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34264 results in Mathematics (general)

Page 98 of 1714

  • 12 - Incidence Applications in Rd

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 172-181

  • Summary

    In Chapter 11 we derived general point-variety incidence bounds in R^d. We now study two applications of these bounds. These applications do not require reading any part of Chapter 11, except for the statement of Theorem 11.3. The first application comes from discrete geometry and is another distinct distances problem. Specifically, it is a distinct distances problem with local properties. The second application is a discrete Fourier restriction problem from harmonic analysis. For simplicity, we only discuss the combinatorial aspect of that problem. This aspect is the additive energy of points on a hypersphere.

Polynomial Methods and Incidence Theory
  • Adam Sheffer
  • Published online:
    17 March 2022
    Print publication:
    24 March 2022
  • The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture, the cap set conjecture, Erdős's distinct distances problem, the joints problem, as well as others, thanks to the introduction of new polynomial methods. There has also been significant progress on a variety of problems from additive combinatorics, discrete geometry, and more. This book gives a detailed yet accessible introduction to these new polynomial methods and their applications, with a focus on incidence theory. Based on the author's own teaching experience, the text requires a minimal background, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front. The techniques are presented gradually and in detail, with many examples, warm-up proofs, and exercises included. An appendix provides a quick reminder of basic results and ideas.

Facets of Algebraic Geometry
  • A Collection in Honor of William Fulton's 80th Birthday
  • Volume 2
  • Edited by Paolo Aluffi, David Anderson, Milena Hering, Mircea Mustaţă, Sam Payne
  • Published online:
    14 March 2022
    Print publication:
    07 April 2022
  • Written to honor the 80th birthday of William Fulton, the articles collected in this volume (the second of a pair) present substantial contributions to algebraic geometry and related fields, with an emphasis on combinatorial algebraic geometry and intersection theory. Featured include commutative algebra, moduli spaces, quantum cohomology, representation theory, Schubert calculus, and toric and tropical geometry. The range of these contributions is a testament to the breadth and depth of Fulton's mathematical influence. The authors are all internationally recognized experts, and include well-established researchers as well as rising stars of a new generation of mathematicians. The text aims to stimulate progress and provide inspiration to graduate students and researchers in the field.

Facets of Algebraic Geometry
  • A Collection in Honor of William Fulton's 80th Birthday
  • Volume 1
  • Edited by Paolo Aluffi, David Anderson, Milena Hering, Mircea Mustaţă, Sam Payne
  • Published online:
    14 March 2022
    Print publication:
    07 April 2022
  • Written to honor the 80th birthday of William Fulton, the articles collected in this volume (the first of a pair) present substantial contributions to algebraic geometry and related fields, with an emphasis on combinatorial algebraic geometry and intersection theory. Featured topics include commutative algebra, moduli spaces, quantum cohomology, representation theory, Schubert calculus, and toric and tropical geometry. The range of these contributions is a testament to the breadth and depth of Fulton's mathematical influence. The authors are all internationally recognized experts, and include well-established researchers as well as rising stars of a new generation of mathematicians. The text aims to stimulate progress and provide inspiration to graduate students and researchers in the field.

Matrix Analysis and Entrywise Positivity Preservers
  • Apoorva Khare
  • Published online:
    10 March 2022
    Print publication:
    31 March 2022
  • Matrices and kernels with positivity structures, and the question of entrywise functions preserving them, have been studied throughout the 20th century, attracting recent interest in connection to high-dimensional covariance estimation. This is the first book to systematically develop the theoretical foundations of the entrywise calculus, focusing on entrywise operations - or transforms - of matrices and kernels with additional structure, which preserve positive semidefiniteness. Designed as an introduction for students, it presents an in-depth and comprehensive view of the subject, from early results to recent progress. Topics include: structural results about, and classifying the preservers of positive semidefiniteness and other Loewner properties (monotonicity, convexity, super-additivity); historical connections to metric geometry; classical connections to moment problems; and recent connections to combinatorics and Schur polynomials. Based on the author's course, the book is structured for use as lecture notes, including exercises for students, yet can also function as a comprehensive reference text for experts.

Pop-Up Geometry
  • The Mathematics Behind Pop-Up Cards
  • Joseph O'Rourke
  • Published online:
    03 March 2022
    Print publication:
    24 March 2022
  • Anyone browsing at the stationery store will see an incredible array of pop-up cards available for any occasion. The workings of pop-up cards and pop-up books can be remarkably intricate. Behind such designs lies beautiful geometry involving the intersection of circles, cones, and spheres, the movements of linkages, and other constructions. The geometry can be modelled by algebraic equations, whose solutions explain the dynamics. For example, several pop-up motions rely on the intersection of three spheres, a computation made every second for GPS location. Connecting the motions of the card structures with the algebra and geometry reveals abstract mathematics performing tangible calculations. Beginning with the nephroid in the 19th-century, the mathematics of pop-up design is now at the frontiers of rigid origami and algorithmic computational complexity. All topics are accessible to those familiar with high-school mathematics; no calculus required. Explanations are supplemented by 140+ figures and 20 animations.

Page 98 of 1714