# Queen's University - Department of Mathematics and Statistics

 Tuesday, November 28 Biostatistics Seminar Time: 1:00 p.m. Place: Bracken 124 (Health Sciences Library) Speaker: Debaraj Sen Title: Inference concerning intraclass correlation for binary responses Abstract Attached Wednesday, November 29 Number Theory Seminar Time: 1:30 p.m. Place: Jeffery 319 1st Speaker: M. Ram Murty Title: Hilbert’s tenth problem over number fields 2nd Speaker: François Séguin Title: A lower bound for the two-variable Artin conjecture 3rd Speaker: Arpita Kar Title: On a Conjecture of Bateman about $r_5(n)$ 4th Speaker: Siddhi Pathak Title: Derivatives of L-series and generalized Stieltjes constants Abstracts Attached Wednesday, November 29 Special Colloquium Time: 3:30 p.m. Place: Jeffery 234 Speaker: Jason Klusowski, Yale University Title: Counting motifs and connected components of large graphs via subgraph sampling Abstract Attached Thursday, November 30 Seminar in Free Probability and Random Matrices Time: 4:00 p.m. Place: Jeffery 319 Speaker: Pei-Lun Tseng, Queen’s University Title: Infinitesimal Laws of non-commutative random variables, II Abstract Attached

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

Tuesday, November 28, 1:00 p.m. Bracken 124 (Health Sciences Library) Biostatistics Seminar

Speaker: Debaraj Sen

Title: Inference concerning intraclass correlation for binary responses

Abstract: In the analysis of several treatment groups for binary outcome data, it is often interesting to determine if the treatments may have stabilizing effects. This inference problem can be done based on the confidence interval for a common intraclass correlation coefficient, and in many applications of epidemiological studies it is preferable by practitioners. Inference procedures concerning the intraclass correlation have been well developed for single-sample problems; little attention has been paid to extend these inference procedures for multiple-sample problems. In this talk, we focus on constructing the confidence interval procedures for a common intraclass correlation coefficient of several treatment groups. We compare different approaches with a large sample procedure in terms of coverage and expected length through a simulation study. An application to a solar protection study is used to illustrate the proposed methods.

Wednesday, November 29, 1:30 p.m. Jeffery 319 Number Theory Seminar

1st Speaker: M. Ram Murty

Title: Hilbert’s tenth problem over number fields

Abstract: Hilbert's tenth problem for rings of integers of number fields remains open in general,

although a conditional negative solution was obtained by Mazur and Rubin assuming some unproved conjectures about the Shafarevich-Tate groups of elliptic curves. In this talk, we highlight how the non-vanishing of certain L-functions is related to this problem. In particular, we show that Hilbert's tenth problem for rings of integers of number fields is unsolvable assuming the automorphy of L-functions attached to elliptic curves and the rank part of the Birch and Swinnerton-Dyer conjecture. This is joint work with Hector Pasten.

2nd Speaker: François Séguin

Title: A lower bound for the two-variable Artin conjecture

Abstract: In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group Z/pZ× for infinitely many p. In a 2000 article, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a 1967 proof by Hooley for the original conjecture. During this talk, we will prove an unconditional lower bound for this two-variable problem. This is joint work with Ram Murty and Cameron Stewart.

3rd Speaker: Arpita Kar

Title: On a Conjecture of Bateman about $r_5(n)$

Abstract: Let $r_5(n)$ be the number of ways of writing $n$ as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum $$\sum_{|j| \leq \sqrt{n}}\sigma(n-j^2)$$ where $\sigma(n)$ is the sum of positive divisors of $n$. We give a proof of Bateman's conjecture in the case $n$ is square-free and congruent to $1$ (mod $4$). This is joint work with Prof. Ram Murty.

4th Speaker: Siddhi Pathak

Title: Derivatives of L-series and generalized Stieltjes constants

Abstract: Generalized Stieltjes constants occur as coefficients of (s-1)^k in the Laurent series expansion of certain Dirichlet series around s=1. The connection between these generalized Stieltjes constants and derivatives of L(s,f) for periodic arithmetical functions f, at s=1 is known. We utilize this link to throw light on the arithmetic nature of L'(1,f) and certain Stieltjes constants. In particular, if p is an odd prime greater than 7, then we deduce the transcendence of at least (p-7)/2 of the generalized Stieltjes constants, {gamma_1(a,p) : 1 \leq a < p }, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath.

Wednesday, November 29, 3:30 p.m. Jeffery 234 Special Colloquium

Speaker: Jason Klusowski

Title: Counting motifs and connected components of large graphs via subgraph sampling

Abstract: Learning properties of large graphs from samples is an important problem in statistical network analysis. We revisit the problem of estimating the numbers of connected components in a graph of $N$ vertices based on the subgraph sampling model, where we observe the subgraph induced by $n$ vertices drawn uniformly at random. The key question is whether it is possible to achieve accurate estimation, i.e., vanishing normalized mean-square error, by sampling a vanishing fraction of the vertices. We show that it is possible by accessing only sublinear number of samples if the graph does not contains high-degree vertices or long induced cycles; otherwise it is impossible. Optimal sample complexity bounds are obtained for several classes of graphs including forests, cliques, and chordal graphs. The methodology relies on topological identities of graph homomorphism numbers, which, in turn, also play a key role proving minimax lower bounds based on constructing random instances of graphs with matching structures of small subgraphs. We will also discuss results for the neighborhood sampling model, where we observe the edges between the sampled vertices and their neighbors.

Thursday, November 30, 4:00 p.m. Jeffery 319

Seminar in Free Probability and Random Matrices

Speaker: Pei-Lun Tseng

Title: Infinitesimal Laws of non-commutative random variables, II

Abstract: In this talk, we will focus on a single type B variable, and introduce the corresponding infinitesimal law. In addition, we will also define the free additive convolution for infinitesimal laws, and to see the relation between the type B laws and the infinitesimal laws.

Seminar website: http://www.mast.queensu.ca/~mingo/seminar/

### 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