College of Liberal Arts and Sciences

# Department of Mathematics

## Algebra Seminar

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Title: Some recent results on arboreal Galois groups [POSTPONED]
Speaker: Robert Benedetto (Amherst College)
Time: Wednesday, January 17, 2018 at 11:15 am
Place: MONT 313Abstract: Let $K$ be a number field, let $f\in K(x)$ be a rational function of degree $d$, and let $a\in K$. The roots of $f^n(z)-a$ are the $n$-th preimages of $a$ under $f$, and they have the natural structure of a $d$-ary rooted tree $T$. There is a natural Galois action on the tree, inducing a representation of the absolute Galois group of $K$ in the automorphism group of $T$. In many cases, it is expected that the image of this arboreal Galois representation has finite index in the automorphism group, but in some cases, such as when f is postcritically finite (PCF), the image is known to have infinite index. In this talk, I'll describe some recent results on arboreal Galois groups of certain polynomials, in both the PCF and non-PCF cases.Comments: This talk has been canceled due to weather and will be rescheduled later.

Title: Schur Super Functors
Speaker: Jonathan Axtell (Sungkyunkwan University)
Time: Wednesday, January 24, 2018 at 11:15 am
Place: MONT 313Abstract: Schur functors provide a convenient way to construct modules of symmetric and general linear groups. Akin, Buchsbaum and Weyman constructed Schur functors over fields of positive characteristic using Hopf algebras of exterior and symmetric powers. We describe a "super-analogueâ of this construction using $mathbb â¤/2mathbb â¤$-graded versions of the above Hopf algebras. The resulting functors then provide representations of corresponding Schur superalgebras.

Title: TBA
Speaker: Christine McMeekin (Cornell University)
Time: Wednesday, February 7, 2018 at 11:15 am
Place: MONT 313Abstract: TBA

Title: Another look at Zagier's formula for multiple zeta values involving 2's and 3's
Speaker: Cezar Lupu (University of Pittsburgh)
Time: Wednesday, February 21, 2018 at 11:15 am
Place: MONT 313Abstract: In this talk, we shall discuss about Zagier's formula for the multiple zeta values, $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ and its connections to Brown's proofs of the conjecture on the Hoffman basis and the zig-zag conjecture of Broadhurst in quantum field theory. Zagier's formula is a remarkable example of both strength and the limits of the motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ as rational linear combinations of products $\zeta(m)\pi^{2n}$ with $m$ odd. The formula is proven indirectly by computing the generating functions of both sides in closed form and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality. By using the Taylor series of integer powers of arcsin function and a related result about expressing rational zeta series involving $\zeta(2n)$ as a finite sum of $\mathbb{Q}$-linear combinations of odd zeta values and powers of $\pi$, we derive a new and direct proof of Zagier's formula in the special case $\zeta(2, 2, \ldots, 2, 3)$. If time allows we discuss a Zagier type formula for special multiple Hurwitz zeta values.

Title: TBA
Speaker: Suh Hyun Choi (University of Connecticut)
Time: Wednesday, March 28, 2018 at 11:15 am
Place: MONT 313Abstract: TBA

Title: TBA
Speaker: Zheng Liu (IAS / McGill University)
Time: Wednesday, April 18, 2018 at 11:15 am
Place: MONT 313Abstract: TBA

Organizer: Liang Xiao