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Schedule and Abstracts
- 9:00 Continental Breakfast
- 9:55 Welcome!
- 10:00 Karen Collins: Graphs and Coloring
- Abstract: This talk will present a series of theorems
and related
open questions about graphs and different kinds of graph
colorings.
- 10:50 Break
- 11:10 Herb Wilf: Recent progress on Ramanujan's partition
congruences
- Abstract: Ramanujan conjectured and then proved that
the number of
partitions of an integer 5n+4 is always divisible by
5. We will describe and
prove some new refinements and extensions of those congruences,
including
recent results of Berkovich and Garvan, Chen, Ji and
myself, and Ken Ono.
- 12:00 Break for Lunch
- 2:00 Ellen Gethner: Unraveling the Chromatic Number
of Thickness-Two Graphs
- Abstract: A graph G is said to have thickness-t if
E(G) can be partitioned into t and no fewer planar graphs.
A longstanding open problem is the following: What is
the largest chromatic number of any thickness-two graph?
Here's what is known so far: The largest chromatic number
of any thickness-two graph is hmmm, where hmmm is one
of 9,10,11, or 12. The "9" is due to exactly one published
example of a 9-critical thickness-two graph. The "12"
is due to a straightforward argument that uses Euler's
Formula.
I will talk about a catalog of new small 9-critical
thickness-two graphs, and a construction that generates
infinitely many 9-critical thickness-two graphs, thus
providing ballast to the "9." In addition I will introduce
new families of thickness-two graphs, (among them, the
"permuted layer graphs") and talk about what is known
so far about these new class of graphs. Finally, I will
give an infinite family of non-trivial graphs for which
both the thickness and chromatic number are known. Many
open questions will be provided throughout the talk.
This is joint work with Debra Boutin (Hamilton College)
and Thom Sulanke (Indiana University).
- 2:50 Break
- 3:10 Bruce Richter: On the Cycle Spaces of an Infinite
Graph
- Abstract: Diestel and Kühn pioneered the study
of the cycle space of infinite graphs. A more general
point of view was taken by Vella and Richter who showed
that many of the earlier results hold for more general
spaces, thereby unifying the cycle spaces introduced
by Diestel and Kühn and by Bonnington and Richter.
In particular, different compactifications of locally
finite graphs yield spaces that have cycle spaces.
In this work, the Vella-Richter approach is pursued
to considering cycle spaces over all fields, not just
Z2. In order to understand "orthogonality"
relations, it is helpful to consider two different cycle
spaces and three different bond spaces. We give an analogue
of the "Edge-tripartition Theorem" of Rosenstiehl
and Read and show how the cycle spaces of different
compactifications of a locally finite graph are related.
This is joint work with Karel Casteels.
- 4:00 Break
- 4:20 Lauren Rose: Piecewise Polynomials with boundary
conditions
- Abstract: For a d-dimensional simplicial complex,
D, embedded in Rd, and a non-negative
integer r, we consider the module of piecewise polynomial
functions on D that are globally Cr. Although
these modules are well studied, there are open problems
even in dimension two. We give a survey of combinatorial,
geometric, and algebraic results, and then discuss what
happens when you add boundary conditions.
- 5:10 End of DMD
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