Gereon Quick

Differential Topology

Details

This book is an advanced introduction to differential topology, a branch of mathematics studying smooth manifolds and smooth maps
between them. It explains how geometric and topological structures interact and develops key tools used in modern geometry and topology.

The text covers:

It is aimed at:

Explanation / Book Description

According to the publisher description:

“Differential topology uncovers the hidden structure of smooth spaces.”

The book begins with the foundations of manifolds and smooth maps, then develops increasingly sophisticated ideas such as:

The author combines:

making the book suitable both for coursework and self-study.

Table of Contents
  1. Introduction
  2. A Brief Introduction to Topological Spaces
  3. Smooth Manifolds
  4. The Inverse Function Theorem, Immersions and Embeddings
  5. Submersions and Regular Values
  6. Transversality
  7. Abstract Smooth Manifolds
  8. Whitney’s Embedding Theorems
  9. Smooth Homotopy
  10. Manifolds with Boundary
  11. Brouwer Fixed Point Theorem
  12. The Brouwer Degree Modulo 2
  13. Tubular Neighbourhoods and Transversality
  14. Intersection Theory Modulo 2
  15. Orientation
  16. The Integer-Valued Brouwer Degree
  17. Pontryagin Construction and Hopf’s Degree Theorem
  18. Vector Fields and the Poincaré–Hopf Index Theorem
  19. Appendix: Solutions to Selected Exercises
  20. References
  21. Index


Mark Bollman

Blackjack Mathematics for Non-Mathematicians

Book Details

This book explains the mathematics behind the casino game blackjack (21) in a way that is accessible to readers without advanced mathematical training. It focuses on probability, strategy, and expected outcomes using mostly high-school-level algebra.

Topics include:

The book is designed for:

Explanation / Description

Publisher descriptions state that the emphasis is on:

“the mathematics that underlies the game”
while remaining understandable to “the educated citizen” with only basic algebra knowledge.

The author explores:

The text reportedly includes:

Available Information About the Table of Contents

A full official Table of Contents was not publicly available in the search results yet, probably because the book is still a forthcoming 2026 release. However, descriptions suggest coverage of chapters related to:

  1. Basics of Blackjack
  2. Probability and Card Counting Concepts
  3. Expected Value and Odds
  4. Basic Strategy Mathematics
  5. Rule Variations and Their Effects
  6. Casino Advantage Analysis
  7. Statistical Decision Making
  8. Blackjack Simulations and Examples
  9. Exercises and Solutions




: Philip J. Davis

The Schwarz Function and Its Applications

Details

This book studies the Schwarz function, an important concept in complex analysis and differential geometry. The subject originates from the work of Hermann Amandus Schwarz on reflection principles for analytic curves.

The book explores:

It is written for readers with background knowledge in:

The text is known for combining:

Explanation / Publisher Description

The publisher summary explains that Schwarz generalized reflection from straight lines and circles to arbitrary analytic arcs. By taking the complex conjugate of Schwarzian reflection, one obtains the Schwarz function, which becomes a powerful analytic tool.

The author emphasizes that the Schwarz function:

The book presents many classical topics from a fresh viewpoint and demonstrates how changing mathematical perspective can generate entirely new problems and theories.

Table of Contents
  1. Prologue
  2. Conjugate Coordinates in the Plane
  3. Elementary Geometric Facts
  4. The Nine-Point Circle
  5. The Schwarz Function for an Analytic Arc
  6. Geometrical Interpretation of the Schwarz Function
  7. Schwarzian Reflection
  8. The Schwarz Function and Differential Geometry
  9. Conformal Maps, Reflections, and Their Algebra
  10. What Figure Is the −1\sqrt{-1} Power of a Circle?
  11. Properties in the Large of the Schwarz Function
  12. Derivatives and Integrals
  13. Application to Elementary Fluid Mechanics
  14. The Schwarz Function and the Dirichlet Problem
  15. Schwarz Functions of Specified Type
  16. Schwarz Functions and Iteration
  17. Dictionary of Functional Relationships
  18. Bibliographical and Supplementary Notes
  19. Bibliograph



    Authors: Alberto Cabada, Francisco Javier Fernández


    Lebesgue Measure and Integration Theory: Foundations and Solved Exercises

    Details

    This book is an advanced undergraduate/graduate-level introduction to Lebesgue measure theory and Lebesgue integration, intended mainly for students in mathematics and physics. It develops the theoretical foundations rigorously while also emphasizing practical understanding through solved exercises.

    Core topics include:

    The book combines:

    Explanation / Description

    Publisher descriptions state that the book provides:

    “the complete theoretical underpinnings”

    of modern measure and integration theory, adapted for advanced undergraduate and graduate students.

    The text begins with basic measure spaces and gradually develops the full machinery of Lebesgue integration. Compared with elementary calculus, Lebesgue theory allows much more powerful convergence theorems and forms the basis of:

    The inclusion of solved exercises makes the book especially suitable for:

    Table of Contents

    The publicly available TOC currently lists:

    1. Measure Spaces and Lebesgue Measure
    2. Measurable Functions
    3. The Lebesgue Integral
    4. The Relationship between Differentiation and Integration on R\mathbb{R}
    5. Product Measures


      Author: Sorin Bangu


      Philosophy of Mathematics after Wittgenstein

      • Publisher: Cambridge University Press
      • Publication Date: June 30, 2026
      • Format: Hardback
      • Pages: 300
      • Language: English
      • ISBN-13: 9781009574891
      Details

      This book examines the philosophy of mathematics through the later philosophy of Ludwig Wittgenstein. The author argues that Wittgenstein’s approach dissolves many traditional philosophical problems about mathematics instead of solving them in the conventional sense.

      The work focuses on:

      • normativism in mathematics
      • rule-following
      • objectivity and skepticism
      • mathematical proofs
      • logicism
      • infinity
      • Cantor’s arguments
      • the relationship between philosophy and mathematics

      The book is aimed at:

      • philosophers of mathematics
      • logicians
      • graduate students
      • researchers interested in Wittgensteinian philosophy.
      Explanation / Description

      The publisher description explains that Wittgenstein wanted philosophical problems to “completely disappear.” Sorin Bangu reconstructs this idea and applies it to classical problems in the philosophy of mathematics.

      Key themes include:

      • Wittgenstein’s “normativist” view of mathematics
      • the therapeutic role of philosophy
      • elimination of traditional metaphysical puzzles
      • reinterpretation of mathematical objectivity
      • analysis of mathematical practice rather than abstract ontology

      The author also discusses how Wittgenstein’s views relate to modern debates concerning:

      • foundations of mathematics
      • formalism
      • logicism
      • mathematical realism
      • skepticism.
      Table of Contents

      The following TOC is publicly available:

      1. Introduction: a disappearing act
      2. Wittgenstein's normativism about mathematics
      3. Normativism as eliminativism: three illustrations
      4. Basic mathematical rules (I): hardening
      5. Basic mathematical rules (II): objectivity, agreement, and skepticism
      6. Advanced rules: proofs and concept-formation
      7. Cantor's proof (I): a “style of thinking”
      8. Cantor's proof (II): reluctant nonrevisionism
      9. Logicism, number and the infinite
      10. Conclusion: a cleared ground
      11. Bibliography