Previous Conferences & Workshops
Exceptional collections and the Néron-Severi lattice for surfaces
On the rationality of the logarithmic growth filtration of solutions of $p$-adic differential equations
Shun Ohkubo
4:30pm|Fine 214, Princeton University
We consider an ordinary linear $p$-adic differential equation
\[Dy=d^ny/dx^n+a_{n-1}d^{n-1}y/dx^{n-1}+\dots+a_0y=0,
a_i\in\mathbb{Z}_p[[x]][p^{-1}]\] whose formal solutions in
$\mathbb{Q}_p[x]$ converge in the open unit disc $|x|<1$. In
1973, Dwork proved that $y$ has a log-growth $n-1$, that is,
$|y|_{\rho}=O((\log{1/\rho})^{1-n})$ as $\rho\uparrow 1$, where
$|y|_{\rho}$ denotes the $\rho$-Gaussian norm of $y$. Moreover,
Dwork defined the log-growth filtration of the solution space of
$Dy=0$ by measuring the log-growth of $y$. Then, Dwork conjectured
that the log-growth filtration can be compared with the Frobenius
slope filtration when $Dy=0$ admits a Frobenius structure.
Recently, some partial results on Dwork's conjecture have been
obtained by André, Chiarellotto-Tsuzuki, and Kedlaya. In this talk,
we discuss the rationality of breaks of the log-growth
filtration.
Moduli of degree 4 K3 surfaces revisited
For low degree K3 surfaces there are several way of constructing
and compactifying the moduli space (via period maps, via GIT, or
via KSBA). In the case of degree 2 K3 surface, the relationship
between various compactifications is well understood by...
On the homology and the tree of $SL_2$ over polynomial rings, and reflexive sheaves of rank 2 on projective spaces I
11:00am|Physics Library, Bloomberg Hall 201
We will first quickly recall basic facts on the tree of SL_2
over a field K with a discrete valuation v, following Serre's book.
We will then generalize the geometric interpretation given in that
book for curves to a higher dimensional situation...
Dimension expanders via rank condensers
Expander graphs are sparse graphs with good connectivity
properties and they have become ubiquitous in theoretical computer
science. Dimension expanders are a linear-algebraic variant where
we ask for a constant number of linear maps that expand...
Finite or infinite? One key to algebraic cycles
Algebraic cycles are linear combinations of algebraic
subvarieties of an algebraic variety. We want to know whether all
algebraic subvarieties can be built from finitely many, in a
suitable sense. We present some recent results and
counterexamples.
On monotonicity testing and boolean isoperimetric type theorems
We show a directed and robust analogue of a boolean
isoperimetric type theorem of Talagrand. As an application, we give
a monotonicity testing algorithm that makes
$\tilde{O}(\sqrt{n}/\epsilon^2)$ non-adaptive queries to a function
$f:\{0,1\}^n...
Symplectic forms in algebraic geometry
Imposing the existence of a holomorphic symplectic form on a
projective algebraic variety is a very strong condition. After
describing various instances of this phenomenon (among which is the
fact that so few examples are known!), I will focus on...
Endoscopy theory for symplectic and orthogonal similitude groups
The endoscopy theory provides a large class of examples of
Langlands functoriality, and it also plays an important role in the
classification of automorphic forms. The central part of this
theory are some conjectural identities of Harish-Chandra...
I will try to explain why one expects there to be a Galois
theory of a certain class of transcendental numbers, called
periods, and illustrate with some simple examples.
