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*NOTE ROOM CHANGE: All courses will meet at the Fields Institute
Stewart Library in the 3rd Floor unless notified otherwise.

Each meeting will run for 90 minutes.

**1) I****ntroduction to Arakelov Geometry
**(**Wednesday & Thursday @ 10 a.m**)

Instructor: **Henri Gillet**, University of Illinois at Chicago

**2) Course on Nevanlinna theory and Diophantine approximation**
(**Tuesday & Wednesday @ 1:00 p.m.**)

Instructor: **Min Ru**, University of Houston

**3) Course on Jet Spaces** (Mini Course In Complex Geometry)(
**Tuesday @ 10:00 a.m. , Thursday @ 1:00 p.m.)**

Instructor:** Pit-Mann Wong**, University of Notre Dame

4) **MAT1191HF Transcendental Methods in Algebraic Geometry**

**Y-T Siu**

Monday and Fridays @ 10:30 a.m. -12 noon

Fields 230

Oct. 20, 27, Nov. 10, 17 [BA2155]

Sept 19, Oct 24, Nov. 14, 21 [BA2175]

Applications of L^2 \partial-estimates and multiplier ideal sheaf
tech niques to problems in algebraic geometry such as the effective
Nullstellensatz, the Fujita conjecture on effective global generation
and very ampleness of line bundles, the effective Matsusaka big
theorem, the deformational invariance of plurigenera, the finite
generation of the canonical ring, and the abundance conjecture.
Will also discuss hyperbolicity problems and the application of
algebraic-geometric techniques to partial differential equations
through multiplier ideal sheaves, especially the global regularity
of the complex Neumann problem, the existence of Hermitian-Einstein
and Kähler-Einstein metrics, and the global nondeformability
of irreducible compact Hermitian symmetric manifolds.

### Introduction to Arakelov Geometry

Instructor: Henri Gillet

**Meeting Times: 90
minutes**

Wednesday & Thursday 10:00 am -11:30 a.m.

The course will begin, following a review of the "non-arithmetic"
theory, with a study of Arakelov's intersection theory on arithmetic
surfaces as developed in Faltings. We will then develop arithmetic
intersection theory for varieties of arbitrary dimesion. The main
results include the relationship between arithmetic intersection
theory, heights and height pairings, the arithmetic Bezout theorem,
the arithmetic Hilbert-Samuel formula, and the arithmetic Riemann-Roch
theorem. The course will continue with the K-theory of Hermitian
vector bundles for general arithmetic varieties and the characteristic
classes for this bundles, the determinant of cohomology and Quillen
metrics, and the arithmetic Grothendieck-Riemann-Roch theorem. Some
applications to problems in number theory will be discussed.

**Syllabus:**

- Review of classical intersection theory
- Intersection theory on regular schemes
- Arakelov's Intersection theory on arithmetic surfaces
- Deligne cohomology and Green currents
- Arithmetic Chow groups and arithmetic intersection theory
- Deformation to the normal bundle in arithmetic intersection
theory
- Bott-Chern forms and Chern Classes of Hermitian vector bundles
- Heights and the arithmetic Bezout theorem
- Heights of Grassmannians and other special varieties
- Determinant of Cohomology and analytic torsion
- Arithmetic Riemann Roch and the arithmetic Hilbert-Samuel formula
- Recent Results and Open questions

**Reading:**

Bost, J.-B.; Gillet, H.; Soulé, C. --Heights of projective
varieties and positive Green forms. J. *Amer. Math. Soc. *7
(1994), no. 4, 903--1027.

Faltings, Gerd --Calculus on arithmetic surfaces. *Ann. of Math.
*(2) 119 (1984), no. 2, 387--424.

Gillet, Henri; Soulé, Christophe --Arithmetic intersection
theory. *Inst. Hautes Études Sci. Publ. Math.* No. 72
(1990), 93--174 (1991).

Gillet, Henri; Soulé, Christophe --Characteristic classes
for algebraic vector bundles with Hermitian metric. *Ann. of Math.
*(2) 131 (1990), 163--238.

Lang, Serge --Introduction to Arakelov theory. *Springer-Verlag,
New York, *1988.

Maillot, Vincent --Un calcul de Schubert arithmétique. (**French**)
[An arithmetic Schubert calculus] *Duke Math. J.* 80 (1995),
no. 1, 195--221.

Soulé, Christophe --Hermitian vector bundles on arithmetic
varieties. *Algebraic geometry---Santa Cruz *1995, 383--419,
Proc. Sympos. Pure Math., 62, Part 1, *Amer. Math. Soc., Providence,
RI, *1997.

Soulé, C. --Lectures on Arakelov geometry. With the collaboration
of D. Abramovich, J.-F. Burnol and J. Kramer. *Cambridge Studies
in Advanced Mathematics, 33. Cambridge University Press, Cambridge,*
1992.

Tamvakis, Harry --Height formulas for homogeneous varieties. *Michigan
Math. J*. 48 (2000), 593--610.

### Course on Nevanlinna theory and Diophantine
approximation

Instructor: Min Ru, University of Houston

**Meeting Times: 90
minutes**

Tuesday & Wednesday 1:00 pm

Diophantine approximation is a tool to study rational points on
algebraic varieties defined over a number field. On the other hand,
Nevanlinna theory studies holomorphic curves in complex algebraic
varieties, especially it studies how well a holomorphic curve intersects
divisors in a complex algebraic variety. It has been observed by
Osgood, Vojta and others that there is a striking correspondence
between statements in Nevanlinna theory and in Diophantine approximation.
The mini-course will cover: Roth’s theorem and Schmidt’s
subspace theorem; Diophantine equations and approximation; the theory
of global and local heights; Faltings’ theorem on abelian varieties;
the classical theory of Nevanlinna on meromorphic functions; The
Ahlfors-Cartan theory of holomorphic curves; holomorphic curves
in Abelian varieties; the complex hyperbolicities and the general
case of Lang’s conjecture.

Syllabus.

- Nevanlinna theory for meromorphic functions, holomorphic curves
in compact Riemann surfaces.
- The theorem of Thue-Siegel-Roth and the theorem of Faltings.
- Nevanlinna’s theory for holomorphic curves, H. Cartan’s
method and Ahlfors’ method.
- The theory of heights.
- Schmidt’s subspace theorem.
- The application to the study of Diophantine equations.
- Holomorphic curves in Abelian varieties and the theorem of Faltings.
- Complex hyperbolic manifolds and Lang’s conjecture.
- A survey of recent developments.

Advanced topics: These will be covered in the seminars.

**Text**:

Min Ru, *Nevanlinna theory and its relation to Diophantine approximation*.
World Scientific Publishing Co., Inc., River Edge, NJ, 2001. xiv+323
pp. ISBN: 981-02-4402-9.

**Here is the list of reference (with *
indicates that they are most related and recommended):**

1. Cowen, M. and Griffith, Ph.(1976) ``Holomorphic curves and metrics
of nonnegative curvature'', *J. Analyse Math*. **29**, 93--153.

2. Demailly, J.P.(1995) ``Algebraic criteria for Kobayashi hyperbolic
varieties and jet differentials'', {\it *Proc. Symp. Pur. Math.
Amer. Math. Soc.*} **62**, Part 2, 285--360.

3.* Green, M.(1975) ``Some Picard theorems for holomorphic maps
to algebraic varieties,'' *Amer. J. Math*. **97**, 43--75.

4. Green, M. and Griffiths,P.(1980) ``Two applications of algebraic
geometry to entire holomorphic mappings'', The Chern Symposium 1979,
Proc. Internat. Sympos.,Berkeley, 1979, Springer-Verlag.

5. Griffiths,P. (1974) "Entire holomorphic mappings in one
and several complex variables", The fifth set of Hermann Weyl
Lectures, given at the Institute for Advanced Study, Princeton,
N. J., October and November 1974. Annals of Mathematics Studies,
No. 85. Princeton University Press, Princeton, N. J.; University
of Tokyo Press, Tokyo, 1976. x+99 pp.

6.* Hayman, W.(1964) *Meromorphic Functions*, Oxford University
Press.

7.* Hindry, M. and Silverman, J.(2000) *Diophantine Geometry:
an Introduction*, Graduate Texts in Mathematics **201**, Springer-Verlag.

8.* Kobayashi, S.(1998) *Hyperbolic Complex Spaces*, Springer-Verlag.

9.* Lang, S.(1983) *Fundamentals of Diophantine Geometry*,
Springer-Verlag.

10.* Lang, S. (1987) *Introduction to Complex Hyperbolic Spaces*,
Springer-Verlag, New

York-Berlin-Heidelberg.

11.* Fujimoto, H.(1993) {\it *Value Distribution Theory on the
Gauss Map of Minimal*

Surfaes in ${\bf R}^m$}, Aspect of Mathematics, E21, Vieweg.

12. Lang, S. and Cherry, W.(1990) *Topic in Nevanlinna theory*,
Lecture Notes in Math. 1433, Springer-Verlag.

13.* Noguchi, J. and Ochiai, T.(1990) *Geometric Function Theory
in Several Complex*

Variables, Transl. Math. Mon. **80**, Amer. Math. Soc., Providence,
R.I..

14. Schmidt, W.M.(1991) *Diophantine approximations and diophantine
equations*, Lecture Notes in Math. 1467. Springer-Verlag.

15. Schmidt, W.M.(1980) *Diophantine approximation*, Lecture
Notes in Math. 785, Springer-Verlag.

16.* Shabat, B.V.(1985) *Distribution of values of holomorphic
mappings*,Translations of Mathematical Monographs Vol. **61**,
American MathematicalSociety.

17.* Siu, Y.T.(1995) ``Hyperbolicity Problems in Function Theory'',
in *Five Decades as a Mathematician and Educator - on the 80th
birthday of Professor Yung-Chow Wong*, ed. Kai-Yuen Chan and
Ming-Chit Liu, World Scientific: Singapore, New Jersey, London,
Hong Kong, pp.409--514.

18. Stoll, W.(1983) *The Ahlfors-Weyl theory of meromorphic maps
on parabolic manifolds*, Lecture Notes in Mathematics **981**,
101--219.

19. Stoll, W.(1985) *Value distribution theory for meromorphic
maps*, Aspects of Math., E7.

20.* Vojta, P.(1987) *Diophantine approximations and value distribution
theory*, Lecture Notes in Math. **1239**, Springer-Verlag.

21. Vojta, P.(1999) ``Nevanlinna theory and Diophantine approximation'',
Several complex variables (Berkeley, CA, 1995--1996), 535--564,
Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge.

22.* Wong, P.M.(1989) ``On the second main theorem of Nevanlinna
theory'', *Amer. J. Math*. 549--583.

23.. Diophantine approximation and abelian varieties. Introductory
lectures. Papers from the conference held in Soesterberg, April
12--16, 1992. Edited by B. Edixhoven and J.-H. Evertse. Lecture
Notes in Mathematics, 1566. Springer-Verlag, Berlin, 1993. xiv+127
pp. ISBN: 3-540-57528-

### Course on Jet Spaces

Instructor: Pit-Mann Wong, University of Notre Dame

**Meeting Times: 90
minutes**

Tuesday 10:00 am & Thursday 1:00 pm

A Mini Course In Complex Geometry

This is an outline of a mini-course in complex geometry. There are
six chapters planned. The content in the rst four chapters are mostly
very well known. There will be no time for complete proofs only
outline and motivations will be given. Explicit references of the
theorems will be given. I shall also try to provide examples to
explain the ideas and clarify the concepts. A reasonable amount
of details will be given in the last two chapters.

1. A brief introduction to Hermitian and Kähler Geometry.

A quick outline of the basic notions of Hermitian connection and
curvature of vector bundles. The concepts of holomorphic bisectional
curvature, Ricci curvature and holomorphic sectional curvature.
Schwarz Lemma. Chern classes of vector bundles.

2. A brief introduction to Complex Analysis and Algebraic geometry.

Brief outline of the concepts of coherent sheaves and sheaf cohomologies.
Characterizations of Stein manifolds via vanishing theorem, via
strictly plurisubharmonic exhaustions and the Stein embedding theorem.
The concepts of ample, big and nef bundles. Kodaira vanishing theorem
and embedding theorem. Riemann-Roch theorem for coherent sheaves.

3. A brief introduction to Complex Finsler Geometry and Intrinsic
Metrics.

Intrinsic metrics, mainly the Kobayashi and the Caratheodory metric,
will be introduced. Positive currents and Lelong numbers. A Finsler
characterization of ample bundles and big bundles will be given.

4. A brief introduction to Nevanlinna Theory.

A brief account of Nevanlinna Theory. Jensen Formula. First Main
Theorem. Crofton Formula. Second Main theorem. Integrated form of
Schwarz Lemma.

5. Holomorphic Jet Bundles.

The basic theory of jet bundles will be introduced with reasonable
amount of details and examples.

6. Applications to Hyperbolic Geometry.

The results in the respective chapters will be applied to resolve
problems in complex hyperbolic geometry.

### Taking the Institute's Courses for Credit

As graduate students at any of the Institute's University Partners,
you may discuss the possibility of obtaining a credit for one
or more courses in this lecture series with your home university
graduate officer and the course instructor. Assigned reading and
related projects may be arranged for the benefit of students requiring
these courses for credit.

### Financial Assistance

As part of the Affiliation agreement with some Canadian Universities,
graduate students are eligible to apply for financial assistance
to attend graduate courses. Application for support now closed.
Two types of support were available: