in the classification theory of higher dimensional algebraic varieties a central role is played by the canonical divisor $k_x$ and its multiples. a famous theorem of y. t. siu states that if ${\pi:x\longrightarrow}$ t is a smooth projective family of varieties, then the plurigenera of the fibres $\lbrace h^0(x_t, mk_{x_t})\rbrace_{m\geq 0}$ are constant in t. despite being an algebraic problem, siu's proof employs methods which are essentially analytic in nature.
after giving an overview of the techniques involved, we outline a path to a possible algebraic proof, based on a reduction to the general type case via the iitaka fibration.

clustering objects into similar clusters is an important practical problem in a wide variety of fields, including statistics, physics, bioinformatics, articial intelligence, and data mining. my thesis focuses on feature selection and semiparametric modeling in clustering. in this talk, i will present three of the projects i have done with my advisor during my phd studies. the first one proposes a hill-climbing approach to sparse clustering, which has been shown to be competitive with existing methods in literature on simulated and real-world datasets. the second one considers a semiparametric mixture model for clustering and we propose a semiparametric em algorithm to fit the model. the third one discusses the difficulty we uncovered in clustering with radial distributions. under mild conditions, we prove that the magnitudes of the norm of observations sampled from a radial distribution are highly concentrated as the dimension becomes large.

compressed sensing or compressive sampling (cs) is a signal
processing technique for efficiently acquiring and reconstructing
sparse signals by solving underdetermined linear systems. in practice,
cs needs to be accompanied by a quantization process. that is, after
sampling the signals, we represent the measurements using discrete
data, e.g. 0s and 1s, and recover the signals from the quantized
measurements. in this talk, i will discuss how to extend the
noise-shaping quantization methods beyond the case of gaussian
measurements to structured random measurements, including random
partial fourier and random partial circulant measurements. this is
joint work with rayan saab

unit root process, as a process with stochastic trend and a generalization
from random walk, is pervasive in physics, economics, and finance. in the
hypothesis test for unit root, bootstrap methods have earned a great deal
of attention. this talk will present three applications of bootstrap
methods. while the first application focuses on classic unit root process,
which contains only stochastic trend, the second and the third tackle
seasonal unit root process, which include simultaneously stochastic trend
and stochastic seasonality.

i will address two conjectures: a generalization of
kurihara's conjecture on higher fitting ideals, and a special case of the
breuil-schneider conjecture for certain de rham representations.

we present a proof of the fact that given a hodge
isometry psi between the rational second cohomology of two kahler k3
surfaces $s_1$ and $s_2$, we can find a finite sequence of k3 surfaces and
analytic (2, 2)-classes supported on successive products, such that
the isometry psi is the convolution of these classes. the proof of
this fact implies that for projective k3 surfaces $s_1$, $s_2$ the class
of psi is algebraic. this proves a conjecture of i. shafarevich.

in this talk i will introduce the notion of
multivariable $(\phi, \gamma)$-modules and explain their relationship
to representations of direct products of galois groups. i will then
describe my recent work with kiran kedlaya aimed at proving that all
such multivariable $(\phi, \gamma)$-modules are overconvergent.

in this talk, we provide a brief description of the stock market and internet search engines. we then suggest some efficient methods to gathering historical stock price data and google search data. additionally, we propose using a test that we created to explore the relationship, if any, of stock prices and the popularity of google searches. finally, we share our results from the test and discuss the possibility of using the popularity of google searches to predict future stock price movement.

in this talk i will discuss some of the details of the proof that all ordinary $(\phi, \gamma)$-modules are overconvergent and examine what additional arguments are necessary to prove this statement in the multivariable setting. along the way we will encounter a restatement of drinfeld's lemma in the context of diamonds. part 1 of this talk is being given during this week's rtg colloquium.

i will discuss recent approaches to working with formal schemes, including details from both proofs in my thesis and in the work of others.