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WEEKLY CALENDAR |
| Mon 24 |
3:10 pm, room SC 1404. Graph Theory and Combinatorics Seminar.
Mark Ellingham.
Trades and flexible graph embeddings.
A cellular embedding of a graph in a surface may be regarded as a
double covering of its edges by closed walks. Given two distinct
embeddings of the graph, we must delete some set C1 of closed walks,
and add a second set C2, covering each edge the same number of times as
C1, to change one embedding into the other. Conversely, given two sets
C1 and C2 of closed walks covering each edge the same number of times,
we may ask whether they represent the difference between two surface
embeddings. We show that if they satisfy some obvious necessary
conditions, then they do. This means, for example, that all triple
system trades satisfying some obvious necessary conditions can be used
to construct flexible triangulations, triangulations having more than
one embedding in some surface.
4:10-5 pm, room 1432. All-Algebra Seminar. Mai Gehrke, New Mexico State University. Canonicity, relational semantics, and duality. Canonical extensions supply a means of completing algebras that have a bounded lattice reduct. We call such algebras lattice expansions (LEs) or lattices with additional operations. LEs play an important role in some parts of logic and computer science as semantics for various (mostly propositional non-classical) logics. Canonical extension is defined for all LEs and recent results guarantee canonicity (i.e preservation of equational properties by canonical extension) for a wide spectrum of LEs. The canonical extension of a lattice is essentially a complete-lattice-theoretic description of its representation via topological duality. Thus the theory of canonical extensions is closely related to this important topic for LEs. Also, whenever relational semantics make sense for a (propositional) logic, canonical extensions are complex algebras of such relational structures, and often logical completeness is proved using canonicity. In this talk we will give an overview of the theory of canonical extensions and sketch the relationship of this topic to the development of topological dualities and to proving completeness of relational semantics for propositional logics. We will illustrate these relationships with duality theory for algebras such as MV-algebras and with Sahlqvist theory for distributive modal logic. |
| Tue 25 |
2:30 pm, room 1425.
Graduate students' tea.
All math personnel are invited.
3:10-4 pm in SC-4309. Faculty Seminar for Graduate Students. Mike Neamtu. Mathematical Issues in Computer Aided Geometric Design. In this expository talk I will highlight several difficulties that mathematicians face in the area of geometric modeling of curves and surfaces. These include the topics of parametric interpolation, geometric continuity, constrained approximation, and the construction of bivariate splines, all which are of fundamental importance for geometric design in many industrial settings. 3-4:30 pm, room 1403. Noncommutative Geometry and Operator Algebras Seminar. Ed Saff, Vanderbilt University. Minimal Riesz Energy Points on Manifolds. We discuss asymptotics (as N goes to infinity) for minimal Riesz s-energy N-point configurations on the union of smooth manifolds. Recent work with Doug Hardin leads to some very general results in the case when s is greater than the Hausdorff dimension of the manifold. Motivation for the investigation is the question of distributing many points on a sphere and best-packing problems. |
| Wed 26 |
3:10 pm, room 1431. Analysis
and Biomathematics Seminar.
Andrea Bertozzi, of
Duke University.
New Challenges for Hydrodynamics: Microfluidics,
Imaging Science, and Mobile Sensors.
This talk will showcase three
new research areas involving mathematical fluid dynamics.
Microfluidics is a rapidly growing field being driven by new technological applications in the medical, materials, and chemical sciences. Surface tension effects (Marangoni stresses) are important on these scales. We consider the basic physics of surface tension gradients (used to move liquids) in conjunction with body forces on fluids and show that the ensuing dynamics can yield multiple shock structures involving undercompressive waves. In the field of imaging science, Image inpainting involves filling in part of an image or video using information from the surrounding area. We introduce a class of automated methods for digital inpainting using ideas from classical fluid dynamics. The main idea is to think of the image intensity as a 'stream function' for a two-dimensional incompressible flow. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results. An emerging area of mobile sensor control is the design of algorithms for multiple unmanned vehicles. Taking ideas from mathematical biology, we consider swarming algorithms for fluid-like motion based on simple rules for self-propulsion and local interaction. Applications range from mine detection algorithms to perimeter patrol and gradient searching. 4-5 pm, room 1431. Approximation Theory Seminar. Tanya Sorokina. A trivariate box macro-element. 4:35 pm, room 1403. Group Theory and Topology Seminar. Speaker? Topic? Abstract? (for more info contact Mark Sapir) --> Instructors please announce to your classes: 6-7 pm, room 1206. Undergraduate Seminar in Mathematics. (With free pizza!) David Petersen, graduate student. Do You Know Your Roots? Irrational, imaginary, complex... These words usually represent ideas that are hard (if not impossible) to wrap your mind around. In this week's talk, David Petersen will help us take a look at some of the ways we can understand these "impossible" ideas applied to numbers. |
| Thu 27 |
3:10 in room 1403.
Special Seminar in Graph Theory and Combinatorics.
Luis Goddyn,
Simon Fraser University.
Coloring/flow duality on wide embedded graphs.
We are concerned with a refinement of the graph chromatic number
called the circular chromatic number. The classical dual relation
between flows and colorings in plane graphs breaks down for graphs on
higher surfaces. Specifically, if X is a surface different from the
sphere, then the circular chromatic number of a graph G embedded on X
may be strictly greater than the circular flow number of the surface
dual map G*. We show that these two parameters are nearly equal provided
that X is orientable and G has large edge-width. (The edge-width is
the length of a shortest circuit in G which is not contractible on the
surface.) For non-orientable surfaces, the appropriate dual notion is
Bouchet's "biflow number".
As an application of this, we show that there are gaps in the range of possible circular chromatic numbers for certain generic classes of embedded graphs. For example a triangulation G of X has circular chromatic number either at least 4 or at most 3+epsilon where epsilon depends only on X and the edge-width of G. A similar consequence holds for even-faced embedded graphs. This is joint work with M. DeVos, B. Mohar, D. Vertigan and X. Zhu. 4:10 pm in 1431. Colloquium. Zhong-Jin Ruan, University of Illinois at Urbana-Champaign. Operator Spaces: A Natural Non-commutative Quantization of Functional Analysis. An operator space is a norm closed subspace of bounded operators on some Hilbert space together with a distinguished "matrix norm." Morphisms between operator spaces are "completely bounded linear maps." Operator space theory is a natural non-commutative quantization of functional analysis (Banach space theory). In this talk, I will first discuss some fundamental results in operator spaces, and then discuss some interesting applications to operator algebras and non-commutative harmonic analysis. (Tea at 3:30 pm in room 1425.) |
| Fri 28 |