Organizatori

• Prof. dr. Răzvan Liţcanu
• CS. Mihai Pavel

Avand ca pretext discutarea diferitelor variante ale Teoremei Riemann-Roch, seminarul isi propune o trecere in revista a proprietatilor suprafetelor Riemann si, mai general, ale varietatilor complexe/algebrice/aritmetice si ale fibratelor vectoriale pe acestea. Seminarul este deschis atat colegilor, cat si studentilor interesati.

Programul seminarului

Week 1 - Riemann surfaces (Vineri 27 martie, ora 13 - Amfiteatrul Al. Myller)

• Definition of Riemann surfaces
• Examples: sphere, torus, elliptic curves, etc.
• Holomorphic functions

Week 2 - Divisors and line bundles

• Divisors on a Riemann surface
• Degree of a divisor
• Meromorphic functions
• Line bundles

Week 3 - Statement of the Riemann-Roch theorem

• Spaces of meromorphic functions with prescribed poles
• Dimension of spaces of sections
• Statement of the theorem

Week 4 - Applications of the Riemann-Roch theorem

• Canonical divisor
• Genus of a curve
• Differential forms on Riemann surfaces

Week 5 - Vector bundles on Riemann surfaces

• Definition of vector bundles
• Rank and degree
• Examples: trivial bundle, tangent bundle, etc.

Week 6 - Riemann-Roch for vector bundles

• Euler characteristic of a bundle
• Statement of the Riemann-Roch theorem for vector bundles
• Examples

Week 7 - Hermitian metrics on vector bundles

• Hermitian metrics on complex vector bundles
• Curvature
• Degree interpreted via curvature

Week 8 - Arithmetic curves and Arakelov divisors

• Analogy between curves and number fields
• Arithmetic divisors
• Metrics

Week 9 - Hermitian line bundles in Arakelov geometry

• Hermitian line bundles
• Arithmetic degree
• Arithmetic intersection numbers

Week 10 - Arithmetic Riemann-Roch theorem

• Arithmetic Euler characteristic
• Statement of the arithmetic Riemann-Roch theorem
• Application