Organizer: Yuchan Lee
Speaker: Yuchan Lee
Abstract: I introduce Arthur-Selberg's role in Langlands program and provide a breif overview for the simple trace formula and its applications.
Speaker: Dowan Kwon
Goal:
Recall the definition of automorphic forms of a connected reductive group G;
Algebraic definition of the automorphic representation;
Adelic quotient of G and its L^2-space;
Definition of the discrete automorphic representation and comparison of two defintions of automorhpic representation;
References: Getz and Hahn - An introduction to automorphic representations, Sections 4-6.
Speaker: Junny Jo
Goal:
Hilbert space and generalized basis
An example assocaited with SL_2(Z)\H
Eisenstein theory and Langlands' decomposition
L^2-expansion of the function in L^2([G]^1)
References: Nelson - Notes on Lecturs on the spectral theory of Eisenstein series, Section 1.
Labesse - The Langlands specrtral decomposition, Sections 1-4.
Goldfeld - Trace formula for SL(2,R), Section 3 (especially, Theorem 3.30).
Mœglin, Waldspurger - Spectral Decomposition and Eisenstein Series: A paraphrase of the scriptures, Chapter II, V, VI.
Note
Speaker: Tom Huh
Goal:
Recall about the direct integral expression of L^2([G])
When [G] becomes compact quotient: anisotropic reductive groups
Kernel function of R(f) and trace
Spectral and Geometric expansions
(If time allows) considering the kernel function for non-compact quotient and derive the convergence problem
References: Kaletha and Taïbi - A breif introduction to the trace formula and its stabilization, Sections 3-4.
Getz and Hahn - An introduction to automorphic representations, Sections 9, 16, 18.
Speaker: Jungtaek Hong
Goal:
The general expansion of kernel function (including non-compact quotient case)
Problem of divergence - How two control
State Deligne-Kazhdan trace formula and prove
Application: globalization theorem
References: Gelbart - Lectures on the Arthur-Selberg trace fomrmula, Section 5.
Arthur - An introduction to the trace formula, Section 7 with note.
Speaker: Chenyu Wang
Goal:
Introduction of GL_2 and D^X
Statement of JL correspondence
Strategy: comparison of two trace fromulas
Isolate the contribution of one specific representation (using a linear independence of automorphic characters and multiplicity one theorem)
References:
Speaker: Yuchan Lee
References: Arthur - An introduction to the trace formula, Sections 5-10, 12
Speaker:
Goal
Introduce coarse geometric expansion (O-equivalence classes)
Describe the contribution of unramified o in terms of weighted orbital integrals
Local behavior of weighted orbital integrals
Fine geometric expansion
Refernces: Arthur - An introduction to the trace formula, Sections 10-11, 16-19
Speaker:
Goal
Introduce coarse spectral expansion (cusp data, Langlands decomposition)
Describe the contribution of unramified \chi in terms of weighted characters
Fine spectral expansion
Refernces: Arthur - An introduction to the trace formula, Sections 12-15, 20-21