Department of

Mathematics


Seminar Calendar
for events the day of Thursday, October 30, 2014.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, October 30, 2014

11:00 am in 241 Altgeld Hall,Thursday, October 30, 2014

The zero distribution of polynomials with a three-term recurrence

Khang Tran   [email] (Truman State University)

Abstract: For any fixed positive integer $n$, we study the zero distribution of a sequence of polynomials $H_{m}(z)$ satisfying the rational generating function \[ \sum_{m=0}^{\infty}H_{m}(z)t^{m}=\frac{1}{1+B(z)t+A(z)t^{n}} \] where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients. We show that for all large $m$, the zeros of $H_{m}(z)$ which satisfy $A(z)\ne0$ lie on the fixed real algebraic curve given by \[ \Im\frac{B^{n}(z)}{A(z)}=0\qquad\mbox{and}\qquad0\le(-1)^{n}\Re\frac{B^{n}(z)}{A(z)}\le\frac{n^{n}}{(n-1)^{n-1}} \] and are dense there as $m\rightarrow\infty$.

12:30 pm in 464 Loomis Laboratory,Thursday, October 30, 2014

Some aspects of large-N vector models and their higher-spin holography

Anastasios Petkou (Aristotle University of Thessaloniki Physics)

1:00 pm in Altgeld Hall 243,Thursday, October 30, 2014

Totally Disconnected, Locally Compact Groups

George Willis (Newcastle)

Abstract: Locally compact groups in general and the structure of connected groups will be brie y surveyed in the rst part of the talk. The second part of the talk will review recent developments in the structure theory of totally disconnected, locally compact groups. There are three strands in this work: the scale function and related ideas; a theory of decomposition into simple pieces; and a local theory. These three strands promise to combine to produce a much richer understanding of totally disconnected groups than we have at present. View talk at http://youtu.be/Sr55vamxQdI

2:00 pm in 243 Altgeld Hall,Thursday, October 30, 2014

Two Fractal Metric Space Dimensions

Claudio DiMarco (Syracuse University)

Abstract: There are many ways to define dimension for metric spaces. Some dimensions measure size, others connectivity, and some consider both. Balka, Buczolich and Elekes (2011) modified the notion of Hausdorff dimension to include topological considerations (such as connectivity). We use that same strategy to modify the notion of conformal dimension. For a metric space X, this usually amounts to considering a basis for the topology on X, then determining the conformal dimension of the boundaries of the basis elements.

3:00 pm in 243 Altgeld Hall,Thursday, October 30, 2014

On the Edge Homomorphism of a Spectral Sequence

Sankar Dutta (UIUC Math)

Abstract: The purpose of these talks is to present a connection between the non-vanishing of a specific edge homomorphism of a spectral sequence originating from the associativity property of Hom and Tensor product and several homological conjectures. Ramifications of this observation will be discussed.

4:00 pm in 245 Altgeld Hall,Thursday, October 30, 2014

Drawings of complete graphs

Gelasio Salazar (Instituto de Fisica, Universidad Autónoma de San Luis Potosí, Mexico)

Abstract: Using the Jordan Curve Theorem, it is an easy exercise to show that if $n\ge 5$, then the complete graph $K_n$ on $n$ vertices cannot be drawn in the plane without edge crossings. It is natural to ask: what is the minimum number of crossings of edges in a drawing of $K_n$ in the plane? Using graph theory terminology, this question reads: what is the "crossing number" cr$(K_n)$ of $K_n$? In the late 1950s, the British artist Anthony Hill developed an interest in drawing $K_n$ with as few edge crossings as possible. He eventually came up with a general construction to draw $K_n$ with exactly $H(n):=(1/4)\lfloor{n/2}\rfloor \lfloor{(n-1)/2}\rfloor \lfloor{(n-2)/2}\rfloor \lfloor{(n-3)/2}\rfloor $ crossings. No drawings with fewer crossings have ever been found, and the widely believed Hill's Conjecture $cr(K_n) = H(n)$ has become one of the most important open problems in Topological Graph Theory. An important variant of this version is the {\em rectilinear crossing number}, in which we require that the edges be drawn as straight segments. In this talk we will review the history of these problems, and give some unexpected and important connections to problems in geometric probability and convex geometry. Finally, we will survey the state-of-the-art of these open problems, including some recent progress obtained for the crossing number of $K_n$ borrowing from classical techniques in discrete geometry.