Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory - Brad Rodgers (Queen's University)

Tuesday, March 5th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Brad Rodgers (Queen's University)

Title: The distribution of traces of powers of matrices over finite fields.

Abstract: Consider a random N by N unitary matrix chosen according to Haar measure. A classical result of Diaconis and Shashahani shows that traces of low powers of this matrix tend in distribution to independent centered gaussians as N grows. A result of Johansson shows that this convergence is very fast -- superexponential in fact. Similar results hold for other classical compact groups. This talk will discuss analogues of these results for N by N matrices taken from a classical group over a finite field, showing that as N grows traces of powers of these matrices equidistribute superexponentially. A little surprisingly, the proof is connected to the distribution in short intervals of certain arithmetic functions in F_q[T]. This is joint work with O. Gorodetsky.

Department Colloquium - Alessandro Portaluri (University of Turin)

Alessandro Portaluri (University of Turin)

Friday, March 1st, 2019

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

Speaker: Alessandro Portaluri (University of Turin)

Title: Existence and stability results in Celestial Mechanics

Abstract: Is the solar system stable? This is maybe one of the oldest open question in dynamical systems. It is still a lively and very active research field starting from Newton, Lagrange, Maxwell, Poincare and Birkhoff (only to mention a few of them) who proved several astonishing results in this direction.

A lot of useful techniques to tackle this problem, were developed during the last decades: KAM theory, symplectic and contact methods, interval arithmetic, etc. One more (variationally oriented) piece we add to this arsenal: the index theory!

In this talk we introduce some new ideas behind many recent results on this topic and we discuss some new perspectives and challenges on singular (weak force) Lagrangian problems. We show the existence of equivariant periodic orbits and we investigate some (in)stability results for a plethora of periodic motions via symplectic techniques.

Prof. Portaluri (University of Turin) obtained his PhD in 2004 from the University of Turin, then worked at the University of Milano-Bicocca and the University of Salento before going back to Turin in 2012. He is currently on sabbatical at Queen's. His research interests include index theory for ordinary and partial differential operators via Maslov index and spectral flow techniques, and linear stability analysis in singular Lagrangian systems.

Math Club - Neha Prabhu (Queen's University)

Thursday, February 28th, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Neha Prabhu (Queen's University)

Title: The one-sentence proof.

Abstract:  Which primes can be written as a sum of two squares? The answer to this classical question has been known for centuries and many different elaborate proofs have been discovered. In this talk, we present Don Zagier's sensational "one-sentence" proof.

Number Theory - Anup Dixit (Queen's University)

Tuesday, February 26th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: An extremal property of general Dirichlet series.

Abstract: A general Dirichlet series is given by $F(s)=\sum_{n=1}^{\infty} a_n/ \lambda_n^s$. Our goal is to realize $F(s)$ as a unique boundary element for a class of functions. In other words, is there a naturally occurring class of functions, for which $F(s)$ is the unique solution to an extremal problem? In this talk, we undertake this question and as a consequence show that an $L$-function satisfying a certain growth condition is uniquely determined by its degree, conductor, and its residue at $s=1$.

Special Colloquium - Kaitlyn Hood (MIT)

Kaitlyn Hood (MIT)

Friday, February 15th, 2019

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

Speaker: Kaitlyn Hood (MIT)

Title: Modeling Laminar Flow: Hairy Surfaces and Particle Migration

Abstract: Hairy surfaces are ubiquitous in nature and exhibit a wide range of functions, such as sensing or feeding in marine crustaceans. New fabrication techniques allow for functional engineered structures at the scale of hairs. However, flows at the intermediate Reynolds numbers relevant to these structures give rise to nonlinear equations of motion, which combined with the fine structure of hair arrays, can become numerically intractable. I derive a simple design principle for engineering hairy surfaces. This principle centers on the boundary layer depth of a single hair over a range of Reynolds numbers, which renders numerical calculations feasible in many geometries. Similarly, particles or cells suspended in flow are an essential component to lab-on-a chip technology for medical diagnostics. High speeds in these devices give rise to intermediate Reynolds numbers, and a range of nonlinear but deterministic behavior. I develop a mixed asymptotic and numerical model to predict the migration of particles across streamlines, and verify this theory against experimental data.

Kaitlyn Hood is an NSF Postdoctoral Fellow at Massachusetts Institute of Technology. She received her Ph.D. in Applied Mathematics at the University of California, Los Angeles, 2016. Her research interests include mathematical modelling of hydrodynamics, numerical techniques for nonlinear and singular PDEs, and studying the role of inertia in laminar flows.

Math Club - Ram Murty (Queen's University)

Thursday, February 14th, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Ram Murty (Queen's University)

Title: The Basel Problem.

Abstract:  In 1650, Pietro Mengoli posed the problem of explicitly evaluating $$1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots $$ and this had come to be known as the Basel problem. The solution is $\pi^2/6$ and it was discovered by Euler almost a century later in 1734, when Euler was 28 years old.

Euler's proof, though correct, was far from rigorous and had to await further developments in complex analysis to put it on a sure footing. We will give a slick proof that uses only ideas from first year calculus.

Special Colloquium - Weiwei Hu (Oklahoma State University)

Weiwei Hu (Oklahoma State University)

Wednesday, February 13th, 2019

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Weiwei Hu (Oklahoma State University)

Title: Theoretical and Computational Issues in Control and Optimization of Fluid Flows.

Abstract: In this talk, we mainly focus on control and optimization of a thermal fluid modeled by the Boussinesq equations. This work was motivated by the design and operation of low energy consumption buildings. We investigate the problem of feedback stabilization of a fluid flow in natural convection, which is important in the theory of hydrodynamical stability. In particular, we are interested in stabilizing a possible unstable steady state solution to the Boussinesq equations in a two dimensional open and bounded domain. The challenge of stabilization of the Boussinesq equations lies in the stabilization of the Navier-Stokes equations and its coupling with the convection-diffusion equation for temperature. We show that a finite number of controls acting on a portion of the boundary through Neumann/Robin type of boundary conditions is sufficient to locally stabilize the full nonlinear equations, where the problems of sensor placement and observer designs will also be addressed. Numerical results are provided to illustrate the idea and suggest areas for future research.

In the end, we briefly introduce our current work on optimal control designs for the Boussinesq equations with zero diffusivity and its application to control of optimal transport and mixing via flow advection. The challenges in numerical implementation will be discussed.

Weiwei Hu is an Assistant Professor of Mathematics at Oklahoma State University. She obtained her Ph.D. in Mathematics from Virginia Tech, Blacksburg. Her research interests include mathematical control theory of partial differential equations, optimal control of transport and mixing via fluid flows, and mathematical fluid dynamics.

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, February 12th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: Distribution of special values of $L$-series attached to Erdos functions.

Abstract: Inspired by Dirichlet's theorem that $L(1,\chi) \neq 0$ for a non-principal Dirichlet character $\chi$, S. Chowla initiated the study of non-vanishing of $L(1,f)$ for a rational valued, $q$-periodic arithmetical function $f$. In this context, Erdos conjectured that $L(1,f) \neq 0$ when $f$ takes values in $\{ -1, 1\}$. This conjecture remains open in the case $q \equiv 1 \bmod 4$ or when $q > 2 \phi(q) + 1$. In this talk, we discuss a density theoretic approach towards this conjecture and the distribution of these values.

Special Colloquium - Ian Tobasco (University of Michigan)

Ian Tobasco (Michigan)

Monday, February 11th, 2019

Time: 4:30 p.m.  Place: Jeffery Hall 234

Speaker: Ian Tobasco (University of Michigan)

Title: The Cost of Crushing: Curvature-driven Wrinkling of Thin Elastic Shells.

Abstract: How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on wrinkling patterns formed by thin floating shells, we develop a rigorous method (via Gamma-convergence) for evaluating the cost of crushing to leading order in the shell's thickness and other small parameters. The observed patterns involve regions of well-defined wrinkling alongside totally disordered regions where no single direction of wrinkling is preferred. Our goal is to explain the appearance of such "wrinkling domains". Our analysis proves that energetically optimal patterns maximize their projected planar area subject to a shortness constraint. This purely geometric variational problem turns out to be explicitly solvable in many cases of interest, and a strikingly simple scheme for predicting wrinkle patterns results. We demonstrate our methods with concrete examples and offer comparisons with simulation and experiment. This talk will be mathematically self-contained, not assuming prior background in elasticity or calculus of variations. The photo is credited to Joey Paulsen and Yousra Timounay of Syracuse University.

Ian Tobasco is a James Van Loo Postdoctoral Fellow at The University of Michigan. He received his PhD at the Courant Institute of Math. Sciences, New York University in 2016. His research interests include calculus of variations and partial differential equations, with specific interests in elasticity theory, fluid dynamics, and spin glasses.

Special Colloquium - Giusy Mazzone (Vanderbilt)

Giusy Mazzone (Vanderbilt)

Friday, February 8th, 2019

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

Speaker: Giusy Mazzone (Vanderbilt)

Title: On the Stability and Long-time Behavior of Fluid-Solid Systems.

Abstract: Consider the inertial motion of a coupled system constituted by a rigid body with a cavity completely filled by a viscous incompressible fluid. In 1885, Zhukovskii conjectured that "the motions of the coupled system about its center of mass will eventually be rigid motions and, precisely, permanent rotations, no matter the size and the shape of the cavity, the viscosity of the liquid and the initial movement of the system'". I will present a proof of Zhukovskii's conjecture for a very broad class of motions. We will see how the fluid stabilizing effect suggested by Zhukovskii occurs when Navier-type (no- and partial-slip) conditions are imposed on the fluid-solid interface. A nonlinear stability analysis shows that equilibria (permanent rotations with the fluid at a relative rest with respect to the solid) associated with the largest moment of inertia are asymptotically, exponentially stable. All other equilibria are normally hyperbolic and unstable in an appropriate topology. Moreover, every Leray-Hopf solution to the time-dependent problem converges to an equilibrium at an exponential rate in the $L_q$-topology, $ q\in (1,6) $, for every fluid-solid configuration.

Giusy Mazzone is an Assistant Professor (NTT) of Mathematics at Vanderbilt University. She has received a PhD in Mathematics at Universita del Salento, Lecce, Italy, in 2012 and a second PhD in Mechanical Engineering and Material Sciences from the University of Pittsburgh, Pennsylvania, in 2016. Her research interests include mathematical analysis of fluid dynamics, applications of partial differential equations in fluid mechanics, and the study of stability and asymptotic behavior of fluid-solid systems.