we consider the problem of classifying completely unlabelled data using convex inequalities that contain absolute values of the data. this allows each data point to belong to either one of two classes by entering the inequality with a plus or minus value. using such absolute value inequalities in support vector machine classifiers, unlabelled data can be successfully partitioned into two classes that capture most of the correct labels dropped from the data. inclusion of partially labelled data leads to a semisupervised classifier. computational results include unsupervised and semisupervised classification of the wisconsin breast cancer wisconsin (diagnostic) data set.

following colding and minicozzi, we consider the entropy of (hyper)-surfaces in euclidean space. this is a numerical measure of the geometric complexity of the surface. in addition, this quantity is intimately tied to to the singularity formation of the mean curvature flow which is a natural geometric heat flow of submanifolds. in the talk, i will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses. this is all joint work with l. wang.

this talk is about representation learning with a nontrivial geometry of
variables. a convolutional neural network can be viewed as a statistical
machine to detect and count features in an image progressively through a
multi-scale system. the constructed features are insensitive to nuance
variations in the input, while sufficiently discriminative to predict
labels. we introduce the haar scattering transform as a model of such
a system for unsupervised learning. employing haar wavelets makes it
applicable to data lying on graphs that are not necessarily pixel grids.
when the underlying graph is unknown, an adaptive version of the
algorithm infers the geometry of variables by optimizing the
construction of the haar basis so as to minimize data variation. given
time, i will also mention an undergoing project of flow cytometry data
analysis, where histogram-like features are used for comparing empirical
distributions. after ``binning'' samples on a mesh in space, the problem
can be closely related to feature learning when a variable geometry is
present.

high-dimensional temporal dependent data arise in a wide range of disciplines.
the fact that the classical clt for i.i.d. random vectors may fail in
high dimensions makes high-dimensional inference notoriously difficult.
more challenges are imposed by temporal and cross-sectional dependence.
in this talk, i will introduce the high-dimensional clt for temporal dependent
data. its validity depends on the sample size $n$, the dimension $p$, the
moment condition and the dependence of the underlying processes. an example
is taken to appreciate the optimality of the allowed dimension $p$. equipped
with the high-dimensional clt result, we have a new sight on many problems
such as inference for covariances of high-dimensional time series which can
be applied in the analysis of network connectivity, inference for multiple
posterior means in mcmc experiments as well as kolmogorov-smirnov test for
high-dimensional dependent data. i will also introduce an estimator for
long-run covariance matrices and two resampling methods, i.e., gaussian
multiplier resampling and subsampling, to make the high-dimensional clt more
applicable. our work is then corroborated by a simulation study with a
hierarchical model.

a basic problem in kahler geometry, going back to calabi in the 50's, is to find kahler metrics with the best curvature properties, e.g., einstein metrics. such special metrics are minimizers of well known functionals on the space of all kahler metrics h. however these functionals become convex only if an adequate geometry is chosen on h. one such choice of riemannian geometry was proposed by mabuchi in the 80's, and was used to address a number of uniqueness questions in the theory. in this talk i will present more general finsler geometries on h, that still enjoy many of the properties that mabuchi's geometry has, and i will give applications related to existence of special kahler metrics, including the recent resolution of tian's related properness conjectures.

a number of questions in galois theory can be phrased in the following
way: how large (in various senses) can the galois group g of an
extension of the rational numbers be, if the extension is only allowed
to ramify at a small set of primes? if we assume that g is abelian,
class field theory provides a complete answer, but the question is
open is almost every nonabelian case, since there is no known way to
systematically and explicitly construct such extensions in full
generality.

however, there have been some programs that are gaining ground on this
front. while the problem above is natural and the objects are
classical, we will see that to answer certain questions about the
“largeness” of this galois group, it seems necessary to use techniques
involving automorphic forms and their representation-theoretic
avatars. in particular, it will turn out that some recent results on
“harmonic” families of automorphic forms translate to the fact that
such number fields, despite not being explicitly constructible by
known methods, turn out to “exist in abundance” and allow us to find
bounds on the sizes of such galois groups.

massive data are often contaminated by \underline{outliers} and heavy-tailed errors. to address this challenge, we propose the adaptive huber regression for robust estimation and inference. the key observation is that the robustification parameter should adapt to sample size, dimension and moments for optimal \underline{tradeoff} between bias and robustness. our framework is able to handle heavy-tailed data with bounded $(1+\delta)$-th moment for any $\delta>0$. we establish a sharp phase transition for robust estimation of regression parameters in both finite dimensional and high dimensional settings: when $\delta \geq 1$, the estimator achieves sub-gaussian rate of convergence without sub-gaussian assumptions, while only a slower rate is available in the regime $0<\delta <1$ and the transition is smooth and optimal. as a consequence, the \underline{nonasymptotic bahadur} representation for finite-sample inference can only be derived when the second moment exists. numerical experiments lend further support to our obtained theories.

tautological relations are certain equations in the chow ring of
the moduli space of curves. i will discuss a family of such relations,
first conjectured by a. pixton, that arises by studying moduli spaces of
ramified covers of the projective line. these relations can be used to
recover a number of well-known facts about the moduli space of curves, as
well as to generate very special equations known as topological recursion
relations. this is joint work with various subsets of s. grushevsky, f.
janda, x. wang, and d. zakharov.

first suggested by witten in the early 1990's, the
landau-ginzburg/calabi-yau correspondence studies a relationship between
spaces of maps from curves to the quintic 3-fold (the calabi-yau side) and
spaces of curves with 5th roots of their canonical bundle (the
landau-ginzburg side). the correspondence was put on a firm mathematical
footing in 2008 when chiodo and ruan proved a precise statement for the
case of genus-zero curves, along with an explicit conjecture for the
higher-genus correspondence. in this talk, i will begin by describing the
motivation and the mathematical formulation of the lg/cy correspondence,
and i will report on recent work with shuai guo that verifies the
higher-genus correspondence in the case of genus-one curves.

this is part of a series lectures studying stability of symplectic forms
on noncompact manifolds. the case of compact manifolds is well understood
thanks to seminal work of jurgen moser in the 1960s.