13.
Holomorphic triangle invariants and the topology of symplectic four-manifolds - Ozsváth, Peter; Szabó, Zoltán
This article analyzes the interplay between symplectic geometry in
dimension $4$ and the invariants for smooth four-manifolds
constructed using holomorphic triangles introduced in [20].
Specifically, we establish a nonvanishing result for the
invariants of symplectic four-manifolds, which leads to new proofs
of the indecomposability theorem for symplectic four-manifolds and
the symplectic Thom conjecture. As a new application, we
generalize the indecomposability theorem to splittings of
four-manifolds along a certain class of three-manifolds obtained
by plumbings of spheres. This leads to restrictions on the
topology of Stein fillings of such three-manifolds.
14.
Restriction and Kakeya phenomena for finite fields - Mockenhaupt, Gerd; Tao, Terence
The restriction and Kakeya problems in Euclidean space have
received much attention in the last few decades, and they are
related to many problems in harmonic analysis, partial
differential equations (PDEs), and number theory. In this paper we
initiate the study of these problems on finite fields. In many
cases the Euclidean arguments carry over easily to the finite
setting (and are, in fact, somewhat cleaner), but there are some
new phenomena in the finite case which deserve closer study.
15.
Reduction of the Hurwitz space of metacyclic covers - Bouw, Irene I.
We compute the stable reduction of some Galois covers of the
projective line branched at three points. These covers are
constructed using Hurwitz spaces parameterizing metacyclic covers.
The reduction is determined by a certain hypergeometric
differential equation. This generalizes the result of Deligne and
Rapoport on the reduction of the modular curve $X(p)$.
16.
Noncommutative projective curves and quantum loop algebras - Schiffmann, Olivier
We show that the Hall algebra of the category of coherent sheaves
on a weighted projective line over a finite field provides a
realization of the (quantized) enveloping algebra of a certain
nilpotent subalgebra of the affinization of the corresponding
Kac-Moody algebra. In particular, this yields a geometric
realization of the quantized enveloping algebra of elliptic (or
$2$-toroidal) algebras of types $D_4^{(1,1)}$, $E^{(1,1)}_6$,
$E^{(1,1)}_7$, and $E_{8}^{(1,1)}$ in terms of coherent sheaves on
weighted projective lines of genus one or, equivalently, in terms
of equivariant coherent sheaves on elliptic curves.
17.
Groups of intermediate subgroup growth and a problem of Grothendieck - Pyber, László
Let $f$ be a function such that for every $\varepsilon
> 0,\ n^{\log n} \leq f(n) \leq n^{\varepsilon n}$ holds if $n$ is
sufficiently large. Suppose that $\log f(n) / \log n$ is
nondecreasing. Using sequences of finite alternating groups, for
every such $f$ we construct a $4$-generator group $\Gamma$ such
that $s_n(\Gamma)$, the number of subgroups of index at most $n$
in $\Gamma$, grows like $f(n)$.
¶
This essentially completes the investigation of the ``spectrum''
of possible subgroup growth types and settles several questions
posed by Lubotzky, Mann, and Segal.
¶
As a by-product we obtain continuously many nonisomorphic
$4$-generator residually finite groups with isomorphic profinite
completions.
¶
Our construction also sheds some light on...
19.
Frobenius amplitude and strong vanishing theorems for vector bundles - Arapura, Donu
The primary goal of this paper is to systematically exploit the
method of Deligne and Illusie to obtain Kodaira-type vanishing
theorems for vector bundles and, more generally, coherent sheaves
on algebraic varieties. The key idea is to introduce a number that
provides a cohomological measure of the positivity of a coherent
sheaf called the Frobenius or F-amplitude. The F-amplitude enters
into the statement of the basic vanishing theorem, and this leads
to the problem of calculating, or at least estimating, this
number. Most of the work in this paper is devoted to doing this in
various situations.
1.
