Departmental Colloquium 2011-2012
Thursdays, 4:15 PM, JWB 335
September 15:
Speaker:
Chris Johnson ,
The Scientific Computing and Imaging Institute at the University of Utah
Title: Image-Based Biomedical Modeling, Simulation, and
Visualization
Abstract: Increasingly, biomedical researchers need to build functional computer models from images (MRI, CT, EM, etc.). The "pipeline" for building such computer models includes image analysis (segmentation,
registration, filtering), geometric modeling (surface and volume mesh generation), large-scale simulation (parallel computing, GPUs),
large-scale visualization and evaluation (uncertainty, error). In my presentation, I will present research challenges and software tools for
image-based biomedical modeling, simulation and visualization and discuss their application for solving important research and clinical problems in
neuroscience, cardiology, and genetics.
October 27:
Speaker:
Andrei Caldararu,
University of Wisconsin, Madison
Title: The Pfaffian-Grassmannian derived equivalence
Abstract: String theory relates certain seemingly different manifolds through a relationship
called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a
mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several
distinct spaces can be mirrors to a given one. When this happens it is expected that certain
invariants of these "double mirrors" match up. For a long time the only known examples of double
mirrors arose through a simple construction called a flop, which led to the conjecture that this would
be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to
this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not
related by flops, but are derived equivalent, and therefore are expected to arise through a double
mirror construction. The talk will be accessible to a wide audience, in particular to graduate
students. There will even be several pictures!
November 17:
Speaker:
Eitan Tadmor,
University of Maryland
Title: A new model for self-organized dynamics:
from particle to hydrodynamic descriptions
Abstract: Self-organized dynamics is driven by "rules of engagement", which
describe how each agent interacts with its neighbors. They consist of
long-term attraction, mid-range alignment and short-range repulsion. Many
self-propelled models are driven by the balance between these three forces,
which yield emerging structures of interest. Examples range from consensus
of voters and traffic flows to the formation of flocks of birds or school of
fish, tumor growth etc.
We introduce a new particle-based model, driven by self-alignment, which
addresses several drawbacks of existing models for self-organized dynamics.
The model is independent of the number of agents: only their geometry in
phase space is involved. We will explain the emerging flocking behavior of
the proposed model in the presence of non-symmetric interactions which decay
sufficiently slow, and discuss the difficulties of tracing graph
connectivity otherwise.
The methodology is based on the new notion of active sets, which carries
over from particle to kinetic and hydrodynamic descriptions, and we discuss
the unconditional flocking at the level of hydrodynamic description.
December 8:
Speaker:
Keith Conrad, University
of Connecticut
Title: Why is the Riemann Hypothesis Important?
Abstract:
All mathematicians have heard that the Riemann Hypothesis is a significant open problem, but why is it such a big
deal?
One can name plenty of older unsolved problems in number theory that get far less publicity. We care about the Riemann
Hypothesis because it is connected to a large number of significant questions. I will discuss the history, scope, and
range of consequences of the Riemann Hypothesis, with the aim of convincing mathematicians outside number theory that
it deserves to be counted as one of the most important unsolved problems in mathematics.
January 12: SPECIAL COLLOQUIUM
Speaker:
Hailin
Sang, Indiana University, Bloomington
Title: Autoregressive model selection with simultaneous sparse
coefficient estimation
Abstract: In this talk we study a sparse coefficient estimation procedure
for autoregressive (AR) models based on penalized conditional maximum
likelihood. The penalized conditional maximum likelihood estimator (PCMLE) thus
developed has the advantage of performing simultaneous coefficient estimation
and model selection. Mild conditions are given on the penalty function and the
innovation process, under which the PCMLE satisfies a strong consistency and
oracle property, respectively. Two penalty functions, least absolute shrinkage
and selection operator (LASSO) and smoothly clipped average deviation (SCAD),
are considered as examples, and SCAD is shown to have better performances than
LASSO. At the end, we provide a simulation study and an application of this
method to a historical price data of the US Industrial Production Index for
consumer goods, and the result is very promising.
This is a joint work with Yan Sun.
January 19: SPECIAL COLLOQUIUM
Speaker:
Jon Chaika, Universtiy
of
Chicago
Title: Interval exchange transformations
Abstract: Interval exchange transformations are invertible, piecewise
order preserving isometries of the unit interval with finitely many
discontinuities. Starting from rotations of the circle, which they
generalize, this talk will present their connections to flows on flat
surfaces, rational billiards and symbolic coding. Recent results on
diophantine approximation for interval exchange transformations will be
presented.
January 26: SPECIAL COLLOQUIUM
Speaker:
Hongxiao Zhu, Duke University
Title: Bayesian graphical models for multivariate functional data
Abstract: In a broad variety of application areas there is interest in inferring the dependence structure
in multivariate functional data. For data in vector form, conditional independence relationships between
variables can be inferred through allowing zeros in the precision matrix through a Gaussian graphical model.
Bayesian methods can be used to allow unknown locations of zeros, with a hyper inverse-Wishart prior chosen for
the covariance. We generalize these methods to define a new class of Gaussian process graphical models for
multivariate functional data. We focus on models with decomposable graph structures with a single precision
matrix encoding the conditional independence between the functions. We also discuss the more general class of
non-decomposable graphs. Properties of the proposed process are considered, and two efficient Algorithms are
developed for posterior computation relying on Markov chain Monte Carlo. The methods are evaluated through
simulation studies and applied to Electroencephalography (EEG) data in neuroscience.
January 31: SPECIAL COLLOQUIUM
NOTE different day: Tuesday
Speaker:
Marisa Eisenberg, Ohio State University
Title: Structural Identifiability Methods for Modeling Human Disease
Abstract: Connecting models with data to yield predictive results
requires a variety of parameter estimation, identifiability, and
uncertainty quantification techniques. Identifiability analysis
addresses the question of whether it is possible to uniquely recover
the model parameters from a given set of data. In this talk, I will
discuss my recent work developing identifiability methods using tools
from computational differential algebra and systems theory. I will
also present some applications to problems in human disease, including
cholera, thyroid hormone regulation, and cancer.
February 2:
Speaker:
John Morgan, Stony Brook University
Title: Geometrization of 3-manifolds
Abstract: The Geometrization Conjecture for 3-manifolds was formulated
by Thurston in the early 1980s. `Geometric' manifolds are defined to
be smooth manifolds that admit complete, finite-volume locally
homogeneous Riemannian metrics. Such manifolds are local homogeneous
spaces and thus can be enumerated in terms of Lie groups and finite
co-volume lattices in them. Thurston's geometrization conjecture
states that every compact 3-manifold is constructed from geometric
ones by simple geometric operations. This conjecture is a vast
generalization of the Poincare Conjecture, which can be reformulated
as saying that every closed, simply connected 3-manifold is geometric,
which means that it has a round (constant positive curvature) metric.
Such a manifold is easily seen to be homeomorphic to the 3-sphere.
Perelman used Ricci flow techniques to study 3-manifolds and to prove
the geometrization conjecture. There are two completely separate parts
of the argument: Ricci flow with surgery deforms the metric and does
connected sum decompositions. Eventually (i.e., for sufficiently large
time) under this flow the metric on a 3-manifold becomes one that
decomposes into 2 pieces: one piece on which the metric is converging
smoothly to a complete hyperbolic metric and another piece on which
the metric is locally volume-collapsed on a scale set by the negative
part of the Riemannian curvature. These pieces are connected via
incompressible tori. All of this is analytic/geometric relying on deep
estimates about Ricci flow. To complete the proof of the
geometrization conjecture, one must deal with the piece of the second
type -- the locally volume-collapsed piece. For that one uses a wider
class of spaces on which curvature bounded below makes sense, the
so-called Alexandrov spaces. The local Alexandrov limits of the
locally volume-collapsed piece are Alexandrov balls of dimension 1 or
2 and are completely understood.
From this one constructs nice local models which mesh well on the
overlaps and from these one can show that the
second piece is a disjoint union of pieces that fiber (including
Seifert fiber) over one- and two-dimensional manifolds, completing the
proof of the Geometrization Conjecture
In this talk we will describe the precise statement of the
Geometrization Conjecture, briefly summarize how Ricci flow on a
3-manifold behaves as time goes to infinity. We then pass to a
description of the locally volume-collapsed pieces in terms of their
weak geometric limits and show how this is enough information to
determine the topological type of the collapsing piece of the
manifold.
February 3: SPECIAL COLLOQUIUM
NOTE different day and time: Friday, 4pm
Speaker:
Dongbin Xiu, Purdue University
Title: A Flexible Stochastic Collocation Algorithm on Arbitrary Nodes via Interpolation
Abstract: Stochastic collocation method have become the dominating methods for uncertainty
quantication and stochastic computing of large and complex systems. Though the idea has
been explored in the past, its popularity is largely due to the recent advance of employing
high-order nodes such as sparse grids. These nodes allow one to conduct UQ simulations with
high accuracy and efficiency.
The critical issue is, without any doubt, the standing challenge of "curse-of-dimensionality".
For practical systems with large number of random inputs, the number of nodes for stochastic
collocation method can grow fast and render the method computationally prohibitive. Such
kind of growth is especially severe when the nodal construction is structured, e.g., tensor
grids, sparse grids, etc. One way to alleviate the difficulty is to employ adaptive approach,
where the nodes are added only in the region that is needed. To this end, it is highly
desirable to design stochastic collocation methods that work with arbitrary number of nodes
on arbitrary locations. Another strong motivation is the practical restriction one may face.
In many cases one can not conduct simulations at the desired nodes.
In this work we present an algorithm that allows one to construct high-order polynomial
responses based on stochastic collocation on arbitrary nodes. The method is based on
constructing a "correct" polynomial space so that multi-dimensional polynomial interpolation
can be constructed for any data. We present its rigorous mathematical framework,
its practical implementation details, and its applications in high dimensions.
February 7: SPECIAL COLLOQUIUM
NOTE different day: Tuesday
Speaker:
Justin Kao, Massachusetts Institute of Technology
Title: Modeling the dynamics of fluid boundaries
Abstract: Fluid flows with free boundaries present the challenges of highly
nonlinear dynamics, multiple scales, and complex physics due to interfaces
between two substances. I examine some approaches to describing and
understanding the behavior of these fluid boundaries, and discuss work on
the rupture of thin liquid films, the evolution of wave-generated sand
ripples, and other examples.
February 9:
Speaker:
Eric Opdam,
University of Amsterdam
Title: Harmonic analysis on p-adic reductive groups and homological algebra
Abstract: The Fourier decomposition of an L^2-function on a p-adic semi simple group G
contains contributions from spectral series of various dimensions. These spectral series
consist of irreducible representations of G which have "moderate growth behavior at
infinity'', the so-called tempered irreducible representations. These tempered irreducible
representations are therefore the relevant representations of G to consider when studying the
harmonic analysis on G.
Tempered representations also have striking algebraic properties. With Maarten Solleveld we
have proved an explicit formula for the space of higher extensions between irreducible tempered
representations. This has applications to the computation of the so-called elliptic pairing of
admissible characters of G (the so-called "Kazhdan orthogonality conjecture'').
With Dan Ciubotaru and Peter Trapa we considered an explicit construction of the (limits of)
discrete series as the index of a Dirac operator (for graded affine Hecke algebras). This gives
an expression of the aforementioned elliptic pairing in terms of index theory.
February 16: SPECIAL COLLOQUIUM
Speaker:
Jianfeng Lu, New York University
Title: Multiscale analysis of solid materials: From electronic structure models to continuum
theories
Abstract: Modern material sciences focus on studies on the microscopic scale. This calls for
mathematical understanding of electronic structure and atomistic models, and also their connections to
continuum theories. In this talk, we will discuss some recent works where we develop and
generalize ideas and tools from mathematical analysis of continuum
theories to these microscopic models. We will focus on macroscopic
limit and microstructure pattern formation of electronic structure models.
March 1:
Speaker:
Jeremy Quastel,
University of Toronto
Title: TBA
Abstract: TBA
March 8:
Speaker:
Chris Jones,
University of North Carolina
Title: TBA
Abstract: TBA
March 20:
NOTE different day: Tuesday
Speaker:
Andrea Bertozzi,
University of California Los Angeles
Title: TBA
Abstract: TBA
March 22:
Speaker:
Heinz-Otto
Peitgen , Fraunhofer MEVIS, University of Bremen, Germany
Title: TBA
Abstract: TBA
April 10: Distinguished Lecture Series
NOTE different day: Tuesday
Speaker:
Howard Masur,
University of Chicago
Title: Billiards in polygons, and the SL(2,R) action on moduli spaces
of translation surfaces
Abstract: An appealing example of a dynamical system is billiards in
a polygon. The most studied examples are when the vertex angles are
rational multiples of pi. By an unfolding procedure the billiard flow
gives rise to a straight line flow on what is called a translation
surface. It turns out that the dynamics of the flow on an individual
surface is intimately related to the study of the orbit of the
surface under the action of the group SL(2,R) on the moduli space of
all translation surfaces.
I will survey some of the major developments in this subject.
April 12:
Speaker:
Lisa Fauci,
Tulane University
Title: TBA
Abstract: TBA
April 19:
Speaker:
Dejan Slepcev,
Carnegie Mellon University
Title: TBA
Abstract: TBA