Princeton University

Art History Lecture Series, Orientations in Renaissance Art

Alexander Nagel
New York University
December 9, 2013
In this lecture, Alexander Nagel, Professor of Fine Arts at the Institute of Fine Arts at New York University, illustrates some ways in which art produced during the Renaissance period points ­eastward towards Constantinople, towards the Holy Land, and to places further east, even as far as China. Nagel focuses on the forms this "orientation" took between 1492-1507, years during which new lands were being discovered, to great fanfare, but were still believed to belong to the continent of Asia.

Feynman categories, universal operations and master equations

Ralph Kaufmann
Purdue University; Member, School of Mathematics
December 6, 2013
Feynman categories are a new universal categorical framework for generalizing operads, modular operads and twisted modular operads. The latter two appear prominently in Gromov-Witten theory and in string field theory respectively. Feynman categories can also handle new structures which come from different versions of moduli spaces with different markings or decorations, e.g. open/closed versions or those appearing in homological mirror symmetry. For any such Feynman category there is an associated Feynman category of universal operations.

Genus of abstract modular curves with level \(\ell\) structure

Ana Cadoret
Ecole Polytechnique; Member, School of Mathematics
November 21, 2013
To any bounded family of \(\mathbb F_\ell\)-linear representations of the etale fundamental of a curve \(X\) one can associate families of abstract modular curves which, in this setting, generalize the `usual' modular curves with level \(\ell\) structure (\(Y_0(\ell), Y_1(\ell), Y(\ell)\) etc.). Under mild hypotheses, it is expected that the genus (and even the geometric gonality) of these curves goes to \(\infty\) with \(\ell\). I will sketch a purely algebraic proof of the growth of the genus - working in particular in positive characteristic.

Calabi-Yau mirror symmetry: from categories to curve-counts

Tim Perutz
University of Texas at Austin
November 15, 2013
I will report on joint work with Nick Sheridan concerning structural aspects of mirror symmetry for Calabi-Yau manifolds. We show (i) that Kontsevich's homological mirror symmetry (HMS) conjecture is a consequence of a fragment of the same conjecture which we expect to be much more amenable to proof; and, in ongoing work, (ii) that from HMS one can deduce (some of) the expected equalities between genus-zero Gromov-Witten invariants of a CY manifold and the Yukawa couplings of its mirror.

Independence of \(\ell\) and local terms

Martin Olsson
University of California, Berkeley
November 14, 2013
Let \(k\) be an algebraically closed field and let \(c:C\rightarrow X\times X\) be a correspondence. Let \(\ell \) be a prime invertible in \(k\) and let \(K\in D^b_c(X, \overline {\mathbb Q}_\ell )\) be a complex. An action of \(c\) on \(K\) is by definition a map \(u:c_1^*K\rightarrow c_2^!K\). For such an action one can define for each proper component \(Z\) of the fixed point scheme of \(c\) a local term \(\text{lt}_Z(K, u)\in \overline {\mathbb Q}_\ell \).