Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium

Department Colloquium - Henri Darmon (McGill)

Henri Darmon (McGill)

Friday, March 15th, 2019

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

Speaker: Henri Darmon (McGill)

Title: Kronecker's Jugendtraum: a non-archimedean approach.

Abstract: Kronecker's Jugendtraum ("dream of youth") seeks to construct abelian extensions of a given field through the special values of explicit analytic functions, in much the same way that the values of the exponential function $e(x) = e^{2\pi i x}$ at rational arguments generate all the abelian extensions of the field of rational numbers. Modular functions like the celebrated $j$-function play the same role when the field of rational numbers is replaced by an imaginary quadratic field. An extensive literature, both classical and modern, is devoted to the special values of modular functions at imaginary quadratic arguments, known as singular moduli; and modular functions have also been central to such modern developments as Wiles' proof of Fermat's Last Theorem and significant progress on the Birch and Swinnerton--Dyer conjecture arising through the work of Gross--Zagier and Kolyvagin. I will describe some recent attempts, in colaboration with Jan Vonk, to extend Kronecker's Jugendtraum to real quadratic fields by replacing modular functions by mathematical structures called "$p$-adic modular cocycles". While they are still poorly understood, these objects exhibit many of the same rich arithmetic properties as modular functions.

Prof. Darmon obtained his PhD from Harvard in 1991, then went on to Princeton before coming to McGill University in 94 where he is now a James McGill Professor. Prof. Darmon received many awards and distinctions, including a Sloan Research Award in 96, the Prix Andre Aisenstadt in 97, the Killam Research Fellowship in 2008, and both the Cole prize and the CRM-Fields-PIMS prize in 2017. Prof. Darmon was elected fellow of the Royal Society of Canada in 2003.

Department Colloquium - Maria Chudnovsky (Princeton University)

Fields Queen's Distinguished Lecture Series, Maria Chudnovsky (Princeton University)

Friday, March 8th, 2019

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

Speaker: Maria Chudnovsky (Princeton University)

Title: Detecting odd holes.

Abstract: A hole in a graph is an induced cycle of length at least four; and a hole is odd if it has an odd number of vertices. In 2003 a polynomial-time algorithm was found to test whether a graph or its complement contains an odd hole, thus providing a polynomial-time algorithm to test if a graph is perfect. However, the complexity of testing for odd holes (without accepting the complement outcome) remained unknown. This question was made even more tantalizing by a theorem of D. Bienstock that states that testing for the existence of an odd hole through a given vertex is NP-complete. Recently, in joint work with Alex Scott, Paul Seymour and Sophie Spirkl, we were able to design a polynomial time algorithm to test for odd holes. In this talk we will survey the history of the problem and the main ideas of the new algorithm.

Prof. Chudnovsky (Princeton) is a leading researcher in graph theory and combinatorics. She received her B.A. and M.Sc. form the Technion, and her PhD from Princeton University in 2003. Before returning to Princeton, she was a member of the IAS, a Clay Math. Inst. research fellow, and a Liu Family Professor of IEOR at Columbia University. For her joint work proving the strong perfect graph theorem, she was awarded the Ostrowski foundation research stipend in 2003 and the Fulkerson prize in 2009. In 2012, she was awarded the MacArthur Foundation Fellowship, and was an invited speaker at the 2014 ICM. Prof. Chudnovsky was also named one of the "brilliant ten" young scientists by the Popular Science magazine.

Fields Lecture Series - Maria Chudnovsky (Princeton University)

Fields Queen's Distinguished Lecture Series, Maria Chudnovsky (Princeton University)

Thursday, March 7th, 2019

Time: 5:30 p.m.  Place: Jeffery Hall 127

Speaker: Maria Chudnovsky (Princeton University)

Title: Parties, Doughnuts and Coloring: Some Problems in Graph Theory

Abstract: A graph is a mathematical construct that represents information about connections between pairs of objects. As a result, graphs are widely used as a modeling tool in engineering, social sciences, and other fields. The paper written by Leonhard Euler in 1736 on the Seven Bridges of Konigsberg is often regarded as the starting point of graph theory; and we have come a long way since. This talk will survey a few classical problems in graph theory, and explore their relationship to the fields of research that are active today. In particular, we will discuss Ramsey theory, graph coloring, perfect graphs, as well as some more recent research directions.

Prof. Chudnovsky (Princeton) is a leading researcher in graph theory and combinatorics. For her joint work proving the strong perfect graph theorem, she was awarded the Ostrowski foundation research stipend in 2003 and the Fulkerson prize in 2009. In 2012, she was awarded the MacArthur Foundation Fellowship, and was an invited speaker at the 2014 ICM. Prof. Chudnovsky was also named one of the "brilliant ten" young scientists by the Popular Science magazine.

Images from Dr. Maria Chudnovsky's Lecture - Mar. 7th, 2019

Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)

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.

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.

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.

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.

Special Colloquium - Haoran Li (UC Davis)

Haoran Li (UC Davis)

Monday, February 4th, 2019

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

Speaker: Haoran Li (UC Davis)

Title: High-Dimensional General Linear Hypothesis Tests via Spectral Shrinkage.

Abstract: In statistics, one of the fundamental inferential problems is to test a general linear hypothesis of regression coefficients under a linear model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. The testing problem is well-studied when the sample size is much larger than the dimension but remains underexplored under high dimensional settings. Various classical invariant tests, despite their popularity in multivariate analysis, involve the inverse of the residual covariance matrix, which is inconsistent or even singular when the dimension is at least comparable to the degree of freedom. Consequently, classical tests perform poorly.

In this talk, I seek to regularize the spectrum of the residual covariance matrix by flexible shrinkage functions. A family of rotation-invariant tests is proposed. The asymptotic normality of the test statistics under the null hypothesis is derived in the setting where dimensionality is comparable to the sample size. The asymptotic power of the proposed test is studied under a class of local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, a family of tests involving ridge-type regularization is constructed.

Haoran Li is a Ph.D. candidate in Statistics at the University of California, Davis.
His research interests include high dimensional statistics, random matrix theory, and high dimensional time series.

Special Colloquium - Yanglei Song (UIUC)

Yanglei Song (UIUC)

Friday, February 1st, 2019

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

Speaker: Yanglei Song (UIUC)

Title: Asymptotically optimal multiple testing with streaming data.

Abstract: The problem of testing multiple hypotheses with streaming (sequential) data arises in diverse applications such as multi-channel signal processing, surveillance systems, multi-endpoint clinical trials, and online surveys. In this talk, we investigate the problem under two generalized error metrics. Under the first one, the probability of at least $k$ mistakes, of any kind, is controlled. Under the second, the probabilities of at least $k_1$ false positives and at least $k_2$ false negatives are simultaneously controlled. For each formulation, we characterize the optimal expected sample size to a first-order asymptotic approximation as the error probabilities vanish, and propose a novel procedure that is asymptotically efficient under every signal configuration. These results are established when the data streams for the various hypotheses are independent and each local log-likelihood ratio statistic satisfies a certain law of large numbers. Further, in the special case of iid observations, we quantify the asymptotic gains of sequential sampling over fixed-sample size schemes.

Yanglei Song is a Ph.D. Candidate in Statistics at the University of Illinois, Urbana-Champaign. His current research interests include multiple testing with streaming data, sequential change-point detection and high dimensional U-statistics.

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