Department of

Mathematics


Seminar Calendar
for events the day of Thursday, October 11, 2018.

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Thursday, October 11, 2018

11:00 am in 241 Altgeld Hall,Thursday, October 11, 2018

Higher-dimensional frontiers in continued fractions

Joseph Vandehey (Ohio State Math)

Abstract: Abstract: We will discuss results from a long-running project between myself and Anton Lukyanenko to understand the connections between continued fractions and hyperbolic geometry in higher dimensions. Our recent breakthrough has allowed us to consider the connections between complex continued fractions and three-dimensional real hyperbolic space, quaternionic continued fractions and five-dimensional real hyperbolic space, octonionic continued fractions and nine-dimensional real hyperbolic space, and also Heisenberg continued fractions and two dimensional complex hyperbolic space. We will discuss the implications of these connections on a variety of number theoretic problems, including rational approximation and the study of algebraic irrationals, and many new problems these results have led us to. This talk will have connections to dynamical systems and hyperbolic geometry.

2:00 pm in 241 Altgeld Hall,Thursday, October 11, 2018

Combinatorial methods for ergodic proofs

Joseph Vandehey (Ohio State Math)

Abstract: Normal numbers are numbers whose digits display certain typical statistical properties. One early result about normal numbers says that if 0.a_1a_2a_3... is normal, then so is 0.a_ka_{k+\ell}a_{k+2\ell}.... That is, selection along arithmetic progressions preserves normality. By applying deep tools from ergodic theory, Kamae and Weiss have shown that the only sequences along which selection preserves normality are those of low complexity. We will show that part of this result may be proved using combinatorics and analyze these types of problems more broadly.

3:00 pm in 345 Altgeld Hall,Thursday, October 11, 2018

Schubert polynomials and flow polytopes

Avery St. Dizier   [email] (Cornell University)

Abstract: The flow polytope associated to an acyclic graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. We will discuss a family of subdivisions of certain flow polytopes and an invariant of these different subdivisions. We will then explain how this invariant leads to certain Schubert and Grothendieck polynomials. This connection implies interesting results about the Newton polytopes of these polynomials, some of which are known to hold generally. This is joint work with K. Meszaros. There will be a pre-talk on the same day at 12:30 PM in 441 Altgeld Hall.