Novikov conjecture

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1: 21.9 Integrable Equations

… ►Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4). The first part of this conjecture was established in Krichever (1976); the second part was proved in Shiota (1986). …

2: Bibliography K

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A proof of Askey’s conjectured $q$-analogue of Selberg’s integral and a conjecture of Morris. SIAM J. Math. Anal. 19 (4), pp. 969–986.

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ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

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§17.13.

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A proof of the $q$-Macdonald-Morris conjecture for $B{C}_n$ . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt

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§17.14.

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Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

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ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt

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§16.23(iii).

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Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).

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Notes:

In Russian. English translation: Funct. Anal. Appl. 23(1989), no. 1, pp. 19–33

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ISSN 0374-1990, Document, MathReview (Fedor Malikov), ZentralBlatt

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§20.12(ii).

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3: Bibliography Q

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A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.

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External Links:

ISSN 0168-9673, Document, MathReview (Dmitrii Petrovich Novikov), ZentralBlatt

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§32.11(i).

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4: 16.23 Mathematical Applications

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§16.23(iii) Conformal Mapping

►The Bieberbach conjecture states that if ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is a conformal map of the unit disk to any complex domain, then $|a_n|\le n|a_1|$. In the proof of this conjecture de Branges (1985) uses the inequality …

5: David M. Bressoud

… ► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

6: 20.12 Mathematical Applications

… ►This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …

7: 27.13 Functions

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§27.13(ii) Goldbach Conjecture

… ►This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. … ►The current status of Goldbach’s conjecture is described in the Wikipedia. … ►Hardy and Littlewood (1925) conjectures that when $k$ is not a power of 2, and that $G\left(k\right)\le 4k$ when $k$ is a power of 2, but the most that is known (in 2009) is for some constant $c$. …

8: 17.14 Constant Term Identities

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Zeilberger–Bressoud Theorem (Andrews’ $q$-Dyson Conjecture)

… ►Macdonald (1982) includes extensive conjectures on generalizations of (17.14.1) to root systems. These conjectures were proved in Cherednik (1995), Habsieger (1986), and Kadell (1994); see also Macdonald (1998). …

9: Bibliography Y

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Partitions with difference conditions and Alder’s conjecture. Proc. Natl. Acad. Sci. USA 101 (47), pp. 16417–16418.

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External Links:

ISSN 1091-6490, FullText, Document, MathReview (Scott D. Ahlgren), ZentralBlatt

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§26.10(iv).

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10: 18.38 Mathematical Applications

… ►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …