The Unity of Mathematics: In Honor of the Ninetieth Birthday by Pavel Etingof, Vladimir S. Retakh, I. M. Singer

By Pavel Etingof, Vladimir S. Retakh, I. M. Singer

A tribute to the imaginative and prescient and legacy of Israel Gelfand, the invited papers during this quantity mirror the solidarity of arithmetic as an entire, with specific emphasis at the many connections one of the fields of geometry, physics, and illustration idea. Written by way of prime mathematicians, the textual content is commonly divided into sections: the 1st is dedicated to advancements on the intersection of geometry and physics, and the second one to illustration conception and algebraic geometry. subject matters comprise conformal box thought, K-theory, noncommutative geometry, gauge conception, representations of infinite-dimensional Lie algebras, and numerous points of the Langlands program.Graduate scholars and researchers will reap the benefits of and locate idea during this large and detailed paintings, which brings jointly basic ends up in a few disciplines and highlights the rewards of an interdisciplinary method of arithmetic and physics.

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Ek )/Gkm , which makes the existence of the map QMaps(Y, T; E1 , . . , Ek )) → QMaps(Y, T; E1 ⊗ · · · ⊗ Ek ) obvious. We know that when Y is smooth, QMaps(Y, T; E1 , . . , Ek ) is proper; therefore, to check that our map is finite, it is enough to show that the fiber over every geometric point of the scheme QMaps(Y, T; E1 ⊗ · · · ⊗ Ek ) is finite. Suppose that we have a line bundle L on Y with an injective map κ : L → OY ⊗ (E1 ⊗ · · · ⊗ Ek ), such that there exists an open subset U ⊂ Y such that κ|U is a bundle map corresponding to a map U → T.

7 The relative version Suppose now that Y itself is a flat family of projective schemes over some base X, and E is a vector bundle on Y, with T ⊂ P(E) a closed subscheme. The scheme of maps Maps(Y, T) assigns to a test scheme S over X the set of maps σ : Y × S → T, such that the composition Y × S → T → Y is the projection on the X X first factor. By definition, Maps(Y, T) is also a scheme over X. 1, we obtain a scheme QMaps(Y, T; E), which is quasi-projective over X. As before, if Y ⊂ Y is a closed subscheme, flat over X, and σ : Y → T is a X-map, we can define a locally closed subscheme QMapsa (Y, T; E)Y ,σ of based maps.

Speculating in another direction, I note that the Jones polynomial is naturally a character of the circle, the integers being the multiplicities of the irreducible representations. One may ask where the circle comes from. Now the knots studied by Jones are traditional ones in R3 and we have an S 2 at ∞, on which SO(3) acts. Moreover, the equivariant K-theory of S 2 is given by the character ring of the circle KS0(3) (S 2 ) ∼ = R(S 1 ). Here S 1 appears as the isotropy group of the action (and is unique up to conjugation).

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