By Sheldon Katz

ISBN-10: 0821836870

ISBN-13: 9780821836873

Maybe the main well-known instance of ways rules from sleek physics have revolutionized arithmetic is the best way string thought has ended in an overhaul of enumerative geometry, a space of arithmetic that all started within the eighteen enormous quantities. Century-old difficulties of enumerating geometric configurations have now been solved utilizing new and deep mathematical thoughts encouraged by way of physics! The ebook starts with an insightful advent to enumerative geometry. From there, the aim turns into explaining the extra complicated parts of enumerative algebraic geometry. alongside the way in which, there are a few crash classes on intermediate themes that are crucial instruments for the coed of recent arithmetic, akin to cohomology and different subject matters in geometry. The physics content material assumes not anything past a primary undergraduate direction. the focal point is on explaining the motion precept in physics, the belief of string thought, and the way those without delay bring about questions in geometry. as soon as those themes are in position, the relationship among physics and enumerative geometry is made with the creation of topological quantum box concept and quantum cohomology.

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Extra info for Enumerative Geometry and String Theory (Student Mathematical Library, Volume 32)

Example text

Hence rprsrt fixes each of these four elements. By (iii) the claim follows. Thus if we put cp : p<—> rp then, by the above presentation, (R, cp) is a representation of IS. The last sentence in the statement of the proposition is rather clear. D In certain circumstances the geometry sUFifS,^ possesses some further automorphisms. , that G S R := R/Z(R). Let v : r — i > f be the natural homomorphism of R onto R. 2 (iii)) possesses a subgroup other than R which also acts regularly on the point-set of siS'^S, R).

Since the geometric triangles are contractible with respect to 8, (y) are adjacent in 0 , which means that the element a := q>(x)q>{y) belongs to the set (p(Tl) of generators. Since <5(a) = tp(z) we have a =

Furthermore, SC\R = Z(R), [R,S] = 1 and RS = H. This shows that H is the central product of R and S. 3)). In particular the automorphism of H which swaps the two central product factors R and S is an automorphism of sUFifS, R) and we obtain the following. 2) suppose that G S R : = R/Z(R). 2 (the central product of two copies of R extended by the automorphism which swaps the central factors) is an automorphism group ). 4) occurs when ^ is isomorphic to or

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Enumerative Geometry and String Theory (Student Mathematical Library, Volume 32) by Sheldon Katz

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