By Selman Akbulut, Sema Salur (auth.), Özgür Ceyhan, Yu. I. Manin, Matilde Marcolli (eds.)

ISBN-10: 0817648305

ISBN-13: 9780817648305

In contemporary a long time, quantization has resulted in fascinating purposes in a variety of mathematical branches. This quantity, made from examine and survey articles, discusses key themes, together with symplectic and algebraic geometry, illustration idea, quantum teams, the geometric Langlands software, quantum ergodicity, and non-commutative geometry. a variety of themes with regards to quantization are coated, giving a glimpse of the huge topic. The articles are written by way of distinct mathematicians within the box and replicate next advancements following the mathematics and Geometry round Quantization convention held in Istanbul.

List of Contributors:

S. Akbulut R. Hadani

S. Arkhipov ok. Kremnizer

Ö. Ceyhan S. Mahanta

E. Frenkel S. Salur

K. Fukaya G. Ben Simon

D. Gaitsgory W. van Suijlekom

S. Gurevich

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**Example text**

We have a map from Pic Z to the discrete 2-group Z which sends a 1-dimensional graded vector space to its degree. This map induces a functor between graded Gm -gerbes and Z-torsors. We can now repeat the entire story with Z-graded gerbes. For instance, instead of a determinant theory we will get a graded determinant theory. The Z-torsor corresponding to it will be the well known dimension torsor of dimension theories. A dimension theory for a 1-Tate space is a rule of associating an integer to each lattice satisfying conditions similar to those of a determinant theory.

3, 193–215. 6. Drinfeld, Vladimir Infinite-dimensional vector bundles in algebraic geometry: an introduction. The unity of mathematics, 263–304, Progr. , 244, Birkh¨ auser Boston, Boston, MA, 2006. 2-Gerbes and 2-Tate Spaces 35 7. Kato, Kazuya Existence theorem for higher local fields. Invitation to higher local ﬁelds (Mnster, 1999), 165–195 (electronic), Geom. Topol. , 3, Geom. Topol. , Coventry, 2000. 8. Lurie, Jacob Higher topos theory. Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009.

We wish to compute γ(f, g). We have the following theorem. Theorem 6. For every non-constant f, g ∈ C + we have: γ(f, g) = μ(g) . μ(f ) (15) Proof. We start with the following deﬁnition. r γ ∗ (f, g) := inf{ |f r ≥ g s , r ∈ Z , s ∈ N}. s (16) We claim that γ ∗ = γ. Indeed, denote by T the set of numbers which satisfy the condition of the right hand side of (16). Assume that rs ∈ T . Then the equivalence of the deﬁnitions follows from the inequality r ≥ γs (f, g) ≥ sγ ∗ (f, g). We claim that a suﬃcient condition for f r ≥ g s is that rμ(f ) − sμ(g) ≥ 2πn.

### Arithmetic and Geometry Around Quantization by Selman Akbulut, Sema Salur (auth.), Özgür Ceyhan, Yu. I. Manin, Matilde Marcolli (eds.)

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