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Sarnak P. — Some applications of modular forms
Sarnak P. — Some applications of modular  forms

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Название: Some applications of modular forms

Автор: Sarnak P.

Аннотация:

The theory of modular forms and especially the so-called ‘Ramanujan Conjectures’ have recently been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications. In order to keep the presentation reasonably self-contained, Professor Sarnak begins by developing the necessary background material in modular forms. He then considers the solution of three problems: the Ruziewicz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs: ‘expander graphs’ and ‘Ramanujan graphs’; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares. These applications are carried out in detail. The book therefore should be accessible to a wide audience of graduate students and researchers in mathematics and computer science.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1990

Количество страниц: 111

Добавлена в каталог: 09.04.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$L^{2}$ discrepancy      57
$\varepsilon$-good      45 47 51 58
Adjacency matrix      66
Amenable      58
Amenable group      47
bad      64
Bipartite Ramanujan graph      71 74 82
Brandt matrices      85
Cayley graph      74 75 77 86
Chebyshev polynominal      54
Chromatic number      2 61 68 74 84
Circle method      21
Class number      100
Commensurator      27
Congruence subgroup      8
Cusp form      7 14 37
Divisor sum      18
Eisenstein series      16 21 37 88
Enlarger, (n,k,c)      69
Expander graph      61 63 84 86
Expander, (n,k,c)      2 65 68
Expansion coefficient      2 3 84
Expansion graph      73
Expansion property      67
girth      2 61 74 75 77 84
Haar measure      57
Hecke operator      5 23 27 30 41 51 53 58 99
Height function      32
Holomorphic modular form of weight k      6
Independence number      61 67 74
Invariant mean      1 3 45 57 58 85
Kloosterman sum      2 25 33 38 74 89 90 95
Kronecker's first limit formula      33
Kuznietsov trace formula      431
Legendre symbol      12
Lindelof hypothesis      42
Linnik conjecture      33 99
Linnik problem      2 16 27 87 88 90 100
Linnik — Selberg conjecture      26 33
MaaB form      37 41 100
Mobius function      20
Modular form of weight k      13
Modular function of weight k      9
New forms      99
Petersson inner product      22 34 87 89 91
Phragmen — Lindelof principle      40
Poincare series      22 37
Poisson summation      10
Principal congruence subgroup      8
Property T      3 42 58 84
Quaternary form      15
Quaternion      52
Quaternion algebra      86
Quaternion, integral      75 77
Rademacher conjecture      43
Ramanujan conjecture      5 8 9 15 16 26 33 37 41 42 43 46 87 100
Ramanujan graph      1 2 71 74 75 82 85 86
Ramanujan sum      92
Random graph      2 75
Reduced word      53 57
Riemann hypothesis      41
Ruziewicz problem      1 45 58
Salie sum      26 90
Selberg conjecture      42
Selberg transform      50
Selberg — Kloosterman zeta function      38
Selberg's zeta function      42
Shimura lift      99
Singular series      21
Spectral Eisenstein series      20 31
Spectral zeta function      32
Spectrum of graphs      66
Super ergodic      1
Superbly ergodic group      46
Theta functions      10
Weil bound      25 33
Weil Conjectures      41
Weyl sums      101
Whittaker function      39
Zonal function      49
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