Category Archives: Susie

Very old paper: The cohomology ring of the space of rational functions

susie-papersThanks to Claudio Gonzales, who requested it, and to MSRI Librarian Linda Riewe, who found and scanned it, my 1990 MSRI preprint “The cohomology ring of the space of rational functions” is available on my webpage.

Abstract: We consider three spaces which can be viewed as finite-dimensional approximations to the 2-fold loop space of the 2-sphere, Ω2S2. These are Ratk(CP1), the space of based holomorphic maps S2→S2; Bβ2k, the classifying space of the braid group on 2k strings; and Ck(R2, S1), a space of configurations of k points in R2 with labels in S1. Cohen, Cohen, Mann, and Milgram showed that these three spaces are all stably homotopy equivalent. We show that these spaces are in general not homotopy equivalent. In particular, for all positive integers k with k+1 not a power of 2, the mod 2 cohomology ring of Ratk is not isomorphic to that of Bβ2k or Ck. There remain intriguing questions about the relation among these three spaces.

Since 1990, a few papers have built on this preprint, including:

J. Havlicek. The cohomology of holomorphic self-maps of the Riemann sphere. Math. Z. 218 (1995), 179–190.

D. Deshpande. The cohomology ring of the space of rational functions (2009). arXiv:0907.4412

Photo: Susie the cat in Cambridge c. 2001.

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New paper: Rationality does not specialize among terminal varieties

IMG-20150805-00382I’ve posted a new paper on the arXiv. A limit of rational varieties need not be rational, even if all varieties in the family are projective and have at most terminal singularities. This shows that a result of de Fernex and Fusi’s does not extend to higher dimensions.

Photo: Susie the cat in Westwood.

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New paper: The integral cohomology of the Hilbert scheme of two points

IMG-20140924-00178I’ve posted a new paper on the arXiv. (I haven’t found an apt cat picture, so have just used a photo of Susie the cat, who supervised the writing of the paper closely.)

The Hilbert scheme X^{[a]} of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X. The integral cohomology of X^{[a]} is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X^{[2]} for any complex manifold X, and the integral cohomology of X^{[2]} when X has torsion-free cohomology.

The results of this paper are used in Voisin’s work on the universal CH_0 group of cubic hypersurfaces, because the crucial point there is to study the 2-torsion in the Chow group.

Photo: Susie the cat in Princeton.

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IAS: getting settled

Using the study in Institute housing

As we start the second week of the fall term at the Institute, the backbone for the Topology of Algebraic Varieties program has been settled. (This forms a small subset of all the relevant mathematical activities at the Institute and the university.)

The regular weekly round of program activities is:
Tuesday
11:00 am Preprint seminar, organized by János Kollár
2:00 pm Talk 1 of the double seminar joint with Princeton
3:30 pm Talk 2 of the double seminar joint with Princeton

Wednesday
11:15 am Program seminar

Some specific events of interest are:
Workshop on Fundamental Groups and Periods, IAS, October 13–17
Barry Mazur: Minerva Lectures, Princeton University, October 14, 15, 17
AGNES, University of Pennsylvania, October 31–November 2

Details and more information on program activities are available on the Institute website and on the informal program website.

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Compositio meeting

IMG-20121219-00062

Update on other commitments: supervising the grading of exams
IMG-20121219-00062

Reviewing the course
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December 19, 2012 · 8:10 pm