UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

December 09, 2014

939 Evans Hall

1:30PM: Endomorphism rings and non-commutative resolution of singularities

Hai Long Dao

Let R be a commutative Noetherian ring. It is a classical result that R is regular if and only if it has finite global dimension. In recent years, certain non-commutative rings which are modules-finite over R and has finite global dimension have become objects of intense interests. They can serve as "non-commutative desingularizations" of Spec(R) and have come up in the three-dimensional solution of the Bondal-Orlov conjecture, higher Auslander-Reiten theory and non-commutative minimal model program. Despite all that attention, these objects remain rather mysterious, for examle we do not know fully when they exist, or what global dimensions can occur. In this talk I will describe some very recent work on these questions. Some of the work are joined with E. Faber, C. Ingalls, O. Iyama, R. Takahashi, I. Shipman and C. Vial.

2:45PM: From matrix rigidity to intersection theory; or how I convinced Paolo Aluffi to work on a question in complexity theory

J.M. Landsberg

Signal processing was revolutionized in the 1960's with the fast Fourier transform computation of the discrete Fourier transform using O(n log n) arithmetic operations (v. the naive O(n^2)), which raised the question of how much better one could do. L. Valiant outlined a path to proving there is no O(n) computation of the DFT. Valiant's framework translates to algebraic geometry: it amounts to determining defining equations for certain cones over the variety of rank r nxn matrices. The study of these deceptively simple varieties has led to interesting questions in representation theory and intersection theory and the talk will be focused primarily on this geometry. This is joint work with F. Gesmundo, J. Hauenstein and C. Ikenmeyer.

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