Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 14, 2006.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2006            March 2006             April 2006
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4             1  2  3  4                      1
5  6  7  8  9 10 11    5  6  7  8  9 10 11    2  3  4  5  6  7  8
12 13 14 15 16 17 18   12 13 14 15 16 17 18    9 10 11 12 13 14 15
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30


Tuesday, March 14, 2006

1:00 pm in 345 Altgeld Hall,Tuesday, March 14, 2006

#### Small profinite structures and their generalizations

###### Krzysztof Krupinski (UIUC Math)

Abstract: A profinite structure in the sense of Newelski is a pair $(X,Aut^*(X))$ consisting of a profinite topological space $X$ and a closed subgroup $Aut^*(X)$ (called the structural group) of the group of all homeomorphisms of $X$ respecting the inverse system defining $X$. We say that a profinite structure $(X,Aut^*(X))$ is small if for every natural number $n>0$, there are only countably many orbits on $X^n$ under the action of the structural group. In small profinite structures Newelski introduced a topological notion of independence, which has similar properties to those of forking independence in stable theories, and developed a counterpart of geometric stability theory in this context. I will present this notion of independence and explain why smallness plays an important role here. I will also give some examples of small profinite groups regarded as profinite structures. Then I will talk about my recent ideas concerning generalizations of small profinite structures to the case of: 1) non-small profinite structures; 2) 'compact structures' (i.e. $X$ is a compact metric space and $Aut^*(X)$ is a compact group acting on $X$ continuously); 3) 'Polish structures' (i.e. $X$ is a Polish space and $Aut^*(X)$ is a Polish group acting on $X$ continuously).

1:00 pm in 347 Altgeld Hall,Tuesday, March 14, 2006

#### Polya's Conjecture

###### Richard S. Laugesen   [email] (UIUC Math)

Abstract: We will discuss Polya's conjecture that Weyl's asymptotic estimate is in fact a rigorous upper bound on the eigenvalue counting function, at all finite energies, for the Laplacian with Dirichlet boundary condtions.

1:00 pm in 241 Altgeld Hall,Tuesday, March 14, 2006

#### Asymptotic behavior of the irrational factor

###### Andrew Ledoan (UIUC)

Abstract: In this talk, I will discuss my recent joint work with Professors Alkan and Zaharescu concerning the asymptotic behavior of a new arithmetical function, called the irrational factor function.

2:00 pm in Quad,Tuesday, March 14, 2006

#### Pi Day

Abstract: From 2:00-5:00 p.m. will sale of baked goods and games (including a pi recitation contest, a pie walk, and tattoo table!) At 3:14 p.m., the pie-eating contests begin. Weather permitting, all of these activites will be held on the north end of the quad. If the weather is bad, join us in room 314 Altgeld from 4:00-6:00 p.m.

2:00 pm in 243 Altgeld Hall,Tuesday, March 14, 2006

#### Empty Ellipse Graphs

###### Prof. Jeff Erickson (UIUC Department of Computer Science)

Abstract: Two points in a planar point set P are neighbors in the empty ellipse graph of P if they lie on an axis-aligned ellipse with no point of P in its interior. Empty-ellipse graphs are clearly generalizations of Delaunay (empty-circle) graphs in the plane, but they also capture the local behavior of Delaunay triangulations of dense point sets on smooth surfaces in 3-space.

In the worst case, the empty-ellipse graph can have an edge between every pair of points, but it is usually much sparser. Specifically, if R is the ratio between the largest and smallest pairwise distances in the point set, then the empty-ellipse graph has only O(Rn) edges. Moreover, if the points generated uniformly at random in an axis-aligned rectangle, the empty-ellipse graph has complexity O(n log n) with high probability. Both of these upper bounds are tight up to constant factors.

3:00 pm in 241 Altgeld Hall,Tuesday, March 14, 2006

#### Roman domination in graphs

###### Noah Prince (UIUC Math)

Abstract: A Roman dominating function of a graph G is a vertex weighting f from {0,1,2} such that every vertex assigned 0 has a neighbor assigned 2. The weight of f is the sum over V(G) of the weights on the vertices. The Roman domination number of G, written \gammaR(G), is the minimum weight of a Roman dominating function of G.

In this talk, we prove sharp upper bounds on \gammaR(G) when G is a connected n-vertex graph with minimum degree k, where k\in{1,2}. We also find sharp Nordhaus-Gaddum type bounds. This is joint work with Erin Chambers, Bill Kinnersley, and Douglas West.

3:00 pm in 443 Altgeld Hall,Tuesday, March 14, 2006

#### No meeting this week

Abstract: David Ross will continue March 28 (after spring break) on Young measures.

4:00 pm in Room M 234 Parkland College,Tuesday, March 14, 2006

#### The Leaving Exam System in Ireland and Australia with comparisons to US college entrance and achievement testing systems in the US.

###### Tony Peressini (UIUC Math) and Kathleen Harness (Champaign Schools)

Abstract: While on a visit to Ireland, Tony had a close look at the Leaving Exam system and tutored three Irish high school students who were preparing for the mathematics exams. Kathleen took the Leaving Exams in Australia in preparation for entering a university there. They will discuss the Leaving Exam System from these two diferent perspectives. They will also facilitate a discussion with other seminar participants of comparisons between the Leaving Exam System and US systems of college entrance and achievement exams in mathematics.