Algebraic Geometry Seminar Spring 2015: Difference between revisions
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===Jordan Ellenberg=== | ===Jordan Ellenberg=== | ||
Furstenberg sets and Furstenberg schemes over finite fields | Furstenberg sets and Furstenberg schemes over finite fields | ||
(joint work with Daniel Erman) | |||
We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer. | We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer. | ||
Revision as of 02:41, 16 January 2015
The seminar meets on Fridays at 2:25 pm in Van Vleck B135.
The schedule for the previous semester is here.
Algebraic Geometry Mailing List
- Please join the Algebraic Geometry Mailing list to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).
Fall 2014 Schedule
| date | speaker | title | host(s) |
|---|---|---|---|
| February 20 | Jordan Ellenberg (Wisconsin) | Furstenberg sets and Furstenberg schemes over finite fields | I invited myself |
| February 27 | Botong Wang (Notre Dame) | TBD | Max |
| March 6 | Matt Satriano (Johns Hopkins) | TBD | Max |
| March 13 | Jose Rodriguez (Notre Dame) | TBD | Daniel |
Abstracts
Jordan Ellenberg
Furstenberg sets and Furstenberg schemes over finite fields (joint work with Daniel Erman)
We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer.
Jose Rodriguez
TBA