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October 25, 2006

Geometry for Americans

The new edition of Paul Zeitz's "The Art and Craft of Problem-Solving," aimed at college math majors, adds a chapter titled "'Geometry for Americans' instead of 'Geometry for Dummies' so as not to offend. The sad truth is that most mathematically inclined Americans know very little geometry, in contrast to their luckier peers in Eastern Europe and Asia. . . We call Sections 8.2-8.3 'survival geometry' because they contain a lean but adequate stock of facts and techniques that will allow you to get started on most problems. If you master the facts and lemmas and fearlessly employ the problem-solving ideas presented in the next two sections, you will be able to tackle an impressive variety of challenging questions." Zeitz has been active with many problem-solving competitions, both as a competitor and as a coach; he emphasizes the fun and exhiliration of challenging problems (as opposed to "exercises").

The Art and Craft of Pproblem Solving, by Paul Zeitz. 2nd ed. Hoboken, NJ: John Wiley, 2007. Mathematics Library QA63 .Z45 2007 Link to MnCat Record

October 23, 2006

Fields authors

Two Fields Medalists are represented in the books arriving today:

Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday, edited by Victor Ginzburg. Progress in mathematics v. 253. Boston: Birkhäuser, 2006. Mathematics Library QA564 .A365 2006 Link to MnCat Record

Drinfeld was awarded the Fields Medal in 1990 for his crucial geometric insights into the "Langlands Program," in particular his proof of the Langlands conjecture in the special case of the group GL2 over function fields, and for his outstanding contributions to quantum groups. Contributors to this festschrift include his advisor, Yu. I. Manin ("Iterated integrals of modular forms and noncommutative modular symbols").

And from Terence Tao, awarded a Fields Medal in 2006 "for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory":
Additive Combinatorics, by Terence Tao and Van Vu. New York: Cambridge University Press, 2006. Mathematics Library QA164 .T36 2006 Link to MnCat Record
Contents: 1. The probabilistic method; 2. Sum set estimates; 3. Additive geometry; 4. Fourier analytic methods; 5. Inverse sumset theorems; 6. Graph theoretic methods; 7. The Littlewood-Offord problem; 8. Incidence geometry; 9. Algebraic methods; 10. Szemerédi's theorem for k = 3; 11. Szemerédi's theorem for k > 3; 12. Long arithmetic progressions in sumsets

See Tao's Books page for some “deleted scenes:? Arithmetic Ramsey Theory, Menger’s theorem, Santalo’s inequality, Entropy sumset estimates.

October 10, 2006

Voronov on string topology

Sasha Voronov's string topology lectures, used this term in Math 8390: Topics in Mathematical Physics, have been available on the arXiv (math.GT/0503625), but now have also appeared in book form:
Ralph L. Cohen and Alexander A. Voronov. "Notes on String Topology" in String Topology and Cyclic Homology. Advanced courses in mathematics, CRM Barcelona. Basel; Boston: Birkhauser, 2006, pp. 1-95. Link to MnCat Record

After a 2003 summer school in Almeria, the paper is "the joint account of the two lecture series which focused on string topology (Cohen and Voronov). It discusses the loop product from the original point of view of Chas and Sulllivan, from the Cohen-Jones stable point of view, as well as Voronov's operadic point of view."

The book includes another paper dealing with free loop spaces: Kathryn Hess's lectures on "the construction of algebraic models for computing topological cyclic homology. Starting with the study of free loop spaces and their algebraic models, it continues with homotopy orbit spaces of circle actions, and culminates in the Hess-Rognes construction of a model for computing spectrum cohomology of topological cyclic homology."
Kathryn Hess. "An Algebraic Model for Mod 2 Topological Cyclic Homology" in String Topology and Cyclic Homology. Advanced courses in mathematics, CRM Barcelona. Basel; Boston: Birkhauser, 2006, pp. 97-163. Link to MnCat Record (also available on the arXiv: math.AT/0412271)

October 6, 2006

Hitchhiking a math route

Which areas of mathematics can you get to by hitchhiking?

Infinite Dimensional Analysis: A Hitchhiker’s Guide. Berlin; New York: Springer, 2006. Mathematics Library QA320 .A45 2006 Link to MnCat Record (Link to MnCat Record of 1999 ed.)

The Hitchhiker’s Guide to Calculus: A Calculus Course Companion, by Michael Spivak. Houston, TX: Polished Pebble Press, 1995. Mathematics Library QA303 .S783 1995 Link to MnCat Record

Complex Analysis: The Hitchhiker’s Guide to the Plane, by Ian Stewart and David Tall. Cambridge; New York: Cambridge University Press, 1983. Link to MnCat Record

Apparently only analysts like to hitchhike. Or perhaps only analysts like Douglas Adams.

October 5, 2006

Math feuds

Hal Hellman has written books on "great feuds" in science, technology, and medicine, but when considering a similar theme in mathematics, he thought he might be gravelled for lack of matter. "Mathematics, I felt, is a cold, logical discipline where questions can be decided, if not quickly, at least objectively and decisively. As opposed to, say, politics or religion, or even science, there is little room for human emotions and sensitivities. How could there be feuds in mathematics?" Needless to say, his research turned up more feuds than he could use; those he chose to include are listed below.

Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever, by Hal Hellman. Hoboken, N.J.: John Wiley & Sons, 2006. Mathematics Library QA21 .H45 2006 Link to MnCat Record
Contents: Tartaglia versus Cardano. Solving cubic equations -- Descartes versus Fermat. Analytic geometry and optics -- Newton versus Leibniz. Credit for the calculus -- Bernoulli versus Bernoulli. Sibling rivalry of the highest order -- Sylvester versus Huxley. Ivory tower or real world? -- Kronecker versus Cantor. Mathematical humbug -- Borel versus Zermelo. The "notorious axiom" -- Poincaré versus Russell. The logical foundations of mathematics -- Hilbert versus Brouwer. Formalism versus intuitionism -- Absolutists/platonists versus fallibilists/constructionists. Are mathematical advances discoveries or inventions?

October 2, 2006

Radically elementary mathematics

The intriguing phrase in the title
Diffusion, Quantum Theory, and Radically Elementary Mathematics, edited by William G. Faris. Mathematical notes 47. Princeton, NJ : Princeton University Press, 2006.
TC Mathematics Library QC21.3 .D54 2006 Link to MnCat Record

turns out not to be new:
Radically Elementary Probability Theory, by Edward Nelson. Annals of mathematics studies no. 117. Princeton, N.J.: Princeton University Press, 1987. TC Mathematics Library QA1 .A626 no.117 Link to MnCat Record

The connection between the books is explicit, since the more recent one resulted from a conference in honor of the author of the older one, Edward Nelson (Princeton).

From the former:
"This volume explains diffusion motion and its relation to both nonrelativistic quantum theory and quantum field theory. It also shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis. . . an infinitesimal approach to diffusion and related probability topics that is radically elementary in the sense that it relies only on simple logical principles."

From the latter:
"This work is an attempt to lay new foundations for probability theory, using a tiny bit of nonstandard analysis. The mathematical background required is little more than that which is taught in high school, and it is my hope that it will make deep results from the modern theory of stochastic processes readily available to anyone who can add, multiply, and reason. What makes this possible is the decision to leave the results in nonstandard form. . . .Mathematicians are quite reightly conservative and suspicious of new ideas. They will ask whether the results developed here are as powerful as the conventional results, and whether it is worth their while to learn nonstandard methods. These questions are addressed in an appendix. . . . The purpose of this appendix is to demonstrate that theorems of the conventional theory of stochastic processes can be derived from their elementary analogues by arguments of the type usually described as generalized nonsense; there is no probabiliistic reasoning in this appendix. This shows that the elementary nonstandard theory of stochastic processes can be used to derive conventional results; on the other hand, it shows that neither the elaborate machinery of the conventional theory nor the devices from the full theory of nonstandard analysis, needed to prove the equivalence of the elementary results with their conventional forms, add anything of significance: the elementary theory has the same scientific content as the conventional theory. This is intended as a self-destructing appendix."