Department of

Mathematics


Seminar Calendar
for events the day of Thursday, April 9, 2015.

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Thursday, April 9, 2015

11:00 am in 241 Altgeld Hall,Thursday, April 9, 2015

An effective asymptotic result for the Lebesgue measure of the sum-level sets for continued fractions

Byron Heersink (UIUC Math)

Abstract: For every positive integer $n$, let $C_n$ be the set of real numbers in $[0,1]$ whose continued fraction expansion $[a_1,a_2,...]$ satisfies $a_1+...+a_k = n$ for some $k$. Using results from infinite ergodic theory, Kessebohmer and Stratmann proved that the Lebesgue measure of $C_n$ is asymptotically equivalent to $1/\log_2 n$ as $n\rightarrow\infty$. In this talk, we provide a simplified proof of this result, mostly using basic properties of the transfer operator of the Farey map and Karamata's Tauberian theorem, while avoiding most of the ergodic results in the proof of Kessebohmer and Stratmann. Additionally, we obtain an error term by adapting Freud's effective version of Karamata's theorem to this situation.

1:00 pm in Altgeld Hall 243,Thursday, April 9, 2015

Clairaut's theorem and potential mechanics on metric spaces

Richard Bishop (UIUC Math)

Abstract: For surfaces of revolution Clairaut's theorem gives a first integral for geodesics: $r \cos\theta =$ constant, where $r$ is the distance from the axis to the profile curve and $\theta$ is the angle the geodesic makes with the latitude circles. We have generalized this to warped products $W = B\times_fF$ of metric spaces: along any geodesic $\gamma$ in $W$, $f^2v = b$ is constant, where $v$ is the speed of the projection of $\gamma$ to $F$. When $B, F$ are Riemannian manifolds, the geodesic equations have a known form: $$ \gamma_B'' = c(1/f^3) {\rm grad} f, \qquad (f^2v)' =0,$$ where $\gamma_B$ is the projection to $B$. This has the interpretation that $\gamma_B$ is a trajectory of the potential function $U = c/2f^2$. The fact that the speed of $\gamma$ is a constant $a = \sqrt{b/c}$ becomes the law of conservation of energy $u^2 + 2U = (b/c)^2$, where $u$ is the speed of $\gamma_B$. Hence, for more general metric spaces $B$, Clairaut's theorem makes it reasonable to interpret the projections of geodesics from a warped product $B\times_fF$ to $B$ as the trajectories of the potential function $U = 1/2f^2 :B \to {\bf R}$. Since we also have shown that these trajectories are independent of the choice of $F$, we can simply take $F$ to be the line or a circle. Joint work with Stephanie Alexander

1:00 pm in Altgeld Hall,Thursday, April 9, 2015

Mechanisms of ecological success in ants

Andy Suarez (Animal Biology, UIUC)

Abstract: I study the ecology, evolution and behavior of ants and I am interested in the ways that ants solve problems. My research capitalizes on the developmental and ecological flexibility of ants to investigate how polymorphism and specialization within complex societies contribute to their ecological success. Recent research in my lab also includes using trap-jaw ants as a model system for understanding biomechanical and kinematic principles in animals.

2:00 pm in 140 Henry Administration Bldg,Thursday, April 9, 2015

Parabolas infiltrating the Ford circles

Suzanne Hutchinson (UIUC Math)

Abstract: We define and study a new family of parabolas in connection with Ford circles and discover some interesting properties that they have. Using these properties, we are able to prove some further applications of such a family.

3:00 pm in 243 Altgeld Hall,Thursday, April 9, 2015

D-modules - Part 2

Yi Zhang (UIUC Math)

Abstract: We continue our introduction of D-modules and discuss the length of D-modules.

4:00 pm in 245 Altgeld Hall,Thursday, April 9, 2015

Divisors on curves: from classical to modern geometry

Rahul Pandharipande (ETH Zurich)

Abstract: Every meromorphic function F on a Riemann surface C has a divisor Div(F) of zeros and poles. When is a divisor on C obtained from a meromorphic function? This question underlies much of the classical study of curves. A modern turn in the subject comes by viewing the answer as a cycle over the moduli space of curves. In 2014, Pixton proposed a complete formula for the class of the associated cycle. I will give an overview of the subject and a sketch of the very recent proof of Pixton's formula (joint work with Janda, Pixton, and Zvonkine).