ABSTRACT: This is a continuation of last week's presentation by Angela Harris.

Hilbert's Nullstellensatz states that if F is an algebraically closed field and f, g_1, ... g_m are polynomials in the ring of polynomials in n variables over F, where f vanishes over all common zeros of the g_i's, then there is an interger k and polynomials, h_1, ..., h_m in the ring so that f^k is equal to the sum of the h_ig_i. We will prove a stronger result for the case m=n and each g_i univariate. We will then give applications of the result in several areas, including Combinatorial Number Theory, Graph Theory and Combinatorics.

ABSTRACT: Hilbert's Nullstellensatz states that if F is an algebraically closed field and f, g_1, ... g_m are polynomials in the ring of polynomials in n variables over F, where f vanishes over all common zeros of the g_i's, then there is an interger k and polynomials, h_1, ..., h_m in the ring so that f^k is equal to the sum of the h_ig_i. In the next two seminars we will prove a stronger result for the case m=n and each g_i univariate. We will then give applications of the result in several areas, including Combinatorial Number Theory, Graph Theory and Combinatorics.

ABSTRACT: Abstract:The classification of the finite simple groups is unprecedented in the history of mathematics, for its proof is 15,000 pages long, representing the combined effort of more than 100 mathematicians in over 500 articles published between the late 1940's and the early 1980's. While common knowledge has it that the proof was completed in the early 80's, several gaps were found, one of them serious, and only recently in an article in the AMS Notices has Michael Aschbacher stated that he once again believes that it is a theorem. In this talk we will mention briefly the ultimate result, but will focus more on the long history leading up to the final classification. We will start with Galois in the 1830's and his use of a simple group to prove the insolvability of the quintic, and then discuss the work of Sylow and Burnside. The range problem began around the turn of the century, and we will trace it to the 70's and Marshall Hall's paper on the simple groups of order less than 1,000,000. We will discuss Brauer's pioneering work in the 50's followed by Thompson's odd order and N-groups papers. And of course we will discuss Janko's remarkable discovery of a new sporadic simple group in 1966, the first in 100 years, and an event that sparked an exciting 10 year period when some 21 new sporadic simple groups were discovered. And finally we will look at the work of Daniel Gorenstein, the commander-in-chief or quarterback of the final assault on the problem. I was fortunate to be one of those 100 contributers during perhaps the most exciting period, 1967-1975, and was also fortunate to be a student of Janko. So I will also share some experiences from those years that did not make it into the journals but shed some light on some of the leading characters in the final proof. For the most part, this will not be too technical a talk, so it should be of interest to a diverse mathematic's audience.

ABSTRACT: A strongly regular graph with parameters (v,k, lambda, mu) is a k-regular graph on v vertices with the property that every pair of adjacent vertices have lambda common neighbors, and every pair of nonadjacent vertices have mu common neighbors. A partial difference set is a subset of a group, the existence of which is equivalent to the existence of a stgrongly regular Cayley graph on that group. We will review some basic results and constructions, concentrating on those which arise in Galois Geometries. Our main result is a proof of the nonexistence of abelian partial difference sets with parameters (p^2(p+2)^2, p(p+1)^2, p, p^2+p), where p and p+2 are both prime. Along the way, we prove a curious lemma on matchings in bipartite graphs.

ABSTRACT: You teach calculus and you want to create an exam question with a cubic polynomial having integer coefficients, distinct integral roots, and integral critical values. Can you do it?

Try it now!

If you want to see a complete determination of all such polynomials, come to the Discrete Math Seminar.

ABSTRACT: Abstract: Domination graphs arose from the study of tournaments. Two players u and v are said to dominate in a tournament if for every other player w, either u beats w or v beats w. In general, the digraph does not have to be a tournament. Vertices u and v dominate if for every other vertex w, (u,w) or (v,w) is an arc in D. Given a digraph D=(V,A), the domination graph of D, dom(D), is the graph with vertex set V and an edge between every pair of dominating vertices.

A large cadre of researchers in this area has passed through the halls of CU-Denver. In this pass, we will explore digraphs with isomorphic underlying graphs, UG(D), and domination graphs. The case where the two are equal has been completely characterized. Classes of digraphs where UG(D) is isomorphic but not equal to dom(D) have been found, although this portion of the problem has not, as yet, been fully characterized. This talk is based upon work done with Larry J. Langley of University of the Pacific.

Spectators are encouraged to bring writing utensils. There may be an opportunity to participate.

ABSTRACT: This is the second of a two-part presentation of several topics from finite geometry (aka Galois Geometry). The goal is to introduce ideas and techniques used to prove the following theorem: Any symplectic spread of PG(3,2^e) that contains even one regulus must in fact be a regular spread. These two talks could be considered an introduction to finite geometry. Lecture notes will be provided.

ABSTRACT: This is the first of a two-part presentation of an introduction to several basic topics in finite geometry (aka Galois geometry). Our goal is to proceed efficiently through the basic definitions and some of the proofs of pertinent theorems to reach (by the end of the second talk) a proof of the following theorem: If S is a symplectic spread of PG(3,2^e) that contains even one regulus, then S must be regular spread. This is a significant result in finite geometry with several applications. A basic understanding of vector spaces over a field along with a willingness to imagine that the field could be finite are the only prerequisites for these two lectures. Lecture notes will be available.

ABSTRACT: NOTICE TO ALL GRADUATE STUDENTS: The two Mondays Sept. 25 and Oct. 2 will be used to introduce graduate students to the research areas of the faculty members in discrete mathematics. On each of the two Mondays, each of our three speakers will give a fifteen minute introduction to his area of research. Please come and see what your professors are working on.

ABSTRACT: NOTICE TO ALL GRADUATE STUDENTS: The two Mondays Sept. 25 and Oct. 2 will be used to introduce graduate students to the research areas of the faculty members in discrete mathematics. On each of the two Mondays, each of our three speakers will give a fifteen minute introduction to his area of research. Please come and see what your professors are working on.

ABSTRACT: Reminder: John Schmitt from Middlebury College will speak at 1:00 p.m. on Thursday, 7 Sept. 06.

NOTE: The Discrete Math Seminar will not meet on September 11, 2006.

NOTE: On Sept. 18 Oscar Vega, Rich Lundgren and Mike Jacobson will each give 15 minute talks about their own research.

ABSTRACT: A graph G is said to be F-saturated if G does not contain a copy of F and for any edge e in the complement the graph G+e does contain a copy of F. The minimumsize of n-vertex F-saturated graphs, sat(n, F), is investigated. We give a history of the problem, present recent results (for when F is a cycle or a complete bipartite graph) and discuss many open problems.

Note: Because Monday, 4 September is a holiday, the Discrete Math Seminar will not be held that day. However, we have a visitor, Dr. John Schmitt, who will speak on Thursday during the usual "Proofs from the Book" seminar. Please note the change in time and place for the Discrete Math Seminar this week. ~

ABSTRACT: A graph G is said to have thickness - t if E(G)can be partitioned into t and no fewer planar graphs. So for example, if G is planar, then G has thickness-one. For another example and with the help of Kuratowski's Theorem, it is easy to see that K_5 has thickness at least two. Exercise: Show that K_5 has thickness exactly two.

A longstanding open problem is the following: What is the largest chromatic number of any thickness-two graph?

Here is 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 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 new construction that produces infinitely many 9-critical thickness-two graphs, thus providing ballast to the "9." In addition I will introduce a new family of thickness-two graphs, the "permuted layer graphs" and talk about what is known so far abouot this new class of graphs. And 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 at the end of the talk.