum Ziel: quadratische Gleichungen und Funktionen

This textbook has been designed to support the initial study of Complex Analysis, progressing to Complex Dynamics. It focuses on the fundamental aspects of one-variable complex functions, covering the geometric theory and dynamics of iterations of rational mappings. Following the standard material, the book delves into an extensive range of advanced topics, encompassing the requirements for a one-year graduate-level course or a preliminary exam.In this work, the reader will discover three distinctive characteristics: it simplifies and unifies ideas and concepts that might appear disparate or complicated in real analysis; it contributes to the development of other areas in mathematics; and it showcases relevance for applications in Science and Engineering, with many exercises. Historical notes throughout the text help to contextualize the theory.With its flexible structure, this textbook provides a solid foundation for a first course in Complex Analysis and for a second more advanced course, establishing a robust basis for subsequent studies.

Complex Analysis and Dynamics in One Variable with Applications

Advanced Courses in Mathematics - CRM Barcelona

This book presents the notes originating from five series of lectures given at the CRM Barcelona in 21-25 June, 2021, during the “Higher homotopical structures” programme.Since their introduction 60 years ago, the notions of infinity algebras (Stasheff, Sugawara), higher categories (Boardman-Vogt), operads (May), and model categories (Quillen) have given rise to powerful new tools which made possible the resolution of open problems and prompted revolutions in many domains like algebraic topology (rational homotopy theory, faithful algebraic invariants of the homotopy type of spaces), deformation theory (formality theorems, formal moduli problems), and mathematical physics (quantization of Poisson manifolds, quantum field theories), to name but a few.&nbsp;This theory of higher structures using operadic calculus is currently under rapid development. The aim of this book is to provide the community with an accessible state-of-the-art, while at the same time giving interested researchers and advanced students a brief overview on the subject.

Higher Structures and Operadic Calculus

Series In Algebraic And Differential Geometry

Unstable Homotopy Decompositions

Handbook of Geometry and Topology of Singularities VIII

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