UIUC Dept. of Mathematics Seminar Calendar

     January 2012          February 2012            March 2012     
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Thursday, February 9, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, February 9, 2012

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Submitted by ford.

Jimmy Tseng (UIUC Math)
Bounded Luroth expansions
Abstract: Luroth series expansions are in a family of various expansions of the real numbers, a family which includes continued fractions. Like for continued fractions, every real number can be expressed as a Luroth expansion. Also like for continued fractions, the digits of a Luroth expansion are generated by a self-map, the Luroth map. The digits are, of course, an encoding of the map and give us a geometric way of looking at the expansion. I will give a sketch of the proof of the following result (joint with B. Mance): the set B of numbers with bounded Luroth expansion, bounded continued fraction expansion, and bounded n-ary expansion for every integer n > 1 is a dense set of full Hausdorff dimension. (Each of these conditions on B would form a superset of zero Lebesgue measure.) The proof is based on applying a technique developed by W. Schmidt (or a later variant made by C. McMullen) to the Luroth map,which has infinite distortion, and is adapted from techniques that I developed in 2009 for cases of bounded distortion.

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, February 9, 2012

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Submitted by kapovich.

Catherine Pfaff (Rutgers - Newark)
Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups
Abstract: The main theorem of my thesis emulates, in the context of $Out(F_r)$ theory, a mapping class group theorem (by H. Masur and J. Smillie) that determines precisely which index lists arise from pseudo-Anosov mapping classes. Since the ideal Whitehead graph gives a finer invariant in the analogous setting of a fully irreducible $\phi \in Out(F_r)$, we instead focus on determining which of the 21 connected 5-vertex graphs are ideal Whitehead graphs of ageometric, fully irreducible $\phi \in Out(F_3)$. Our main theorem accomplishes this. The methods we use for constructing fully irreducible $\phi\in Out(F_r)$, as well as our identification and decomposition techniques, can be used to extend our main theorem, as they are valid in any rank. Our methods of proof rely primarily on Bestvina-Feighn-Handel train track theory and the theory of attracting laminations.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, February 9, 2012

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Kevin Wildrick (University of Bern)
Lipschitz constants and differentiability almost everywhere
Abstract: Rademacher's theorem that Lipschitz functions are differentiable almost everywhere forms the backbone of results in function theory, geometric measure theory, and geometric topology. A prime example is Cheeger's theorem regarding the existence of differentiable structures on metric spaces supporting a Poincaré inequality. We will review some classical results and then discuss a version of Rademacher's theorem for the "lower" Lipschitz constant, which detects oscillation only on some sequence of scales tending to zero, rather than on all sequences of scales tending to zero. We also provide an example showing the sharpness of the results and the relationship of differentiability to the capacity of points.

Graduate Geometry and Topology Seminar
2:00 pm in 241 Altgeld Hall, Thursday, February 9, 2012

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Submitted by lukyane2.

Michael DiPasquale (UIUC Math)
Resolutions and Geometry
Abstract: Given a variety X inside of some projective space, one of the primary ways in which algebraic geometers study X is through its homogeneous coordinate ring S_X. One way to unpack the information hiding mysteriously inside of S_X is to study the free resolution of S_X. From this resolution come many fantastic invariants of X, primarily the betti diagram of X from which one can compute the Hilbert function of X and the regularity of X. Many difficult open conjectures for curves relate to the configuration of the betti diagram of the curve. Our modest goal is to see how such a seemingly arcane algebraic object as a resolution can actually reflect the geometry of a variety, primarily by looking at the case where X is a bunch of points in projective space. Interested folks may find David Eisenbud's book Geometry of Syzygies a good read; that is where much of my material will come from.