Previous Conferences & Workshops
Learning with little data
The current successes of deep neural networks have largely come
on classification problems, based on datasets containing hundreds
of examples from each category. Humans can easily learn new words
or classes of visual objects from very few examples...
Recent advances in high dimensional robust statistics
Daniel Kane
It is classically understood how to learn the parameters of a
Gaussian even in high dimensions from independent samples. However,
estimators like the sample mean are very fragile to noise. In
particular, a single corrupted sample can arbitrarily...
Proofs from algorithms, algorithms from proofs
Constructive vs Pure Existence proofs have been a topic of
intense debate in foundations of mathematics. Constructive proofs
are nice as they demonstrate the existence of a mathematical object
by describing an algorithm for building it. In computer...
Integral points and curves on moduli of local systems
Junho Peter Whang
The classical affine cubic surface of Markoff has a well-known
interpretation as a moduli space for local systems on the
once-punctured torus. We show that the analogous moduli spaces for
general topological surfaces form a rich family of log
Calabi...
An asymptotic for the growth of Markoff-Hurwitz tuples
Ryan Ronan
For integer parameters $n \geq 3$, $a \geq 1$, and $k \geq 0$
the Markoff-Hurwitz equation is the diophantine equation \[ x_1^2 +
x_2^2 + \cdots + x_n^2 = ax_1x_2 \cdots x_n + k.\] In this talk, we
establish an asymptotic count for the number of...
Integral points on Markoff-type cubic surfaces
We report on some recent work with Peter Sarnak. For integers
$k$, we consider the affine cubic surfaces $V_k$ given by $M(x) =
x_1^2 + x_2 + x_3^2 − x_1 x_2 x_3 = k$. Then for almost all $k$,
the Hasse Principle holds, namely that $V_k(Z)$ is non...
Markoff surfaces and strong approximation
Markoff triples are integer solutions of the equation
$x^2+y^2+z^2 = 3xyz$ which arose in Markoff's spectacular and
fundamental work (1879) on diophantine approximation and has been
henceforth ubiquitous in a tremendous variety of different
fields...
Diophantine analysis in thin orbits
We will explain how the circle method can be used in the setting
of thin orbits, by sketching the proof (joint with Bourgain) of the
asymptotic local-global principle for Apollonian circle packings.
We will mention extensions of this method due to...
From counting Markoff triples to Apollonian packings; a path via elliptic K3 surfaces and their ample cones
Arthur Baragar
4:30pm|Fine 214, Princeton University
The number of integer Markoff triples below a given bound has a
nice asymptotic formula with an exponent of growth of 2. The
exponent of growth for the Markoff-Hurwitz equations, on the other
hand, is in general not an integer. Certain elliptic K3...
