A short film about abstraction

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My book at 42% 27% discount

totaro-bookUpdate (March 19th): The price is now $61.72, a 27% discount.

As of March 17th, 11:00 pm, Amazon is selling my book for $49.02, a 42% discount. I don’t know why this is or how long it will last, but if you’ve been waiting for a drop in price, now may be the time.

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Stony Brook Mini-school 6 April: Derived Categories and Applications to Birational Geometry (+ workshop 7-11 April)

3406865941_e8c7d1893a_oThere will be a Stony Brook mini-school in geometry, on the topic of derived categories and their applications to birational geometry, taking place on Monday, April 6. Aimed at postdocs and advanced graduate students, it is intended to serve as an introduction to some of the topics to be discussed at the workshop on New Techniques in Birational Geometry that will take place at Stony Brook starting on April 7.

Interested postdocs or graduate students can find information at the mini-school website.

Stony Brook will be able to cover lunch and modest travel expenses for registered participants (the registration deadline is March 27). It would be helpful administratively if participants from a given institution could pool their expenses, so that the organizers don’t need to fill out a large number of separate reimbursement requests.

Photo: Math pirate Paul Sally, in the medium of Peeps, from the University of Chicago Magazine, whose photostream is recommended for many great pictures of the old Seminary Co·op (and more about Paul Sally)

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Chow Groups, Motives, and Derived Categories

scarry-pieThis workshop begins tomorrow, March 9th, at the Institute for Advanced Study. The schedule of talks is posted here (linked from “Agenda”).

One theme of the workshop is the Chow group of algebraic cycles on an algebraic variety, and the related concept of motives. A recent advance is the application of Chow groups to birational geometry, for example showing that very general quartic 3-folds are not stably rational. The workshop also covers derived categories of algebraic varieties, which also have exciting interactions with birational geometry.

The Saturday after the workshop is “Pi Day” in Princeton — that is, in the American style of writing dates: 3/14/15. There are events in town for children and adults.

Drawing by Richard Scarry.

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New paper: Adjoint functors on the derived category of motives

robert-leighton-cat-on-a-couch-dreaming-about-being-on-the-other-side-of-the-couch-new-yorker-cartoonI’ve posted a new paper here (and on the arXiv). Voevodsky’s derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper I study whether the inclusions of three important subcategories of motives have a left or right adjoint. These adjoint functors are useful constructions when they exist, describing the best approximation to an arbitrary motive by a motive in a given subcategory. I find a fairly complete picture: some adjoint functors exist, including a few which were previously unexplored, while others do not exist because of the failure of finite generation for Chow groups in various situations. For some base fields, I can determine exactly which adjoint functors exist.

Drawing by Robert Leighton, from The New Yorker

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New paper: Hypersurfaces that are not stably rational

gorey-catsI’ve posted a new paper here (and on the arXiv). It uses the Chow group of algebraic cycles to study a fundamental question in algebraic geometry: which hypersurfaces are stably rational varieties. The result is that for all d at least about 2n/3, a very general complex hypersurface of degree d and dimension n is not stably rational. This is a wide generalization of Colliot-Thélène and Pirutka’s theorem that very general quartic 3-folds are not stably rational. In a vague sense it uses the same machine as last week’s paper.

Drawing by Edward Gorey via Goreyana

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New paper: Complex varieties with infinite Chow groups modulo 2

UPDATEVideo of a talk in which I focus on the idea of the ‘machine’ mentioned below (from the IAS workshop held 9-13 March 2015).

ourcatsite-com_ray-bradburyI’ve posted a new paper here (and on the arXiv). It contains an outline of a general machine for studying Chow groups mod p of a complex variety. This turns out to be an effective way of attacking finiteness problems about algebraic cycles.

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