At the heart of the Langlands program lies the reciprocity conjecture, which can be thought of as a non-abelian generalization of class field theory. An example is the correspondence between modular forms and representations of the absolute Galois group of Q. This can be realized geometrically in the cohomology of modular curves, making essential use of their structure as algebraic curves.
In this talk, I will describe some techniques involved in the recent work of Harris-Lan-Taylor-Thorne and Scholze, who construct Galois representations associated to systems of Hecke eigenvalues occurring in the cohomology of locally symmetric spaces for GL_n. These are real manifolds which generalize modular curves, but lack the structure of algebraic varieties. I will then focus on a very specific property of these Galois representations: the image of complex conjugation, which can be identified by combining Hodge theory with p-adic interpolation techniques. Finally, I will mention some open problems.Updated on Nov 20, 2014 12:24 PM PST
Created on Sep 03, 2014 09:23 AM PDT
Updated on Sep 03, 2014 09:20 AM PDT
Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
organizer: David Eisenbud
Date: Nov 25
3:45 Bernd Sturmfels: The Hurwitz Form of a Projective Variety
The Hurwitz form of a variety is the discriminant that characterizes linear spaces of complementary dimension which intersect the variety in fewer than degree many points. We study computational aspects of the Hurwitz form, relate this to the dual variety and Chow form, and show why reduced degenerations are special on the Hurwitz polytope.
5:00 Winfried Bruns: Maximal minors and linear powers
Abstract: We say that an ideal I in a polynomial ring S has linear powers if all the powers of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that all ideals defining rational normal scroll have linear powers. (This is joint work with Aldo Conca and Matteo Varbarao, to appear in J. Reine Angew. Math.)Created on Sep 03, 2014 09:28 AM PDT
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