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 Wyman constructed Schur functors over fields of positive characteristic using Hopf algebras of exterior and symmetric powers. We describe a super-analogue of this construction using $\mathbb Z / 2\mathbb Z$-graded versions of the above Hopf algebras. The resulting functors then provide representations of corresponding Schur superalgebras.

Title: On the Distribution of Splitting Behavior in Number Fields Depending on $p$
Speaker: Christine McMeekin (Cornell University)
Time: Wednesday, February 7, 2018 at 11:15 am
Place: MONT 313Abstract: I will discuss a new formula for the density of primes which exhibit a given splitting behavior in certain number fields depending on the prime in question. The dependence of the number field on the prime forces ramification in every case, distinguishing this type of question from the classical results of Dirichlet and Chebotarev on the density of splitting behavior of primes in \textit{fixed} number fields.

Title: Hyperelliptic curves and symplectic Lie algebras
Speaker: Kyu-Hwan Lee (University of Connecticut)
Time: Wednesday, February 14, 2018 at 11:15 am
Place: MONT 313Abstract: In this talk, I will explain a (conjectural) connection between statistics of hyperelliptic curves and weight multiplicities of symplectic Lie algebras.

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: Some recent results on arboreal Galois groups
Speaker: Robert Benedetto (Amherst College)
Time: Wednesday, February 28, 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.

Title: The origin of pictures [POSTPONED]
Speaker: Kiyoshi Igusa (Brandeis University)
Time: Wednesday, March 7, 2018 at 11:15 am
Place: MONT 313Abstract: Gordana Todorov, Jerzy Weyman, Kent Orr and I are working on a book about pictures'' which have gained renewed attention because they are equivalent to scattering diagrams''. This talk is about very old work that I did to introduce pictures and their relation to the cohomology of $\mathrm{GL}(n,\mathbb Z)$. In particular, I will discuss the pictures below (from our book!) and their relationship to $H^3(\mathrm{GL}(n,\mathbb Z),\mathbb Z/2)$. The algebraic side of this story is a fun topic. The cohomology class which detects the exotic element'' of $K_3(\mathbb Z)$ is the degree 3 class which counts the number of times (modulo 2) that commutativity of addition is used to prove that matrix multiplication is associative! This is an old result which I am happy to present in a new light.

Title: The origin of pictures
Speaker: Kiyoshi Igusa (Brandeis University)
Time: Wednesday, March 21, 2018 at 11:15 am
Place: MONT 313Abstract: Gordana Todorov, Jerzy Weyman, Kent Orr and I are working on a book about pictures'' which have gained renewed attention because they are equivalent to scattering diagrams''. This talk is about very old work that I did to introduce pictures and their relation to the cohomology of $\mathrm{GL}(n,\mathbb Z)$. In particular, I will discuss the pictures below (from our book!) and their relationship to $H^3(\mathrm{GL}(n,\mathbb Z),\mathbb Z/2)$. The algebraic side of this story is a fun topic. The cohomology class which detects the exotic element'' of $K_3(\mathbb Z)$ is the degree 3 class which counts the number of times (modulo 2) that commutativity of addition is used to prove that matrix multiplication is associative! This is an old result (about the obstruction to right distributivity in left near-rings) which I am happy to present in a new light.

Title: Finiteness of certain crystalline Galois representations
Speaker: Suh Hyun Choi (University of Connecticut)
Time: Wednesday, March 28, 2018 at 11:15 am
Place: MONT 313Abstract: In 1993, Fontaine and Mazur suggested several conjectures on properties of geometric Galois representations, and one of the conjecture is about the finiteness of geometric Galois representations when the Hodge-Tate weight and level of the representation is fixed. In this talk, we introduce the background and some history of the conjecture and state our result for dimension 2 crystalline Galois representations for absolute Galois group of totally real fields. This is a joint work with Dohoon Choi.

Title: Picture Groups and their Homology
Speaker: Gordana Todorov (Northeastern University)
Time: Wednesday, April 4, 2018 at 11:15 am
Place: MONT 313Abstract: To each Dynkin quiver, using domains of semi-invariants, we associate 'spherical semi-invariant picture' $L(Q)$. To such a picture $L(Q)$ we associate the 'picture group' $G(Q)$. The picture group has the same generators as the unipotent group $U(Q)$ associated to the same quiver, however it has fewer relations. In order to compute the homology of the picture group $G(Q)$, we construct the picture space $X(Q)$ and show that $X(Q)$ has only first homotopy group non-trivial, and that group is actually isomorphic to $G(Q)$, i.e. $X(Q)$ is a $K(\pi,1)$ for $G(Q)$. Using this, we can compute homology of the picture group $G(Q)$ by computing homology of the picture space $X(Q)$. For the quiver of type $A_n$, we show that the homology groups are free abelian groups of ranks given by ballot numbers. Partial results for the quivers of type $D$ will be stated.\\ This is a joint work with Kiyoshi Igusa, Kent Orr and Jerzy Weyman.

Title: Vector-valued conjugate-invariant functions on a semisimple group
Speaker: Liang Xiao (University of Connecticut)
Time: Wednesday, April 18, 2018 at 11:15 am
Place: MONT 313Abstract: We study the space of vector-valued conjugate-invariant functions on a semisimple group. Let $G$ be a simply-connected semisimple algebraic group over $\mathbb C$, and let $T$ be a maximal torus. Restricting a conjugate-invariant function on $G$ to $T$ gives rise to a Weyl group-invariant function on $T$. It turns out that this induces an isomorphism ${\mathbb C}[G]^{G} \cong {\mathbb C}[T]^W$. In this talk, we instead consider the space of functions $f$ on $G$ with values in a representation $V$ of $G$ such that $f(ghg^{-1}) = g(f(h))$ for $g,h \in G$. The analogous Chevalley restriction map is no longer an isomorphism. In this talk, we study certain invariants related to the failure of the Chevalley restriction map being an isomorphism. If time permits, I will briefly discuss the motivation from number theory. This is a joint work with Xinwen Zhu.

Title: $p$-adic L-functions for symplectic groups and Klingen Eisenstein ideal
Speaker: Zheng Liu (IAS / McGill University)
Time: Wednesday, April 25, 2018 at 11:15 am
Place: MONT 313Abstract: The Iwasawa--Greenberg Main Conjecture predicts that the characteristic ideal of the Selmer group of $p$-adic deformation of a motive is generated by the corresponding $p$-adic L-function. One strategy for proving one divisibility is to construct cohomology classes through the congruences between Eisenstein series and cusp forms, which was used by Ribet in proving the converse to Herbrand's theorem, and further developed by Mazur--Wiles, Wiles, Skinner--Urban, etc, for proving the main conjectures for GL(1) and GL(2). A key step in this strategy is to study certain Eisenstein series including constructing $p$-adic families of Klingen Eisenstein series, relating the constant terms to $p$-adic L-functions and analyzing the non-degenerate Fourier coefficients. I will first illustrate the idea of Eisenstein congruences in the GL(1) case, and then explain some results for symplectic groups.

Organizer: Aurel Mihai Fulger