Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, January 17, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 17, 2019

#### What is Carmichael's totient conjecture?

###### Kevin Ford (Illinois Math)

Abstract: A recent DriveTime commercial features a mathematician at a blackboard supposedly solving "Carmichael's totient conjecture". This is a real problem concerning Euler's $\phi$-function, and remains unsolved, despite the claim made in the ad. We will describe the history of the conjecture and what has been done to try to solve it.

12:00 pm in 243 Altgeld Hall,Thursday, January 17, 2019

#### Index properties of random automorphisms of free groups.

###### Ilya Kapovich (Hunter College)

Abstract: For automorphisms of the free group $F_r$, being "fully irreducible" is the main analog of the property of being a pseudo-Anosov element of the mapping class group. It has been known, because of general results about random walks on groups acting on Gromov-hyperbolic spaces, that a "random" (in the sense of being generated by a long random walk) element $\phi$ of $Out(F_r)$ is fully irreducible and atoroidal. But finer structural properties of such random fully irreducibles $\phi\in Out(F_r)$ have not been understood. We prove that for a "random" $\phi\in Out(F_r)$ (where $r\ge 3$), the attracting and repelling $\mathbb R$-trees of $\phi$ are trivalent, that is all of their branch points have valency three, and that these trees are non-geometric (and thus have index $<2r-2$). The talk is based on a joint paper with Joseph Maher, Samuel Taylor and Catherine Pfaff.