#### Diffeology, Categories and Toposes and Non-commutative Geometry Summer School

**Tarih**June 18-29, 2018 (Arrival to the village: 17 June 2018, departure from the Village: 29 June 2018)**Amaç**Over the past decades, physicists have become accustomed to working on infinite dimensional spaces (function spaces, groups of diffeomorphisms ), or on singular spaces (symplectic reductions, moduli spaces ). The notion of space has traversed very radical transformations, both in pure mathematics and in mathematical physics. Over time, several approaches have been identified: traditional functional analysis, non-commutative geometry, diffeology and toposes. The objective of this summer school is to introduce these recent notions of space in mathematics and/or physics (such as diffeologies, schemes, topoi, stacks, spin networks, homotopy types, noncommutative spaces, supermanifolds, etc.). The lectures will be given by:Serap Gürer, Galatasaray University, Istanbul Turkey (Diffeology)Patrick Iglesias-Zemmour, CNRS, Aix-Marseille University, France (Diffeology)Murad Özaydın, University of Oklahoma, USA (Non-commutative Geometry)Urs Scheiber, Academy of the Sciences, Prague, Czechia (Categories and Toposes)**Genel Bilgi****Who may participate:**Graduate, PHD students, and postdocs**Sponsors:**TÜBİTAK**Deadlines:**1 May 2018**Quota:**30 people**Fees:**The normal daily fee is 110 TL; for those staying in tents the fee is 80 TL. This covers accomodation, four meals a day and all the facilities that the Village offer. If you need financial support please write it to application form.**Accommodation:**Undergraduate and graduate students will mostly stay in tents. Experience shows that due to the greater freedom and independence it affords them, older students themselves prefer this arrangement. If you have any doubts about this subject please write to us. Though we can give no guarantees, you can be sure we will do our best to come to an arrangement which is satisfactory to you.

**Kayıt ve başvuru işlemleriyle ilgili sorularınız için**

Please write to cigdemsahin@nesinvakfi.org regarding problems with Maths Village. Otherwise, please write to serapgurer@gmail.com.

Monday June 18, 9AM Opening

M-18: 10-12 PIZ - 14-16 MÖ

T-19: 10-12 US - 14-16 PIZ

W-20: 10-12 MÖ - Free

T-21: 10-12 US - 14-16 PIZ

F-22: 10-12 MÖ - 14-16 US

M-25: 10-12 PIZ - 14-16 MÖ

T-26: 10-12 US - 14-16 PIZ

W-27: 10-12 MÖ - Free

T-28: 10-12 US - 14-16 PIZ

F-29: 10-12 MÖ - 14-16 US

**Syllabuses:**

Patrick Iglesias-Zemmour (piz@math.huji.ac.il): Diffeology

- Category {Diffeology} Set theoretic construction and stability.

- The case of singular quotients and an example or two in infinite dimension.

- Modelling diffeologies (manifolds, orbifolds, manifolds with corners
).

- Differential calculus and De Rham cohomology (with Serap Gürer).

- Fiber bundles and homotopy.

- Moment Maps and integration of closed forms, general prequantization.

Murad Özaydin (mozaydin@math.ou.edu): Non-Commutative Geometry

- Geometry-Algebra dictionary: C*-Algebras, Gelfand-Naimark duality; vector bundles, K-theory, projective modules, Serre-Swan theorem.

- Mathematical methods and models of quantum physics. Gelfand-Naimark-Segal duality.

- Quantum tori, noncommutative quotients.

- Foliations, von Neumann algebras.

- Morita equivalence.

- Hopf algebras and quantum groups.

- Deformation and quantization.

- Noncommutative algebraic geometry: affine, projective and derived.

- Noncommutative Stone-Weierstrass theorem.

- Homological algebra, cyclic cohomology.

- Connes-Chern character: topological index = analytic index.

- Spectral triples.

Urs Schreiber (Urs.Schreiber@gmail.com): Categories and Toposes - Local field theory in a differentially cohesive infinity-topos.

- Categories and limits/colimits.

- Adjoint functors and Kan extension.

- Categories of pre-sheaves.

- Simplicial sets and simplicial pre-sheaves.

- Simplicial sheaves on the site of smooth manifolds.

- Simplicial sheaves on the site of formal manifolds.