abstracts
Qile Chen: A1 curves on quasi-projective varieties
Abstract: Rational curve plays an important role in the study of birational
geometry of projective varieties. A1-curves are the analogues of rational
curves for the study of quasi-projective varieties. In this talk, I will
introduce the joint work with Yi Zhu on the studying of A1-curves, and
A1-connectedness of quasi-projective varieties. To study the geometry of a
quasi-projective variety U, we replace U by a toroidal (or equivalently, a
log smooth) compactification (X,D). Using the theory of stable log maps to
(X,D) developed by Abramovich-Chen-Marcus-Wise and Gross-Siebert, we were
able to produce A1 curves on U from degeneration. This technique provides
many interesting examples of A1-connected varieties. Some applications to
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-space
over a finite field with k-planes. Let S be a subset of Fqn with the
"k-plane Furstenberg property": for every k-plane V, there is a k-plane W
parallel to V which intersects S in at least qc points. We prove that such
a set has size at least a constant multiple of qcn/k. The novelty is the
method; 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 Gm-invariant non-reduced subschemes
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, meaning
that such a curve is contained in the expected number of independent
quadrics. From this we deduce that the general curve of genus g and
degree d in projective space of dimension r is projectively normal if
and only if (r+2)(r+1)/2 is at least 2d-g+1. Our proof uses
techniques 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.
Kristin Lauter: Cryptographic Problems in Algebraic Geometry,
or: Algebraic Geometry Problems in Cryptography.
Abstract: A surprising number of challenging and interesting problems in algebraic
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 conjecture
for higher genus curves: the set of isomorphism classes of genus g
curves defined over K and with good reduction outside of S is
finite. 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.