**BCAM-PV/EHU Graduate School: Introduction To Geometric Analysis: The Atiyah-Singer Index Theorem**
**June 5-23, 2017**
**Objectives:**
Geometric Analysis consists in applying PDE tools in order to prove deep theorems in Geometry. It is one of the areas mathematics with more spectacular results in the last half a century.

The school is primarily aimed for Master and beginning Ph. D. students, and hence prerequisites will be kept to a minimum. The objective is to develop the tools and some working knowledge in Geometric Analysis, that is the interaction of manifold theory with PDE analysis. We hope to bring together students in analysis and geometry and establish a fruitful interaction between them.

We have chosen as a guiding goal of the school, to develop the tools and explain a proof of one of the most classic and striking theorems in geometric analysis, namely the Atiyah-Singer Index Theorem. Besides the importance of the theorem, we feel that developing the tools, assembling a proof and explaining some applications (Hirzebruch-Riemann-Roch and Hirzebruch signature theorems) will help the students in learning useful techniques in geometry and analysis, and learning how they can be combined in order to prove deep results.

The school will last for three weeks. The first two weeks will be devoted to develop basic techniques in geometry and analysis of PDE’s, which are needed for the proof of the theorem, but which are completely central by themselves. In the third one Atiyah-Singer index theorem will be proved, some applications will be derived, and in addition some topics of modern geometric analysis will be surveyed, in order to let the participants taste some topics of current research.

**Prerequisites:**
In geometry and topology we will assume basic knowledge in differentiable manifolds, basics in algebraic topology, including cohomology, Poincare duality and fundamental classes of submanifolds. In analysis we will assume basic real analysis, the basic theory of Hilbert spaces.

**Speakers:**
Enrique ARTAL Universidad de Zaragoza

Laurent BESSIERES, Institut de Mathématiques de Bordeaux

Pedro CARO, BCAM

Javier FERNÁNDEZ de BOBADILLA, BCAM

Vicente MUÑOZ, Universidad Complutense de Madrid

Carlos PÉREZ, BCAM-UPV/EHU

(More speakers TBA).

**Organizers:**
Marisa FERNÁNDEZ, UPV/EHU

Gustavo FERNÁNDEZ ALCOBER, UPV/EHU

Javier FERNÁNDEZ de BOBADILLA, BCAM

Luis VEGA, BCAM-UPV/EHU

Ion ZABALLA,UPV/EHU

**Courses:**
**Weeks 1 and 2:**
1.- Characteristic classes and K-theory (18 hours), by E. Artal and J. Fernandez de Bobadilla: we will develop the theory of Chern classes and of topological K-theory following the treatments in:

J. Milnor, J.Stasheff “Characteristic classes”. Annals of Mathematics Studies 76, (1974).

M.Atiyah “K-Theory”. Westview Press, (1994).

2.- Analysis of elliptic PDE’s (18 hours), by P. Caro and C. Perez: basics on distributions and Fourier transform, Sobolev spaces, elliptic differential and pseudo-differential operators in R^n. Elliptic PDE’s in manifolds and vector bundles, index, elliptic complexes. A list of references for this course is:

**Basic Bibliography:**
Folland, Gerald B. Introduction to partial differential equations. 2nd ed. (English) Princeton, NJ: Princeton University Press. xi, 324 p. (1995).

Cerdà, Joan Linear functional analysis. Graduate Studies in Mathematics, 116. American Mathematical Society, Providence, RI. ISBN: 978-0-8218-5115-9.

Pinsky, M.A. Introduction to Fourier Analysis and Wavelets, Ed. Brooks/Cole, 2002.

Wong, M. W. An introduction to pseudo-differential operators. Second edition. World Scientific Publishing Co., Inc., River Edge, NJ, 1999. x+138 pp.

J. Duoandikoetxea, Fourier Analysis, American Math. Soc., Grad. Stud. Math., Providence, RI, 2000.

**Advanced Bibliography:**
Hörmander, Lars The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis. Reprint of the 2nd edition 1990. Grundlehren der Mathematischen Wissenschaften, 256. Berlin etc.: Springer-Verlag. xi, 440 p. (1990).

Stein, Elias M. Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S.

Murphy. Princeton Mathematical Series. 43. Princeton, NJ: Princeton University Press. xiii, 695 pp. (1993).

Taylor, Michael E. Partial differential equations. I: Basic theory. 2nd ed. Applied Mathematical Sciences 115. New York, NY: Springer (ISBN 978-1-4419-7054-1/hbk; 978-1-4419-7055-8/ebook). xxii, 654 p. (2011).

3.- Some exercise/discussion sessions may be scheduled, at the convenience of the participants.

**Week 3:**
1.- The Atiyah-Singer Index Theorem and its applications (12-15 hours), by V. Muñoz: Atiyah-Singer Index Theorem will be explained and proved, and some of its applications developed (Hirzebruch-Riemann-Roch Theorem and Hirzebruch Signature Theorem). The basic reference are the first and third original papers of Atiyah-Singer

M. F. Atiyah, I. M. Singer. “The index of elliptic operators I” The Annals of Mathematics, Second Series, Volume 87, Issue 3 (1968).

M. F. Atiyah, I. M. Singer. “The index of elliptic operators III” The Annals of Mathematics, Second Series, Volume 87, Issue 3 (1968).

2.- "Ricci flow and applications" L. Bessieres (4 hours).

**Student grants**: a limited number of student grants will be available which will cover lodging and full board.

**Application procedure:**
In order to apply please send an e-mail to

** geometry@bcamath.org** indicating:

1.- Full name.

2.- Institution, in which year of Master/Ph. D/Postdoc you are. Please indicate your specialization.

3.- If you wish to apply for a grant, please, attach a CV which contains the list of subjects you have taken.

4.- Special dietary needs.

Students which are selected to participate need to pay a

**registration fee of 60€**, and

**Registration **is required.

After having received your inscription we will send to your email address all the details to make the transfer.

**Deadline for application is April 10th.**
**Deadline for grant application is March 1st.**
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**P R O G R A M : **

**Monday 5 Jun 2017 **

11:45-13:45 K-Theory and Characteristics clases *E.Artal and J.Fernandez de Bobadilla*

13:45-16:00 Lunch time / free

16:00-18:00 Analysis of elliptic operators *P.Caro and C. Pérez*

**Tuesday 6 Jun 2017 **

11:45-13:45 K-Theory and Characteristics clases *E.Artal and J.Fernandez de Bobadilla*

13:45-16:00 Lunch time / free

16:00-18:00 Analysis of elliptic operators *P.Caro and C. Pérez*

**Wednesday 7 Jun 2017 **

11:45-13:45 K-Theory and Characteristics clases *E.Artal and J.Fernandez de Bobadilla*

13:45-16:00 Lunch time / free

16:00-18:00 Analysis of elliptic operators *P.Caro and C. Pérez*

**Thursday 8 Jun 2017 **

11:45-13:45 K-Theory and Characteristics clases *E.Artal and J.Fernandez de Bobadilla*

13:45-16:00 Lunch time / free

16:00-18:00 Analysis of elliptic operators *P.Caro and C. Pérez*

**Friday 9 Jun 2017 **

11:45-13:45 K-Theory and Characteristics clases *E.Artal and J.Fernandez de Bobadilla*

13:45-16:00 Lunch time / free

16:00-18:00 Analysis of elliptic operators *P.Caro and C. Pérez*

**Monday 12 Jun 2017 **

9:00-11:00 Analysis of elliptic operators *P.Caro and C. Pérez*

11:00-11:30 Coffee break

11:30-13:30 K-Theory and Characteristics clases *E.Artal and J.Fernández de Bobadilla*

13:30-15:30 Lunch time / free

15:30-17:30 tba

**Tuesday 13 Jun 2017 **

9:00-11:00 Analysis of elliptic operators *P.Caro and C. Pérez*

11:00-11:30 Coffee break

11:30-13:30 K-Theory and Characteristics clases *E.Artal and J.Fernández de Bobadilla*

**Wednesday 14 Jun 2017 **

9:00-11:00 Analysis of elliptic operators *P.Caro and C. Pérez*

11:00-11:30 Coffee break

11:30-13:30 K-Theory and Characteristics clases *E.Artal and J.Fernández de Bobadilla*

13:30-15:30 Lunch time / free

15:30-17:30 tba

**Thursday 15 Jun 2017 **

9:00-11:00 Analysis of elliptic operators *P.Caro and C. Pérez*

11:00-11:30 Coffee break

11:30-13:30 K-Theory and Characteristics clases *E.Artal and J.Fernández de Bobadilla*

13:30-15:30 Lunch time / free

15:30-17:30 tba

**Friday 16 Jun 2017 **

9:00-11:00 Analysis of elliptic operators *P.Caro and C. Pérez*

11:00-11:30 Coffee break

11:30-13:30 K-Theory and Characteristics clases *E.Artal and J.Fernández de Bobadilla*

**Saturday 17 Jun 2017 **Tentative Lunch. It will be confirmed

**Monday 19 Jun 2017 **

9:00-10:30 The Atiyah-Singer Index Theorem *V.Muñoz*

10:30-11:00 Coffee break

11:00-13:00 The Atiyah-Singer Index Theorem *V.Muñoz*

13:00-15:00 Lunch time / free

15:00-16:00 Ricci flow and applications *L.Bessieres*

**Tuesday 20 Jun 2017 **

9:00-10:30 The Atiyah-Singer Index Theorem *V.Muñoz*

10:30-11:00 Coffee break

11:00-13:00 The Atiyah-Singer Index Theorem *V.Muñoz*

13:00-15:00 Lunch time / free

15:00-16:00 Ricci flow and applications *L.Bessieres*

**Wednesday 21 Jun 2017 **

9:00-10:30 The Atiyah-Singer Index Theorem *V.Muñoz*

10:30-11:00 Coffee break

11:00-13:00 The Atiyah-Singer Index Theorem *V.Muñoz*

13:00-15:00 Lunch time / free

15:00-16:00 Ricci flow and applications *L.Bessieres*

**Thursday 22 Jun 2017 **

9:00-10:30 The Atiyah-Singer Index Theorem *V.Muñoz*

10:30-11:00 Coffee break

11:00-13:00 The Atiyah-Singer Index Theorem *V.Muñoz*

13:00-15:00 Lunch time / free

15:00-16:00 Ricci flow and applications *L.Bessieres*

**Friday 23 Jun 2017 **

9:00-10:30 The Atiyah-Singer Index Theorem *V.Muñoz*

10:30-11:00 Coffee break

11:00-13:00 The Atiyah-Singer Index Theorem *V.Muñoz*

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