Colloquia — Spring 2006
Thursday, April 20, 2006
| Title |
Moment Analysis of Distributions |
| Speaker |
Dr. Jordan Stoyanov
School of Mathematics & Statistics
University of Newcastle upon Tyne
UNITED KINGDOM
|
| Time |
2:00-3:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor George Yanev |
Abstract
We start with a brief discussion on the importance of the moments when analyzing
probability/statistical distributions from both theoretical and applied points
of view. Then we turn to the question of how, in terms of the moments, to characterize
a distribution as being unique or non-unique. In particular:
Under what conditions can a distribution be determined uniquely by its moments?
This is the classical problem of moments originated in works by Chebyshev,
Markov, Stieltjes with further contribution by Carleman, Cramer, … , Krein,
Akhiezer, … , and more recently by Heyde, Berg, Slud, Lin, Pakes, Gut, …
We describe conditions under which a distribution is uniquely determined
by its moments (M-det) and conditions when it is non-unique (M-indet). The
illustrations involve distributions such as Normal, Log-normal, IG, Gamma,
Poisson, etc. Many of the facts are new, hence not so well-known, and some
of them even look surprising.
Of special interest in statistical applications are the so-called Box-Cox
functional transformations of random data (coming from random variables
or stochastic processes). What can we say about the moment determinacy
or indeterminacy of the distributions involved? Some answers are based
on classical results (Carleman and Cramer), however, we need new ideas
and techniques (Krein, Slud, Lin, Pakes, Gut).
We show next that the non-uniqueness phenomenon can be efficiently studied
by constructing Stieltjes classes (infinite families of distributions all having
the same moments) and calculating their Index of dissimilarity.
More illustrations will be given to show the role of the moment determinacy
when studying analytic properties of distributions (symmetry, unimodality and
infinite divisibility) and also in applied areas such as aerosol sciences,
stochastic financial modelling (Black-Scholes type models), identifiability
for mixture distributions.
It will be clear that Moment Analysis of Distributions is a rich area of
research with important practical implications. There are, however, challenging
open questions and if time permits some of them will be briefly outlined.
The material will be presented in an understandable and (hopefully) attractive
way, addressing the lecture not only to professionals in statistics, probability,
mathematics (theory and/or applications), but also to doctoral and master students
in these areas.
References
- Heyde, C.C. (1963). On a property of the lognormal distribution. J. Royal
Statist. Society Ser. B 25, 392—393.
- Feller, W. (1971). An Introduction to Probability Theory and Its Applications,
Vol. 2, 2nd ed. Wiley, New York.
- Stoyanov, J. (1997). Counterexamples in Probability, 2nd ed. Wiley, Chichester-NY.
- Lin, G.D. (1997). On the moment problem. Statist. & Probab. Letters
35, 85—90.
- Stoyanov, J. (2000). Krein condition in probabilistic moment problems.
Bernoulli 6, 939—949.
- Pakes, A.G., Hung, W.-L., Wu, J.-W. (2001). Criteria for the unique determination
of probability distributions by moments. Austral. & New Zealan J. Statist.
43, 101—111.
- Gut, A. (2002). On the moment problem. Bernoulli 8, 407—421.
- Lin, G.D., Stoyanov, J. (2002). On the moment determinacy of the distributions
of compound geometric sums. J. Appl. Probab. 39, 545—554.
- Stoyanov, J. (2004). Stieltjes classes for M-indeterminate probability
distributions. J. Appl. Probab. 41A, 281—294.
- Stoyanov, J., Tolmatz, L. (2005). Method for constructing Stieltjes classes
for M-indeterminate probability distributions. Appl. Math. & Comput.
165, 669—685.
Friday, April 14, 2006
| Title |
Relative difference sets fixed by inversion |
| Speaker |
Yuqing Chen
Wright State University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Xiang-Dong Hou |
Abstract
If a group G acts on a divisible design regularly and transitively,
the subsets in G that correspond to the blocks of the design are
called relative difference sets relative to a subgroup N of G.
In this talk, I will discuss various techniques of constructing relatives difference
sets that are closed under taking inversion.
Friday, April 7, 2006
| Title |
Walter, me and the odd order paper |
| Speaker |
John Thompson
Graduate Research Professor
University of Florida |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 141 |
| Sponsor |
Professor Arunava Mukherjea |
Abstract
In my lecture, I will outline some of the early results about groups of odd
order, and then discuss in some detail the contribution of Walter Feit to the
proof that groups of odd order are solvable, with special emphasis on his work
on group characters. I won't give any proofs, since it is not that sort of
lecture.
Tuesday, March 28, 2006
| Title |
The ropelengths of physical knots |
| Speaker |
Yuanan Diao
University of North Carolina
Charlotte, NC |
| Time |
2:00-3:00 p.m. |
| Place |
LIF 266 |
| Sponsor |
Professor Masahiko Saito |
Abstract
The ropelength of a knot is the minimum amount of rope needed to tie the knot
(assuming the rope has a unit thickness). An overview of the knot ropelength
problem will be given in this talk. Topics will cover the global minimum ropelength
of nontrivial knots, general lower and upper ropelength bounds for knots and
links, and ropelength bounds for some knot and link classes.
Friday, March 24, 2006
| Title |
Boundary Value Problems Arising From Monge-Ampère
Equations |
| Speaker |
Shouchuan Hu
Missouri State University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Yuncheng You |
Abstract
In seeking radially symmetric convex solutions of a Monge-Ampère
equation, we are led to a boundary value problem which can be studied by various
methods. In this talk, we are going to use a topological approach to establish
some results on the existence, multiplicity and nonexistence of radially symmetric
convex solutions for the Monge-Ampère type equations.
Monday, March 20, 2006
| Title |
Extremal problems for non-vanishing functions in Bergman spaces |
| Speaker |
Catherine Bénéteau
Seton Hall University |
| Time |
3:15-4:15 p.m. |
| Place |
PHY 108 |
| Sponsor |
Analysis Search Committee |
Abstract
In this talk, I will give a brief survey of duality techniques introduced by
S. Ya. Khavinson in 1949 and independently by Rogosinski and Shapiro in 1953.
I will discuss how these techniques were then applied to general extremal problems
for non-vanishing functions in analytic function spaces in the 50's and 60's.
I will also examine the Bergman space context, where such techniques seem to fail.
However, in a wide class of such problems, solutions exist and are unique, and we
are able to obtain some regularity results with surprising twists. In particular,
if we consider the specific problem of minimizing the norm of non-vanishing
Bergman functions whose first two Taylor coefficients are given, the conjectured
form of the extremal is not continuous in the closed disk! This most recent result
is joint work with D. Aharonov, D. Khavinson, and H. Shapiro.
Friday, March 10, 2006
| Title |
Constant Mean Curvature Surfaces and Loop Groups |
| Speaker |
Hongyou Wu
Northern Illinois University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Wen-Xiu Ma |
Abstract
Surfaces of non-zero constant mean curvature, i.e., soap bubbles, are fundamental
surfaces and have many applications. We will discuss the dressing action of
loop groups on these surfaces and present a Weierstrass type representation
of the surfaces. We will also mention constructions of such surfaces with a
non-trivial topology. Many examples of surfaces of non-zero constant mean curvature
will be shown.
Friday, March 3, 2006
| Title |
The Hamiltonian Approach to Solving Differential Equations |
| Speaker |
Jin Yue
Dalhousie University |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 263 |
| Sponsor |
Analysis Search Committee |
Abstract
Theory of differential equations as the mathematical community has known it
since the 17th century (the term “differential equation” was coined by
Leibnitz in 1676) is a vast area of mathematics utilyzing various techniques from
analysis, differential geometry, topology, etc.
In this talk I will discuss how recent developments in the area of Hamiltonian
mechanics can be effectively employed to solve nonlinear ordinary differential
equations. The approach is based on the invariant theory of Killing tensors that,
in turn, subsumes the ideas of classical invariant theory and Cartan's geometry.
Monday, February 27, 2006
| Title |
Concentration Phenomenon in a Singularly Perturbed Quasi-linear
Elliptic Problem |
| Speaker |
Chunshan Zhao
University of Iowa |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 266 |
| Sponsor |
Analysis Search Committee |
Abstract
I will talk about the structure and concentration phenomenon of positive solutions
to a singularly perturbed quasi-linear Neumann problem arising from a biological
model.
In this talk I will first address questions regarding the existence, uniform
boundedness, Harnack-type inequality and asymptotic behaviors of this quasi-linear
Neumann problem. After that I will focus on a class of special solutions, so-called
least-energy solutions and introduce new techniques involved intrinsic variation
methods, exponential decay properties of ground states and the least-energy
solutions. Spiky behaviors of the least-energy solutions and the location of
peak(s) will be investigated.
Monday, February 27, 2006
| Title |
Generating Random Graphs With Given Degrees |
| Speaker |
Joseph Blitzstein
Stanford University |
| Time |
11:00-12:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Discrete Mathematics Search Committee |
Abstract
Random graphs with a given degree sequence have recently been studied as an
alternative to the Erdős-Renyi random graph model. We will discuss algorithms
for generating such graphs, including an algorithm found in joint work with
Persi Diaconis, which uses a combinatorial theorem of Erdős and Gallai
to guarantee that the output has the desired degrees. The algorithm is easy
to implement and use in tandem with sequential importance sampling. Applications
such as studying a real-world food web and approximate enumeration of the number
of graphs with a given degree sequence will be discussed.
Friday, February 24, 2006
| Title |
Asymptotic Results for Discrete Minimum Energy Problems |
| Speaker |
Sergiy Borodachov
Vanderbilt University |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Analysis Search Committee |
Abstract
We consider a generalization of the classical Thomson problem dealing with
the final (equilibrium) positions of N electrons repelling each other on the
surface of a sphere. The potential of the repelling force in our consideration
is assumed, more generally, to be proportional to the reciprocal of the power s > 0
of the distance between points and the points (electrons) are constrained to
a compact rectifiable set embedded in Euclidean space (for which the sphere
is one example). For one-dimensional rectifiable sets, asymptotics for such
minimal energy configurations were obtained by Rakhmanov et al. We will discuss
the history of the question, recent progress for higher-dimensional rectifiable
sets, applications, and related problems.
Monday, February 20, 2006
| Title |
From the Fundamental Theorem of Algebra to Astrophysics:
a Harmonic Journey |
| Speaker |
Dmitry Khavinson
University of Arkansas |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 266 |
| Sponsor |
Analysis Search Committee |
Monday, February 20, 2006
| Title |
Spherical Codes and Delsarte's Method |
| Speaker |
Florian Pfender
TU Berlin
Germany |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 109 |
| Sponsor |
Discrete Mathematics Search Committee |
Abstract
In coding theory, a lot of energy is spend to find good codes. But very often it is hard to give good quality guarantees as upper bounds on sizes of codes are hard to come by. In the 70s, Phillippe Delsarte developed an approach which gives the best known bounds for many problems. In this talk we will take a closer look at this approach and see some recent improvements to the method. This leads to some new upper bounds for kissing numbers in several dimensions, the number of unit balls which can simultaneously touch one central ball.
Friday, February 17, 2006
| Title |
Multivariate Polynomial Spline Approximation |
| Speaker |
Tatiana Sorokina
University of Georgia |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Analysis Search Committee |
Abstract
The talk is designed to provide an overview of multivariate polynomial splines.
We will discuss basic concepts, main research directions, open problems, and
applications. Polynomial spline functions (in their simplest form, piecewise
polynomials) are a well established computational tool. They are used for a
variety of purposes, including approximating curves and surfaces, computer-aided
geometric design, finite elements and solution of differential equations, image
processing, biomedical imaging, etc. In contrast to the univariate case, the
theory of multivariate polynomial splines is yet to be developed.
Friday, February 17, 2006
| Title |
Virtual Knot Theory |
| Speaker |
Louis H. Kauffman
University of Illinois at Chicago |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Nagle Lecture Committee |
Abstract
Virtual knot theory is a generalization of classical knot theory to knot
diagrams that have crossings that are “not really there”
just as an embedding of a non-planar graph must have special crossings when
it is represented in the plane. Knot diagrams with such virtual crossings have
a natural notion of extended topological deformation in the form of combinatorial
rules for changing the diagrams.
This talk will discuss virtual knot theory showing how it can be interpreted
in terms of stabilized embeddings of knots in thickened surfaces. We will discuss
the Jones polynomial for virtual knots, and how this is related to conjectures
about the Jones polynomial for classical knots. If time permits we will discuss
how new algebra (biquandles) is related to virtual knots and how new invariants
such as Khovanov homolgy are related to this theory. The talk will be self-contained.
Thursday, February 16, 2006
| Title |
Natural Exponential Families and Approximation Operators |
| Speaker |
Mourad Ismail
University of Central Florida |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 260 |
| Sponsor |
Analysis Search Committee |
Abstract
The abstract can be found here.
Thursday, February 16, 2006
| Title |
Structure and self-assembly of viral capsids |
| Speaker |
Reidun Twarock
University of York
York, UNITED KINGDOM |
| Time |
2:00-3:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Natasha Jonoska |
Abstract
In a landmark paper Sir Aaron Klug and Don Caspar have established a theory that predicts the surface structures of viruses in terms of a family of polyhedra [1]. It is fundamental in virology and has a broad spectrum of applications, ranging from image analysis of experimental data to the construction of models for the self-assembly of viral capsids (i.e., of the protein shells encapsulating, and hence providing protection for, the viral genome). Despite its huge success, experimental results have provided evidence for the fact that this theory is incomplete, and in particular cannot account for the structure of viruses in the family of Papovaviridae, which are of particular interest for the public health sector because they contain cancer-causing viruses.
Based on group theory and tiling theory we have developed a theory that closes this gap [2, 3]. It leads to a new series of polyhedra, the triacontahedral series [4], that corresponds to the particles observed during self-assembly of the major capsid proteins of viruses in the family of Papovaviridae. Among others, it allows to classify the malformations that may occur during self-assembly (e.g., [5]). The new theory has opened up various areas of application. In this talk, we will focus in particular on our models for the self-assembly of viral capsids and the classification of crosslinking structures, which have been featured recently by Science News (Sept. 2005, Vol. 168, No. 10).
(1) Assembly models:
Our theory for the structural description of viruses encodes the locations of both the capsid proteins and the bonds (dimer- and trimer-interactions) between them, and hence predicts the local bonding structure in terms of the locations of the C-terminal arm extensions of the proteins. We use this information to derive graphs that encode the structure of the intermediate species occurring during self-assembly of the capsid proteins. These graphs are combinatorial objects that are used to derive quantities of interest such as the concentrations of the assembly intermediates, and they hence characterize the assembly process [6]. Moreover, they allow us to determine the dominant pathways of assembly and hence to develop strategies of interference with the assembly process [7].
(2) Crosslinking structures:
Crosslinking structures are additional covalent bonds that provide particular stability to the viral capsids. We have shown that our approach can be used to classify crosslinking structures, and that it provides a theoretical tool to probe whether crosslinking is possible for general types of viruses [8].
References
- Caspar, D.L.D & Klug, A. (1962) Physical Principles in the Construction of Regular Viruses. Cold Spring Harbor Symp. Quant. Biol. 27, pp. 1.
- Twarock, R. (2004), A tiling approach to virus capsid assembly explaining a structural puzzle in virology, J. Theor. Biol. 226, pp. 477.
- Twarock, R. (2005) The architecture of viral capsids based on tiling theory, J. Theor. Medicine 6, pp. 87.
- Keef, T. and Twarock, R. (2005) A novel family of polyhedra as blueprints for viral capsids in the family of Papovaviridae, submitted to J. Math. Biol.
- Twarock, R. (2005) Mathematical models for tubular structures in the family of Papovaviridae, Bull. Math. Biol. 67, pp. 973.
- Keef, T., Taormina, A. and Twarock, R. (2005) Assembly Models for Papovaviridae based on Tiling Theory, Phys. Biol. 2, pp.175.
- Keef, T., Micheletti, C. and Twarock, R. (2005) Master equation approach to the assembly of viral capsids, submitted to J. Theor. Biol.
- Twarock, R. and Hendrix, R. (2005) Crosslinking in Viral Capsids via Tiling Theory, to appear in J. Theor. Biol.
Friday, February 10, 2006
| Title |
Algebraic Aspects of Topological Quantum Computing |
| Speaker |
Eric Rowell
Indiana University, Bloomington |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Discrete Mathematics Search Committee |
Abstract
Quantum computers would take advantage of quantum mechanical phenomena to solve
problems more efficiently than “classical” computers. The topological
quantum computer (TQC) of Freedman/Kitaev uses topological degrees of freedom to
achieve a higher error tolerence than the usual quantum circuit model. Moreover,
their model can be described algebraically as modular categories. As such, many
fundamental questions related to TQCs can be translated into algebraic problems
(interesting in their own right).
In this talk I will give an overview of TQCs and their algebraic conterparts.
As time permits, I will discuss a some problems and results underscoring the
application of algebra to quantum computing.
Friday, February 10, 2006
| Title |
Physical Knot Theory |
| Speaker |
Jorge Calvo
Ave Maria University
Naples, FL |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Mohamed Elhamdadi |
Abstract
Knot theory is the study of simple closed loops embedded in 3-dimensional space
and their deformations in this space. If a loop can be deformed into a perfectly
round flat circle, then we say that loop is an unknot; otherwise, the loop is a
knot. Any deformation is legal as long as it doesn't force one part of the knot
through another (or pull the knot tight so it shrinks down to a single point). In
the classical view of the theory, our knotted loops can be thought of as being
made out of some sort of theoretical string which is infinitesimally thin and
arbitrarily flexible. In this talk, we shall look at some knots which are made out
of “real stuff”. In particular, we shall look at knots made out of
straight rigid “sticks” and flexible pivot joints, and at the
deformations which preserve this piecewise-linear structure. We will then examine
how the mathematics change when we add this new level of rigidity.
Wednesday, February 8, 2006
| Title |
A Hypergraph Method in Arithmetic and Extremal Combinatorics |
| Speaker |
Brendan Nagle
University of Nevada, Reno |
| Time |
3:30-4:30 p.m. |
| Place |
ENG 004 |
| Sponsor |
Discrete Mathematics Search Committee |
Abstract
A so-called Hypergraph Regularity Method was established by V.
Rödl, M. Schacht, J. Skokan and the speaker and, independently, by W. T.
Gowers. This method provides an essential extension to hypergraphs of
Szemerédi's Regularity Lemma for graphs, and yields alternative (and
quantitative) proofs of well-known density theorems of Szemerédi and of
Furstenberg and Katznelson. These proofs are, in fact, derived from the following
corollary of the hypergraph egularity method (formerly a conjecture of Erdős,
Frankl and Rödl (1986), now dubbed the ‘removal lemma’): Every
k-graph $\cH_n^{(k)}$ (on n
vertices) containing only o(nt) copies
of the clique K_t^{(k)} admits a
K_t^{(k)}-free subhypergraph on only o
(nk) fewer edges.
In this talk, we give an overview of the hypergraph regularity method and some
of its applications.
Friday, January 27, 2006
| Title |
The 2-category associated to surfaces embedded in 3-space |
| Speaker |
J. Scott Carter
University of South Alabama |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Masahiko Saito |
Abstract
A 2-category has objects, 1-morphisms, and 2-morphisms. Many of the 2-morphisms
function as if they were identities among morphisms. What if they are not
identities, but are invertible quantities that satisfy higher-order identities? In
this talk, I describe in huristic terms a 2-category whose objects are finite sets
of points on a line, whose morphisms are arcs in the plane, and whose 2-morphisms
interpolate between a pair of families of arcs. I will describe how to decompose
the topography of these things, and why they are categorically interesting.
The talk will be entirely self-contained, and no knowledge beyond calculus will
be assumed.
Thursday, January 26, 2006
| Title |
2-Groups: An introduction to higher-dimensional groups |
| Speaker |
Alissa Crans
Ohio State University/
Loyola Marymount University |
| Time |
2:00-3:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Mohamed Elhamdadi |
Abstract
Group theory plays a prominent role in many branches of science where
symmetries appear. In many contexts where we are tempted to use groups, however,
it is actually more natural to use a richer sort of structure, that of a
higher-dimensional group, or ‘2-group’. A 2-group blends together the notion of a group with that of a category.
Thus, in addition to group elements describing symmetries, a 2-group also has isomorphisms between these, describing symmetries between symmetries. This talk consists of an introduction to higher-dimensional group theory in which we will examine examples of 2-groups and address their contexts and motivation.