I’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
I’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
I’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.
The Institute for Advanced Study is closed from 3:00 pm today (Mon. Jan. 26th) and will cancel all activities tomorrow (Tues. Jan. 27th). This includes the dining hall and IAS shuttle service, as well as math talks. A blizzard is coming. UPDATE (1/27) — No blizzard, but plenty of snow.
Photo: Submarine snow cat from reddit.
The Second Latin American School of Algebraic Geometry and Applications (II ELGA) will take place from June 1st to 12th, 2015, at Hotel La Plage, Cabo Frio, State of Rio de Janeiro.
The goal of this CIMPA research school is to train young mathematicians working in Latin America in some of the most active areas of research in algebraic geometry, as well as to promote greater interaction among researchers and students, and to build a network of collaborations. The first week of the school will consist of four mini-courses covering different aspects of algebraic geometry, all taught in English. (I will be giving a course about the integral Hodge conjecture.) There will also be poster presentations by young researchers and students. The second week will consist of research talks by leading specialists, as well as presentations by young researchers.
Registration is open, with a deadline of January 18th, 2015, for participants from outside Brazil.
Photo: A cat dressed in a fancy costume with its owner at the animal carnival parade at Copacabana beach in Rio de Janeiro, Brazil. AFP photo/Christophe Simon, via Hindustan Times.
Algebraic geometry is fortunate to have the established tradition of a major discipline-spanning meeting every decade. The next such meeting will be held in Utah from July 12th to August 1st, 2015, with a preceding graduate student bootcamp from July 6th to 10th.
Registration is now open, with a deadline of January 15th, 2015.
I am organizing a seminar in the third week, with the following speakers:
Aravind Asok, Joseph Ayoub, Nicolas Bergeron, Patrick Brosnan, Aise Johan de Jong, Moritz Kerz, Bruno Klingler, Max Lieblich, Alena Pirutka, Shuji Saito, Christian Schnell, Zhiyu Tian
Photo from pechanga on instagram, via Millie, the Brave Rock Climbing Cat.
Registration is now open. The Institute does not offer funding for visitors to this workshop; however, registration is free.
This workshop is part of the topical program “The Topology of Algebraic Varieties” taking place during the 2014-15 academic year at the Institute for Advanced Study. 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.
Drawing by Louis Wain