Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory Seminar - Ertan Elma (University of Waterloo)

Monday, February 24th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Ertan Elma (University of Waterloo)

Title: Discrete Mean Values of Dirichlet L-functions.

Abstract: Let $\chi$ be a Dirichlet character modulo a prime number $p \geq 3$ and let $\mathfrak{a}_{\chi}:=(1-\chi(-1))/2$. Define the mean value $$ \mathcal{M}_{p}(s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi \bmod p\\\psi(-1)=-1}}L(1,\psi)L(s,\chi\overline{\psi})$$ for a complex number $s$ such that $s\neq 1$ if $\mathfrak{a} _{\chi}=1.$ Mean values of the form above have been considered by several authors when $\chi$ is the principal character modulo $p$ and $\Re(s)>0$ where one can make use of the series representations of the Dirichlet $L$-functions being considered. In this talk, we will investigate the behaviour of the mean value $\mathcal{M}_{p}(-s,\chi)$ where $\chi$ is a non-principal Dirichlet character modulo $p$ and $\Re(s)>0$. Our main result is an exact formula for $\mathcal{M}_{p}(-s,\chi)$ which, in particular, shows that $$ \mathcal{M}_{p}(-s,\chi)= L(1-s,\chi)+\mathfrak a_\chi 2p^sL(1,\chi)\zeta(-s)+o(1), \quad (p\rightarrow \infty)$$ for fixed $0<\sigma:=\Re(s)<\frac{1}{2}$ and $|\Im s|=o\left(p^{\frac{1-2\sigma}{3+2\sigma}}\right)$.

Dynamics, Geometry, & Groups - Francesco Cellarosi (Queen's)

Friday, February 14th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Francesco Cellarosi (Queen's University)

Title: Denjoy's non-transitive diffeomorphisms of the circle.

Abstract: H. Poincaré proved that an orientation-preserving homeomorphism f of the circle with irrational rotation number \alpha is semi-conjugate to the rotation by \alpha. Moreover, he proved that if the homeomorphism f is transitive, then the semi-conjugacy is a homeomorphism (and hence a conjugacy) and that if f is not transitive, then the semi-conjugacy is not invertible (and hence not a conjugacy). A. Denjoy constructed examples of non-transitive homeomorphisms (in fact, diffeomorphisms) of the circle with arbitrary irrational rotation number. I will review the history of the problem and explain Denjoy's construction.

Department Colloquium - Bill Ralph (Brock University)

Bill Ralph (Brock University)

Friday, February 14th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 128

Speaker: Bill Ralph (Brock University)

Title: Can Mathematics Recognize Great Art?

Abstract: Is there an objective truth hiding within great works of art that only mathematics can detect? In this talk, I'll present evidence for a mathematical aesthetic shared by many great artists across the centuries. We'll look at several striking works of art by artists ranging from Tintoretto to Picasso and use a new statistic to see that they are creating bell curves with their brushes. I’ll also show some of my own attempts to create visual art based on a variety of mathematically oriented techniques. Examples of my work can be viewed at www.billralph.com.

Bill Ralph is in the Faculty of Mathematics and Statistics at Brock University. His mathematical research began in algebraic topology with the study of exotic homology and cohomology theories and their connections with Banach Algebras. After that, he developed a transfer for finite group actions and studied a number that appears in the transfer that he calls the "coherence number" of the group. Lately, he has also been using the Hausdorff dimension of the orbits of dynamical systems to generate mathematical art. The following is an excerpt from the curator's notes from Prof. Ralph's Rodman Hall Museum show:
It is perhaps not surprising that some of the images have a painterly feel to them since the mixing of paint on the palette and the action of the brush on a surface are both processes that can be modeled as chaotic dynamical systems. In a sense, each image is a window into the intersection of the two great universes of mathematics and fine art.

Math Club - Atabey Kaygun

Thursday, February 13th, 2020

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 110

Speaker: Atabey Kaygun

Title: Can one summarize a text without reading it?

Abstract: There is an area of research that lies in the intersection of graph theory, linear algebra, probability, (discrete) stochastic processes, linguistics, literature, history and many areas in humanities.

This talk will be a leisurely excursion into this type of mathematics with examples from ongoing research. As a fun example, I will also demonstrate how one can summarize a text without actually reading it.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, February 12th, 2020

Time: 4:30-5:30 p.m Place: Jeffery Hall 319

Speaker: Mike Roth (Queen's University)

Title: Lazarsfeld-Mukai bundles.

Abstract: The talk will cover the construction of a particular kind of vector bundle on a K3 surface, obtained by starting with a $g^{1}_{d}$ on a curve on that surface.

This vector bundle construction appeared in Lazarsfeld’s proof of the Petri conjecture, but has also been one of the main starting points for investigating Green’s conjecture. In particular, it will be used in future lectures giving a recent proof due to Kemeny of Green’s conjecture for generic curves of even genus.

Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, February 11th, 2020

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 202

Speaker: Jamie Mingo (Queen's University)

Title: A Survey of the Weingarten Calculus.

Abstract:  The Weingarten calculus is a method for calculating the expectation of products of entries of Haar distributed unitary (or orthogonal) matrices. The basic tool is Schur-Weyl duality, however I will present an approach based on Jucys-Murphy operators. The Weingarten calculus started with Don Weingarten in 1978. Many authors have contributed since then. This talk will be based on papers of Collins, Sniady, and Zinn-Justin.

Free Probability and Random Matrices Seminar Webpage:

Department Colloquium - Natalia Stepanova (Carleton University)

Natalia Stepanova (Carleton University)

Friday, February 7th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Natalia Stepanova (Carleton University)

Title: Goodness-of-fit tests based on sup-functionals of weighted empirical processes.

Abstract: A large class of goodness-of-fit test statistics based on sup-functionals of weighted empirical processes is proposed and studied. The weight functions employed are Erdos-Feller-Kolmogorov-Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Csorgo, S. Csorgo, Horvath, and Mason obtained for this type of test statistics, we provide the asymptotic null distribution theory for the class of tests in hand, and present an algorithm for tabulating the limit distribution functions under the null hypothesis. A new family of nonparametric confidence bands is constructed for the true distribution function and it is found to perform very well. The results obtained, together with a new result on the convergence in distribution of the higher criticism statistic, introduced by Donoho and Jin, demonstrate the advantage of our approach over a common approach that utilizes a family of regularly varying weight functions. Furthermore, we show that, in various subtle problems of detecting sparse heterogeneous mixtures, the proposed test statistics achieve the detection boundary found by Ingster and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects. This is joint work with Tatjana Pavlenko (Sweden).

Natalia Stepanova is a Professor of Statistics in the School of Mathematics and Statistics at Carleton University. She has a Ph.D. degree in Statistics from St. Petersburg State University. From 2001--2003 Natalia Stepanova was a Postdoctoral Fellow at Queen's University, supervised by Prof. Boris Levit. Her research interests lie mainly in the area of nonparametric statistics, including high-dimensional statistical inference.

Dynamics, Geometry, & Groups - Giusy Mazzone (Queen's University)

Friday, February 7th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Giusy Mazzone (Queen's University)

Title: On the stability of solutions to semilinear evolution equations.

Abstract: In this talk, we will discuss the stability and long-time behavior of solutions to nonlinear evolution equations in Banach spaces. We are particularly interested in semilinear parabolic systems of PDEs possessing a nontrivial manifold of equilibria, and whose linearization admits zero as a semi-simple eigenvalue. We will provide sufficient conditions that ensure asymptotic (exponential) stability of equilibria. We will also discuss an instability result for normally hyperbolic equilibria, and provide conditions for their attainability. Applications to fluid-solid interaction problems will be presented.

CYMS Seminar - Jia-Wei Guo (National Taiwan University)

Thursday, February 6th, 2020

Time: 3:00 p.m Place: Jeffery Hall 319

Speaker: Jia-Wei Guo (National Taiwan University)

Title: Class number relations arising from intersections of Shimura curves and Humbert surfaces.

Abstract: By considering the intersections of Shimura curves and Humbert surfaces on the Siegel modular threefold, we obtain new class number relations. The result is a higher-dimensional analogue of the classical Hurwitz-Kronecker class number relation. This is a joint work with Yifan Yang.

Math Club - Francesco Cellarosi (Queen's University)

Thursday, February 6th, 2020

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 110

Speaker: Francesco Cellarosi (Queen's University)

Title: Breaking the Fundamental Theorem of Calculus?

Abstract: In this talk we will ‘challenge’ the statement of the Fundamental Theorem of Calculus for the Riemann integral by constructing:

  • a function $g$ such that $G(x)=\int_0^x g(t)dt$ exists for all $x\in\mathbb R$ but $G$ is not differentiable at many points.
  • a differentiable function $F$ such that $\int_0^x F’(t)dt\neq F(x)-F(0)$.

To do so, we will use some ill-behaved functions constructed by B. Riemann and V. Volterra.

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