# Queen's University - Department of Mathematics and Statistics

 Wednesday, October 18 Number Theory Seminar Time: 1:30 p.m. Place: Jeffery 319 1st Speaker: Ravindranathan Thangadurai Title: On a conjecture of Schmid and Zhuang 2nd Speaker: Anup Dixit Title: Value distribution of L-functions Abstracts Attached Wednesday, October 18 Curves Seminar Time: 3:30 p.m. Place: Jeffery 319 Speaker: Mike Roth Title: Proof of the Fulton-Hansen-Deligne connectedness theorem and first applications Abstract Attached Friday, October 20 Department Colloquium Time: 2:30 p.m. Place: Jeffery 234 Speaker: Tianyuan Xu, Queen’s University Title: Fully commutative elements of Coxeter groups and their heaps Abstract Attached Monday, October 23 Geometry and Representation Theory Seminar Time: 4:45 p.m. Place: Jeffery 319 Speaker: Elisabeth Fink, University of Ottawa Title: Labelled geodesics in Coxeter groups Abstract Attached

Items for the Info Sheet should reach Anne ( burnsa@queensu.ca) by noon on Monday. The Info Sheet is published every Tuesday.

Wednesday, October 18, 1:30 p.m. Jeffery 319 - Number Theory Seminar

Title: On a conjecture of Schmid and Zhuang

Abstract: Let $G$ be a finite abelian additive group with exponent of $G$ as $\exp(G)$. A fundamental constant $D(G)$ attached to $G$ (is called `Davenport Constant) is defined to be the least positive integer $t$ such that any sequence $S$ over $G$ of length $t$ has a subsequence whose sum

is the zero element in $G$ (such a subsequence is called zero-sum subsequence). This constant naturally

arises from a question in Algebraic Number Theory. By the structure theorem of finite abelian groups $G$, we know that $$G\cong C_{n_1}+\cdots +C_{n_r}$$ where $n_i >1$ integers and $n_i$ divides $n_{i+1}$. Then it is easy to see that $$D^*(G) := 1+\sum_{i=1}^r(n_i-1) \leq D(G) \leq |G|$$ which implies $D(G) = |G|$ if and only if $G =C_n$. In 1969, Olson proved that $D(G) = D^*(G)$ for all finite abelian $p$-groups.

There is another related constant $\eta(G)$ which is defined as the least positive integer $t$ such that any given sequence $S$ of length $t$ has a zero-sum subsequence of length $\leq \exp(G)$. The value of this constant is known for $G$ with rank $\leq 2$ and in general it is unknown. In 2010, Schmid and Zhuang conjectured that when $G$ is a finite abelian $p$-group having $D(G) \leq 2\exp(G)-1$, then
$$\eta(G) = 2D(G) - \exp(G).$$

In this talk, we prove this conjecture for most of the finite abelian $p$-group satisfying the conditions.

2nd Speaker: Anup Dixit

Title: Value distribution of L-functions

Abstract: Nevanlinna theory establishes that if two meromorphic functions share five values, then they must be the same. Replacing meromorphic functions with L-functions in the Selberg class, M.R. Murty
and V.K. Murty proved that any two L-functions sharing the 0-value counting multiplicity must be the same. Moreover, J. Steuding showed that two L-functions in the Selberg class sharing two values ignoring multiplicity must be the same. In this talk, we show similar results in a larger family of L-functions, namely the Lindelof class. We also show a new uniqueness result in the Lindelof class and therefore the Selberg class.

Wednesday, October 18, 3:30 p.m. Jeffery 319 - Curves Seminar

Speaker: Mike Roth

Title: Proof of the Fulton-Hansen-Deligne connectedness theorem and fist applications

Abstract: We will prove the Fulton-Hansen-Deligne connectedness theorem and give some applications.

Friday, October 20, 2:30 p.m. Jeffery 234 - Department Colloquium

Speaker: Tianyuan Xu

Title: Fully commutative elements of Coxeter groups and their heaps

Abstract: An element in a Coxeter group is called fully commutative if any of its reduced words can be obtained from any other by braid relations that only involve commuting generators. Fully commutative elements can often be studied in terms of their heaps, and are closely connected to the study of symmetric functions, Catalan combinatorics, generalized Temperley-Lieb algebras, and Kazhdan-Lusztig theory. In particular, the fully commutative elements in a symmetric group are exactly the 321-

avoiding permutations, and they index a basis of the usual Temperley-Lieb algebra of type A.
In this talk, we will recall the classification of all Coxeter groups with finitely many fully commutative elements and highlight an interesting connection between heaps of fully commutative elements and Lusztig's a-function on a Coxeter group. Using this connection, we will then classify all Coxeter groups with finitely many elements of a-value 2. (This is a joint result with Richard Green.)

Monday, October 23, 4:45 p.m. Jeffery 319

Geometry and Representation Theory Seminar

Speaker: Elisabeth Fink

Title: Labelled geodesics in Coxeter groups

Abstract: Studying geodesics in Cayley graphs of groups has been a very active area of research over the last decades. We introduce the notion of a uniquely labelled geodesic, abbreviated with u.l.g. These will be studied first in finite Coxeter groups of type An. Here we introduce a generating function, and hence are able to precisely describe how many u.l.g.’s we have of a certain length and with which label combination. These results generalize known results about unique geodesics in Coxeter groups. In the second part of the paper, we expand our investigation to infinite Coxeter groups described by simply laced trees. We use the example of the group D˜ 6 to show the existence of infinite u.l.g.’s in groups which do not have any infinite unique geodesics. We conclude by exhibiting a detailed description of the geometry of such u.l.g.’s and their relation to each other in the group D˜ 6.

### Contact Info

 Department of Math & Stats Jeffery Hall, 48 University Ave. Kingston, ON Canada, K7L 3N6 Phone: (613) 533-2390 Fax: (613) 533-2964 mathstat@mast.queensu.ca Office Hours: 8:30am-12:00pm & 1:00pm-4:30pm