Department of Mathematical
Sciences
Georgia
Southern University
Statesboro, GA 30458-8093
Spring 2005 Lectures
Summary
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| Friday, January 28 | Gary Gruenhage, Auburn University | Frechet-Urysohn for Finite Sets |
| Friday, February 4 | Frederic Mynard, Georgia Southern Univ. | Stability of local topological properties under product |
| Friday, February 11 (3:00 p.m. - 5:00 p.m.) | 1. Hakki Turgay Kaptanoglu, University of Virginia 2. Semra Ozturk Kaptanoglu, University of Virginia |
1. Function theory in diagonal Besov spaces
2. Betti numbers of fixed point sets |
| Thursday, February 17 (CLEC lecture, 7:00 p.m. - 8:00 p.m., Math/Physics 3001) | Ralph Kopperman, City College of CUNY | Not all points are created equal. |
| Friday, February 18 | Ralph Kopperman, City College of CUNY | Partial metrics |
| Friday, February 25 | Haomin Zhou, Georgia Institute of Technology | Variation and PDE techniques in wavelet inpainting |
| Friday, March 4 | ------------- | ------------ |
| Friday, March 11 | Peter Nyikos, Univ. of South Carolina | Elbow room in Banach spaces |
| Friday, March 25 | Renato D.C. Monteiro, School of ISyE, Georgia Institute of Technology | A Generic Iterative Solver-Based Infeasible Primal-Dual
Path-Following Algorithm for Convex Quadratic Programming |
| Friday, April 1 | Wen-Ran Zhang, Dept of Computer Science, Gerogia Southern Univ. | YinYang Bipolar Sets and Dynamic Modus Ponens for Open World Reasoning |
| Friday, April 8 | Len Olsen, Dept of Literature and Philosophy, Georgia Southern Univ. | A Formal Ontology for a Theory of Notation And Some Issues Concerning Identity |
| Friday, April 15 | Francois Ziegler, Georgia Southern Univ. | Cotangents of the Skies: an Introduction to Symplectic Geometry |
| Friday, April 22 | Yingkang Hu, Georgia Southern Univ. | The Return of FoThe Return of Fortran -- Fortran 90/95/2003 |
| 1. (2:30 - 3:30) Tom Kunkle, College of Charleston
(3:30 - 4:00 Coffee Break) 2. (4:00 - 5:00) Carl de Boor, Univ. of Wisconsin, Madison (Distinguished Lecture Series) |
1. Multivariate Interpolants with Bounded Derivatives 2. Ideal Interpolation | |
| Friday, April 29 | Krystyna Kuperberg , Auburn University | Wild arcs in dynamics |
| Thursday, May 5 (11:00 a.m.- 12: 00
noon), MP 3311 |
Jun Luo, Zhongshan Univ. (currently visiting Queens College, CUNY) | On boundary structure of plane tiles |
| Tuesday, June 28 (3:00 - 4:00 p.m.) | CNRS Marseille and Hebrew University of Jerusalem | Some Aspects of Diffeologies |
Details of Spring 2005 lectures
1. Friday, January 28, 2005, 3:00 p.m. - 4:00
p.m. Room: MP 3314 (Visiting
Lecture).
Speaker: Gary Gruenhage, Auburn University.
Title: Frechet-Urysohn for finite sets.
Abstract: A fundamental convergence property of topological spaces is the
following, called the Frechet-Urysohn property: whenever
a point p is in the
closure of a set A, there is a sequence of points of A converging to p. We
will discuss a natural generalization of this
property which in effect
replaces the points of A by finite sets. This property has important
connections to the analysis of the Frechet-Urysohn property in products, to
convergence in topological groups, and to certain topological games.
2. Friday, February 4, 2005, Room: MP
3314
Speaker: Frederic Mynard, Georgia Southern Univ.
Title: Stability of local topological properties under product.
Abstract: Local topological properties can be interpreted as properties of
neighborhood filters. A general scheme to study the stability under
product
of various properties of filters is presented, and applied to topological
product theorems.
3. Friday, February 11, 2005, 3:00 p.m. - 5:00 p.m. Room: MP
3314. (Visiting Lectures).
1. Speaker: Hakki Turgay Kaptanoglu, Univ. of Virginia.
Title: Function theory in diagonal Besov spaces.
Abstract: Diagonal Besov spaces Bpq of
analytic functions on the disc or the ball generalize the weighted Bergman
spaces
Aqp, q>-1, to all real values of the
parameter q. As usual, 0<p<=infinity. The
Bpq spaces include many other
function spaces as
special cases. We first discuss elementary properties of functions
in Bpq . Then we give a complete
description of bounded Bergman projections from the Lebesgue classes
onto Bpq and their right inverses. If
time remains,
we mention Carleson measures and a characterization of
Toeplitz operators on Bpq . The results are
natural extensions of
those known for weighted Bergman spaces.
2. Speaker: Semra Ozturk Kaptanoglu, Univ. of Virginia.
Title: Betti numbers of fixed point sets.
Abstract: Let G be a group and X be a space on which G
acts. For a subgroup L in G, the subspace XL is the
set of points in X fixed by the action of L. Capturing
information on XL from the equivariant cohomology of X is
a problem attacked by Borel-Atiyah-Quillen-Hsiang using localization. On the
other hand the G-action on X induces a kG-module structure on its cohomology
groups in each dimension where K is a field and kG is the group algebra. For
suitable G and X , we will show how to get information on the Betti numbers
of XL using the kL-module structure of M.
4. CLEC Lecture. Thursday, February 17, 2005, 7:00 p.m. - 8:00 p.m. Room: 3001 Math/Physics Building.
Speaker: Ralph Kopperman, City College of CUNY. (CLEC speaker).
Title: Not all points are created equal.
Abstract: The computer screen looks like a rectangle in the plane, but under
a magnifying glass it breaks up into an array of about a million
bright dots
(pixels). It could save memory space to store boundaries and colors of regions,
rather than the color of each pixel. This turns out
to be impossible to do
if the screen is thought of as a set of equal points, but possible and useful if
it is thought of as a space where some
points are more equal than others.
5. Friday, February 18, 2005, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Ralph Kopperman, City College of CUNY
Title: Partial metrics.
Abstract: Generally accepted metric axioms, such as d(x,y)=d(y,x) and
d(x,x)=0 seem "right", but there are
extremely natural situations in which
they fail. We discuss these failures, the theory that arises from them,
and
applications (in computing) of generalized metrics for which these axioms fail.
6. Friday, February 25, 2005, 3:00 p.m. - 4:00 p.m. Room: MP 3314 (Visiting Lecture).
Speaker: Haomin Zhou, School of Mathematics, Georgia Tech.
Title: Variation and PDE Techniques in Wavelet Inpainting.
Abstract: In this talk, I will present a recent work (collaborated with Tony
Chan (UCLA) and Jackie Shen (Minnesota)) on image inpainting in wavelet
domain. The problem is closely related to the classical image inpainting,
with the difference being that the inpainting regions that we are interested in
are in the wavelet domain, that brings new challenges to the
reconstructions, as there is no geometrically well defined inpainting
region in the pixel domain, and the damage is inhomogeneous. We propose
new variational models, especially total variation minimization in conjunction
with wavelets for the image inpainting problems in the wavelet domain. The
models lead to PDE's, which are Euler-Lagrange
equations of the variational
formulations, in the wavelet domain and can be solved numerically. The proposed
models can have effective and automatic control over geometric features of the
inpainted images including sharp edges, even in the presence of substantial
loss of wavelet coefficients, including in the low frequencies.
7. Friday, March 4 , 2005, Room: MP
3314 (Cancelled due to job
interview.)
8. Friday, March 11, 2005, 3:00 p.m. -
4:00 p.m., Room: MP 3314 (Visiting
Lecture)
Speaker: Peter Nyikos, Department of Mathematics, University of South Carolina
Title: Elbow room in Banach spaces
Abstract: Separable Banach spaces have the following unexpected property. If
a family of subsets is locally finite in the norm then
each member can be
expanded to a set that is open in the weak topology, in such a way that the
family of expansions is still locally
finite in the norm. This seems to be
new even for Hilbert space, where we can even go further and expand to a locally
finite collection of sets open in the natural product topology (thinking of
members of Hilbert space as functions from the natural numbers to the
reals). We can do this for all lp spaces
except loo , which is not a separable Banach space. In
fact, there is a countable closed discrete subset of loo which
cannot be expanded to a locally finite collection of weakly open sets. Other
examples, counterexamples and an application to Erdos space will be presented.
9. Friday, March 25, 2005, 3:00 p.m. -
4:00 p.m., Room: MP 3314 (Visiting
Lecture)
Speaker: Renato D.C. Monteiro, School of ISyE, Georgia Institute of Technology Tech
Title: A Generic Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex Quadratic Programming
Abstract: We develop an interior-point primal-dual long-step
path-following algorithm for convex quadratic programming (CQP) whose search
directions are computed by means of an iterative linear system solver. We
propose a new linear system, which we refer to as the augmented normal equation
(ANE), to determine the primal-dual search directions. Since the condition
number of the matrix associated with the ANE may become large for degenerate CQP
problems, we provide a family of preconditioners (e.g. partial-update, maximum
weight basis, etc.) to better condition this matrix. We establish a uniform
bound on the number of iterations required for the iterative solver to obtain a
sufficiently accurate solution to the ANE. Since the iterative solver can only
generate an approximate solution to the ANE, this solution does not yield a
primal-dual search direction satisfying all equations of the primal-dual Newton
system. We propose a unified approach to compute an inexact primal-dual search
direction so that the Newton equation corresponding to the primal residual is
satisfied exactly, while the one corresponding to the dual residual contains a
manageable error which allows us to establish a polynomial bound on the number
of outer iterations of our method.
10. Friday, April 1, 2005, 3:00 p.m. -
4:00 p.m., Room: MP 3314
Speaker: Wen-Ran Zhang, Dept of Computer Science, Gerogia Southern Univ.
Title: YinYang Bipolar Sets and Dynamic Modus Ponens for Open World Reasoning
Abstract: Yinyang bipolar sets, bipolar lattices, dynamic modus ponens, and
equilibrium relations are introduced which form a theory of bipolar machinery.
YinYang bipolar sets and modus ponens build a bridge from a linear, static, and
closed world to a non-linear, dynamic, and open world of equilibria or
non-equilibria, that provide an effective means for bipolar information fusion
and visualization. It is also shown that equilibrium relations as bipolar
generalization of equivalence relations induce hard partitions or bipolar sets.
Application examples are illustrated in cognitive mapping and bipolar diagnostic
analysis.
11. Friday, April 8, 2005, 3:00 p.m. -
4:00 p.m., Room: MP 3314
Speaker: Len Olsen, Dept of Literature and Philosophy, Georgia Southern Univ.
Title: A Formal Ontology for a Theory of Notation And Some Issues Concerning Identity
Abstract: The central issue of this presentation concerns the identity
conditions for entities. The theory of notation, like all other theories,
presupposes the existence of certain entities. A catalogue of these
entities (or types of entities) is called an ontology. My talk will focus
on the possibility of using mereology as the basis of a formal ontology for the
theory of notation. Mereology, otherwise known as the calculus of
individuals or the logic of parts and wholes, is offered as an alternative
to set theory.
12. Friday, April 15, 2005, 3:00 p.m. -
4:00 p.m., Room: MP 3314
Speaker: Francois Ziegler, Department of Mathematical Sciences, Georgia Southern Univ.
Title: Cotangents of the Skies: an Introduction to Symplectic Geometry.
Abstract: Symplectic geometry was born in the 19th century as the geometry of
classical mechanics, ray optics, and line complexes. Much later, in the
1960s, mathematicians began to realize that it is also the geometry of
quantum mechanics and representation theory. In this talk I will attempt
an
elementary tour of these ideas and techniques, using as a running example the
manifold of oriented straight lines in R3 (a.k.a. light rays).
13. Friday, April 22, 2005, 3:00 p.m. -
4:00 p.m., Room: MP 3314
Speaker: Yingkang Hu, Department of Mathematical Sciences, Georgia Southern Univ.
Title: The Return of Fortran -- Fortran 90/95/2003.
Abstract: Fortran, which first appeared in the mid 1950's, is one of the oldest high-level languages, and is a very successful one. Some say its structure is superior for vector computation. But it took a long break after its ANSI 1978 version (best known as Fortran 77), while new languages, such as Java, had emerged, and other old ones, such as C/C++, had evolved greatly. Many think Fortran is a dinosaur.
Well, old things die hard. Fortran came back in three new versions, Fortran 90/95, and just last November, Fortran 2003, with lots of new features. It is now the best language for scientific computation in my opinion. (Java and C/C++ are better for system and Internet programming.) Some of the new features are:
1. Contains Fortran 77;
2. Allows free format. Counting columns is no
longer needed;
3. Dynamic memory allocation. (Arrays can be allocated after
the program inputs their dimensions);
4. Recursive calling of subprograms;
5. High-level operations, such as matrix assignment/addition/multiplication,
dot product, norm, and max/min of vectors/matrices, are intrinsic now;
6. It
is modular;
7. Pointers installed;
8. Object-Oriented Programming is
supported by Fortran 2003.
Code for matrix/vector calculation can be shortened greatly, but still remains readable. This is one of the main reasons people love Matlab, it's no longer a monopoly.
(Visiting Lecture)
Speaker: Tom Kunkle, College of Charleston
Title: Multivariate Interpolants with Bounded Derivatives.
Abstract: Let n and d be natural numbers and consider the following problem. Construct a smooth interpolant F to function values f given at points m in Rd, where F depends locally and linearly on f, and F's derivatives of total degree n are bounded by a constant C times the corresponding divided differences of f. Here C may depend on d and n, but must be independent of f and m.
Favard [1] gave an optimal solution to the problem in case d=1 by bounding each of two consecutive derivatives by the corresponding divided differences and allowing m to be any discrete set in R. Such freedom would not be possible in the multivariate setting. In fact, if m is a tensor product grid in Rd, i.e., the Cartesian product of d increasing sequences of real numbers, then such an interpolation scheme exists if and only if m has uniform spacing in each of the coordinate directions.
When m is a subset of a tensor product grid with uniform spacing, the interpolation problem may or may not have a solution, depending on the geometry of the subset.
[1] Favard, J., Sur l'interpolation, J. Math. Pures Appl. \/ 19
(1940), 281--300.
Speaker: Carl de Boor, Univ. of Wisconsin, Madison
Title: Ideal Interpolation
Abstract: Starting from G. Birkhoff's definition of `ideal interpolation' (as
a linear projector whose kernel is a polynomial ideal), multivariate polynomial
interpolation is explored.
15. Friday, April 29, 2005, 3:00 p.m. - 4:00 p.m., Room: MP 3314 (Visiting Lecture)
Speaker: Krystyna Kuperberg, Auburn University
Title: Wild arcs in dynamics
Abstract: One of the most interesting aspects of dynamical systems are the topological invariants, the geometric side of the configuration of trajectories, and the nature of the individual trajectories.
A trajectory of a flow on a 3-manifold is wild if the closure of at least one of its two semi-trajectories is a wild arc. A trajectory is 2-wild if the closure of each its two semi-trajectories is a wild arc. For example, the closure of a 2-wild trajectory may be the wild arc described by Artin and Fox (see MR0027512). A 2-wild trajectory may be also homoclinic, with the limit sets equal and consisting of a fixed point.
We describe a method of embedding wild trajectories in flows on 3-manifolds
and obtain the following: every boundaryless 3-manifold admits a flow with
a discrete set of fixed points and such that every non-trivial trajectory is
2-wild. Every closed 3-manifold admits a flow with exactly one fixed point and
such that every non-trivial trajectory is homoclinic and 2-wild.
16. Thursday, May 4, 2005, (Visiting Lecture)
Speaker: Jun Luo, Zhongshan University, China (currently visiting Queens College, City Univ. of New York)
Title: On boundary structure of plane tiles
Abstract: We consider the boundary of a planar tile or components of
its interior. Particularly, we obtain local connectivity of the whole boundary
whenever the tile itself is connected and locally connected, as an
application of a new characterization of local connectivity. Also, we will
comment on a necessary and sufficient condition for interior
components of a tile to be bounded by a simple closed curve.
17. Tuesday, June 28, 2005, 3:00 p.m. - 4:00 p.m., Room: MP 3314
Speaker: Patrick Iglesias, CNRS Marseille and Hebrew University of Jerusalem
Title: Some Aspects of Diffeologies
Abstract: Diffeology is a differential geometric framework introduced to deal with differential objects for which topology is not of a big help. It is generally presented as an extension of the classical theory of manifolds, but it is much more than that. It is a completely new point of view on mathematical objects which support "differentiable parametrizations."
The diffeology category is extremely stable under standard operations of sums, products, restrictions and quotients, and its applications run from singular spaces to infinite dimensional spaces.
I will give a general presentation of the theory and try
to illustrate it through the famous examples of "irrational tori." An
irrational torus is defined as the quotient of an ordinary torus by a proper
dense subgroup. Irrational tori are topologically trivial, but
surprisingly rich from the diffeological viewpoint.
If you have any question regarding the colloquium, please
e-mail Dr.
Sze-Man Ngai: ngai@gsu.mat.GeorgiaSouthern.edu.
( Last updated: June 23, 2005.)