Qile Chen: curves on quasi-projective varietiesA^{1}Abstract: Rational curve plays an important role in the study of birational geometry of projective varieties. -curves are the analogues of rationalA^{1}curves for the study of quasi-projective varieties. In this talk, I will introduce the joint work with Yi Zhu on the studying of -curves, andA^{1}-connectedness of quasi-projective varieties. To study the geometry of aA^{1}quasi-projective variety U, we replace U by a toroidal (or equivalently, alog smooth) compactification (X,D). Using the theory of stable log maps to(X,D) developed by Abramovich-Chen-Marcus-Wise and Gross-Siebert, we wereable to produce curves on A^{1}U from degeneration. This technique providesmany interesting examples of -connected varieties. Some applications toA^{1}problems from rationally connectedness and arithmetic geometry over function fields of curves will be discussed. Jordan Ellenberg: Furstenberg sets and Furstenberg schemes Abstract: The study of extremal configurations of points and subspaces sits at the boundary between combinatorics, harmonic analysis, and number theory; since Dvir's 2008 resolution of the Kakeya conjecture over finite fields, it has been clear that algebraic geometry is also part of the story. We prove a theorem of Kakeya type for the intersection of subsets of n-spaceover 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 Wparallel to V which intersects S in at least q points. We prove that such^{c}a set has size at least a constant multiple of q ^{cn/k}. The novelty is themethod; we prove that the theorem holds, not only for subsets of affine space, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G-invariant non-reduced subschemes_{m}supported at a point. This is joint work with Daniel Erman. Dave Jensen: Tropical Independence and the Maximal Rank Conjecture for Quadrics Abstract: The maximal rank conjecture, which has roots in the work of Noether and Severi in the late 19th and early 20th centuries, predicts the Hilbert function of the general embedding of a general curve. In recent joint work with Sam Payne, we show that this conjecture holds for the Hilbert function evaluated at m=2, meaningthat such a curve is contained in the expected number of independent quadrics. From this we deduce that the general curve of genus g anddegree d in projective space of dimension r is projectively normal ifand only if ( r+2)(r+1)/2 is at least 2d-g+1. Our proof usestechniques from tropical and nonarchimedean geometry. Kiran Kedlaya: Some applications of group theory to the arithmetic of abelian varieties Abstract: Given an abelian surface over a number field, its geometric endomorphisms can all be realized over an extension field of degree at most 48; moreover, this bound is best possible. This result, from our work with Fité-Rotger-Sutherland, is ultimately a statement of group theory; we will explain this assertion, and indicate what is expected (and possibly can be proved) for abelian varieties of arbitrary dimension. or: Algebraic Geometry Problems in Cryptography. geometry arise from applications in cryptography. This talk will survey a number of these problems and explain their importance in industry and society. Brian Lehmann: Positivity for curves
Abstract: The best way to capture the geometry of a divisor is to study the asymptotic behavior of sections of multiples of the divisor. This leads to a rich theory of "positivity" relating asymptotic and intersection-theoretic invariants. I will discuss recent progress in understanding the analogous picture for curves. Much of this work was done jointly with Mihai Fulger or Jian Xiao. Jun Li: Mixed-Spin-P fields and Wall crossing of GW theory of CY quintics. Abstract: We will introduce the notion of MSP fields which is a field theory that provides a geometric model for the wall-crossing of the GW theory of quintic CY and the FJRW theory of the Fermat quintic, envisioned by Witten. This theory provides a class of vanishing results, which coupled with virtual localization formulas, gives a class of polynomial relations among the GW and FJRW invariants of quintics. This is a joint work with H.-L. Chang, W.-P. Li and C.-C. Liu.
Yiwei She: The (unpolarized) Shafarevich conjecture for K3 surfaces
Abstract: Let K be a number field, S a finite set of places of K,and g a positive integer. Shafarevich made the following conjecturefor higher genus curves: the set of isomorphism classes of genus gcurves defined over K and with good reduction outside of S isfinite. Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces and Zarhin removed the necessity of specifying a polarization. Building on the work of Faltings and Andre and using technical advances by Madapusi Pera, we prove the unpolarized Shafarevich conjecture for K3 surfaces. |