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Enumerative Geometry and String Theory

4.4 out of 5 stars 5 customer reviews
ISBN-13: 978-0821836873
ISBN-10: 0821836870
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Editorial Reviews


"The most accessible portal into very exciting recent material." ---- CHOICE Magazine

"The book contains a lot of extra material that was not included in the original fifteen lectures. It is a nicely and intuitively written remarkable little booklet covering a huge amount of interesting material describing a beautiful area, where modern mathematics and theoretical physics meet. It can give inspiration to teachers for a lecture series on the topic as well as a chance for self-study by students." ---- EMS Newsletter

"It is a welcome addition to the spectrum of available references on the topic and ideal for someone between undergraduate and beginning graduate education who wants to know more about this exciting field or for more advanced students who would like to see how the pieces of the puzzle fit together." ---- Mathematical Reviews

Product Details

  • Paperback: 206 pages
  • Publisher: American Mathematical Society (April 19, 2006)
  • Language: English
  • ISBN-10: 0821836870
  • ISBN-13: 978-0821836873
  • Product Dimensions: 0.5 x 5.8 x 8.5 inches
  • Shipping Weight: 8.8 ounces (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Best Sellers Rank: #1,877,058 in Books (See Top 100 in Books)

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Customer Reviews

Top Customer Reviews

Format: Paperback Verified Purchase
Enumerative geometry can be viewed in the non-rigorous "classical" setting of the Italian geometers, in the rigorous modern setting of sheaf theory and algebraic geometry, or in the non-rigorous setting of high-energy physics and string theory. Modern mathematics insists upon rigorous formulation for all of its constructions, so it has appropriately rejected the Italian and physicist setting for enumerative geometry. But the move to put the results of the Italian geometers on a rigorous basis resulted in much of the current field of algebraic geometry, esoteric as it may be. A similar movement is now occurring in the attempt to make rigorous some highly interesting predictions in enumerative geometry coming from physics. This has proven to be a challenge, since anyone involved in it must understand not only the mathematics behind enumerative geometry but also the physics behind string theory. The author of this book is one of the few that does have this understanding, and he has passed on some of his insights in this short but illuminating book.

The main issue in the learning of advanced mathematics, particularly an esoteric subject like enumerative geometry, has centered on the proper method by which to motivate the central concepts. To better appreciate these concepts, it is better to present many examples of them, preferably in an historical context, and then illustrate the properties that these examples have in common. One can then show how the concepts arose from abstracting or generalizing over these concepts.
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Format: Paperback
Katz's "Enumerative Geometry and String Theory" is to my knowledge a unique book, as it attempts to explain the recent and surprising connections between classical enumerative geometry and modern physics, 2 seemingly disparate fields. The intended audience were undergraduate mathematics students, who were likely to be unfamiliar with both the mathematics and physics involved, and the aim was to give them a taste of this rapidly evolving and exciting new field. Thus the goal was to provide motivation and an outline on how to go about learning all the vast material that study in this area requires, rather than actually teaching this material, or at least, that is how the book reads (perhaps the lectures were more informative).

Enumerative geometry is an area of mathematics originating in the 19th-century that seeks to answer questions such as "how many conics in the plane contain x number of distinct points and are tangent to y general lines?" The subject progressed by first rephrasing everything in the more general language of intersections of hypersurfaces in complex projective space (the author explains the reasons for these generalizations), then the problems were reduced to integrating certain cohomology classes over submanifolds, and finally formulas from string theory were derived to compute these integrals relatively easily, and this is more or less the order in which this material is presented in the book.

The first 3 chapters are devoted to describing enumerative geometry, using simple examples as motivation, first from the classical perspective and then in modern algebraic geometric terminology.
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By nehiker on December 17, 2007
Format: Paperback Verified Purchase
This is a nice, informal, introduction to enumerative geometry and string theory. The first three chapters give a flavor of the former, indicating connections between algebra and geometry and motivating the use of complex numbers and projective spaces. The concrete question "How many lines in the plane pass through k fixed points and are tangent to 5-k fixed general lines?" is raised and answered here. Chapters 4-6 introduce standard tools of algebraic topology (general topology, manifolds, basic algebraic topology); more enumerative geometry, including an introduction to excess intersection theory, follows in Chapters 7-9. For example, the author introduces rudiments of Schubert calculus (intersection theory of cycles on Grassmannians) and uses it to determine the number of lines that lie on a typical quintic 3-fold in P^4. Chapters 10-13 concern physics, and the final chapter is meant to indicate relations between the two subjects.

The book certainly achieves its aim of giving a feeling for enumerative geometry and string theory, but I do not feel it indicates the connections between the two in a meaningful way; perhaps, this would be too much to expect from such a short book. Overall, the discussion is very geometric, though for some reason the author chooses to introduce vector bundles in a rather formal way. Another, minor, drawback of the book is that the author references entire books; more specific references would often be helpful.

The book would make a great text for a one-semester advanced undergraduate seminar, with each chapter taking up about a week for discussion. While graduate students (and beyond) are likely to be familiar with the material covered in Chapters 4-6, they will likely learn many new interesting things in the rest of the book, without putting in much effort.
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