Mini Course / Dynamics Learning Seminar

from Tuesday
January 01, 2019 to Friday
May 31, 2019
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Wednesday
February 20, 2019

2:30 PM - 3:30 PM
Math Tower P-131
Byung-Geun Oh, Hanyang University
Combinatorial Gauss-Bonnet Theorem and its applications

In this talk we will start with the concept of combinatorial curvature on planar graphs. After brief explanation for some progress related to combinatorial curvature, the main topic of this talk will come in, the "combinatorial Gauss-Bonnet theorem". Definitely it is the combinatorial counterpart to Gauss-Bonnet theorem in differential geometry. We will especially focus on the Gauss-Bonnet formula involving boundary (left) turns, since we found at least two reasonable applications of it.

The first application is related to the He-Schramm conjecture [1] about types of disk circle packing, which was later proved by Repp [2]. During the talk a statement stronger than the He-Schramm conjecture(i.e., Repp's theorem) will be presented, and one will see that the stronger version can be proved in a simpler way.

The next application is about isoperimetric constants on planar graphs. Suppose a given planar graph has faces and vertices whose degrees are at least $p$ and $q$, respectively, where $p$ and $q$ are natural numbers such that $1/p + 1/q < 1/2$.Then it is natural to guess that the isoperimetric constant of this graph is at least that of the $(p,q)$-regular graph, the $q$-regular planar graph all of whose faces have the same degree $p$. This `guess' was in fact conjectured by Lawrencenko, Plummer, and Zha [3], for which we could give an affirmative answer using the combinatorial Gauss-Bonnet theorem. A sketch of the proof will be given if time allows.

[1] Z. He and O. Schramm, Hyperbolic and parabolic packings, Discrete Comput. Geom. 14 (1995), no. 2, 123-149.

[2] A. Repp, Bounded valence excess and the parabolicity of tilings, Discrete Comput. Geom. 26 (2001), no. 3, 321-351.

[3] S. Lawrencenko, M. Plummer, and X. Zha, Isoperimetric constants of infinite plane graphs, Discrete Comput. Geom. 28 (2002), no. 3, 313-330.


Wednesday
March 06, 2019

2:30 PM - 3:30 PM
Math Tower P-131
Jonguk Yang, University of Michigan
TBA

TBA


Wednesday
March 27, 2019

2:30 PM - 3:30 PM
Math Tower P-131
Jonathan Fraser, University of St Andrews
Dimensions of Kleinian limit sets

The dimension theory of geometrically finite Kleinian groups and their limit sets has a rich and interesting history, with the first calculation of Hausdorff dimension going back to seminal work of Patterson and Sullivan from the 1970s and 80s. There are many different (but related) notions of dimension but, nevertheless, many of the most popular coincide in this setting. In particular, the Hausdorff, box-counting, and packing dimensions of a Kleinian limit set are all given by the Poincare exponent of the group. I will discuss recent work concerning the Assouad dimension, which is not necessarily given by the Poincare exponent in the presence of parabolic points.


Wednesday
April 17, 2019

2:30 PM - 3:30 PM
Math Tower P-131
John Milnor, Stony Brook University
TBA

TBA


Show events for:
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