﻿ 2022 대수기하-정수론 겨울학교
 2022 대수기하-정수론 겨울학교       2022.01.16-21       소노벨 천안
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Langlands functoriality conjecture: Langlands-Shahidi method and Trace formula

- Abstract:

Langlands functoriality conjecture is one main conjecture in Langlands program and it describes a relationship between automorphic (or admissible) representations of two different groups. It was posed in a famous (17 hand-written page) letter of Robert Langlands to Andre Weil in 1967. We explain two main methods to construct Langlands functoriality conjecture: Converse theorem and trace formula. We plan to cover the following subjects:

(1) Linear algebraic groups with basic examples

(2) L-functions and Langlands-Shahidi methods

(3) Converse theorem by Cogdell and Piatetski-Shapiro

(4) Arthur's trace formula

Moduli spaces of pointed stable curves of genus zero

- Abstract:

Easiest examples of algebraic varieties include points and projective line. The moduli space of n ordered distinct points in projective line up to projective equivalence has been a topic of intensive research since the 19th century. A natural compactification was constructed by Deligne and Mumford in late 1960s and explicit constructions were provided by Knudsen, Keel, Kapranov and others in 1980s and 1990s. Its cohomology was computed in mid 1990s by Manin, Getzler, Keel and others. There is an obvious action of the symmetric group permuting the marked points and it is a topic of on-going research to investigate the representation theory of its cohomology. The goal of this lecture series is to provide a survey on these topics which may offer a glimpse into actual research in modern algebraic geometry.

In this lecture series, I hope to cover the following topics.

(0) Minumum basics on algebraic geometry (history, affine varieties, projective varieties, morphisms of varieties),

(1) Points on projective space,

(2) Moduli functor and moduli space,

(3) Compactification of the moduli space of points on projective line by stable curves,

(4) Explicit constructions,

(5) Cohomology of the moduli space,

(6) Representation theory on the cohomology.

Selberg trace formula and its applications

- Abstract:

The Selberg trace formula for a hyperbolic compact Riemann surface relates the eigenvalue spectrum of the Laplacian to lengths of closed geodesics. As a prototype of its descendants that describe spectral data in terms of geometric data, it is an indispensable tool for analyzing the structure of the spectrum of the Laplacian. Its generalization plays a central role in the theory of automorphic forms and the Langlands program. The goal of these lectures is to give an introduction to the classical Selberg trace formula and to provide hands-on experience in how to use it by putting some examples into the computer.

We will cover the following topics:

- Selberg trace formula for compact Riemann surfaces

- spectrum of the Bolza surface

- Selberg trace formula for PSL(2, Z)

- Jacquet-Langlands correspondence

References

Hejhal (1976), The Selberg trace formula for PSL(2,R). Vol. I, Springer-Verlag

Bergeron (2016), The Spectrum of Hyperbolic Surfaces, Springer

Moduli space of vector bundles

- Abstract:

In algebraic geometry, when we try to understand the nature of objects, it is sometimes very important and useful to study, not just the individual object itself, but the collection of them. To achieve this philosophy it is natural to try to construct this family as an algebraic scheme (or just variety, or even stack). In this lecture series, we summarize the basic notions to construct moduli spaces.

(0) algebraic geometry 101 (sheaf cohomology and vector bundle)

(1) GIT construction

(2) moduli space of vector bundles

(3) examples

Possible references would include the following:

(a) S. Mukai, Introduction to invariants and moduli

(b) P. Nestead, Introduction to moduli problems and orbit spaces

(c) I. Dolgachev, Lectures on invariant theory