Thursdays, 4:30-5:30pm, Thomas Hunter Hall, Room 412, (currently hosted in hybrid format: Zoom Link for Remote Access).
HCAM was initiated by the late John Arthur Loustau, a former professor of the Mathematics and Statistics Department here at Hunter, with his then post-doc Emmanuel Asante-Asamani in 2018. John had an eclectic mix of mathematical interests, each of which he pursued with gusto and depth. In a career spanning nearly 50 years, he began his journey in Commutative Algebra, transitioned afterwards to Computer Science, then Numerical Analysis, and eventually into Mathematical Biology. John recounted once a vacation he took with his family to Reno long ago. His father, a hardware merchant who also did plumbing and electric work, had taken him to the School of Mining Engineering at University of Nevada and told John that he had once dreamt of enrolling there when he was younger. Nevertheless, as John fondly recalled,He was a fine applied mathematician.
In the spirit of John's open-mindedness and willingness to foster and maintain a diverse community of mathematics, HCAM hosts speakers across a wide range of disciplines, from both academia and industry. HCAM also showcases the work of rising graduates of the Applied Math MA program at Hunter and regularly hosts Applied Math MA alumni to share their post-graduate experiences with current students. If you are interested in giving a talk at this seminar, please contact me at vrmartinez-at-hunter-dot-cuny-dot-edu.
Fall 2022 Schedule
September 29 Zoom (Recording) (Slides)
Alex Ely Kossovsky (Devry College of New York)
Mathematical Perspectives in the Emergence of Physics
This presentation briefly explores humanity's first major scientific achievement, namely the discovery of modern physics during the late Renaissance era, and demonstrates the decisive role of mathematics and rudimentary data analysis in facilitating this multi-generational accomplishment. Galileo's profound insight about motion and vertical acceleration, the role of his exhaustive pendulum experiments and studies, and especially the splitting of projectile motion into its vertical accelerating component versus its horizontal inertial component as the most significant inspiration for Newton's mechanics. In addition, the inspirational history of how mathematical advances like logarithms -discovered in the early 1600s - paved the way for this remarkable scientific advance in physics shall be explored, detailing how logarithms led Kepler to the discovery of his Third Law by facilitating arithmetical computations and by hinting at power-law relationships. Kepler's planetary statistical discovery of his Third Law relates the square of the time period for one full orbit around the sun to the cube of the planet's distance from the sun, namely Period2 = K*Distance3, and this remarkable discovery was courageously based on merely six data points, corresponding to the periods and distances of the six planets known at that era. In addition, the presentation shall briefly explore the role of Newton in midwifing the birth of science with his grand synthesis of Kepler's celestial data analysis and Galileo's terrestrial experiments. Lastly, the rise and fall of Bodes' Law shall be examined, and its ambitious but failed attempt to fit the orbital distances of the planets into an exact mathematical expression shall be presented as an illustrative example of the inability to apply rigid and exact mathematical formulas to probabilistic and chanced events, such as the chaotic process of star and planet formation from the random distribution in space of gas and dust particles into much larger entities via the force of gravity. Reference book: The Birth of Science, Springer Nature Publishing. Alex Ely Kossovsky. Aug 2020. ISBN-10: 3030517438.
October 6 Zoom (Recording)
Eviatar Bach (Caltech, Division of Geological and Planetary Sciences)
Towards the combination of physical and data-driven forecasts for Earth system prediction
Due to the recent success of machine learning (ML) in many prediction problems, there is a high degree of interest in applying ML to Earth system prediction. However, because of the high dimensionality of the system, it is critical to use hybrid methods which combine data-driven models, physical models, and observations. I will present two such hybrid methods: Ensemble Oscillation Correction (EnOC) and the multi-model ensemble Kalman filter (MM-EnKF). Oscillatory modes of the climate system are one of its most predictable features, especially at intraseasonal timescales. It has previously been shown that these oscillations can be predicted well with statistical methods, often with better skill than dynamical models. However, they only represent a portion of the signal, and a method for beneficially combining them with dynamical forecasts of the full system has not previously been developed. Ensemble Oscillation Correction (EnOC) is a method which corrects oscillatory modes in ensemble forecasts from dynamical models. I will show results of EnOC applied to forecasts of South Asian monsoon rainfall, outperforming the state-of-the-art forecasts on subseasonal-to-seasonal timescales. A more general method for combining multiple models and observations is multi-model data assimilation (MM-DA). MM-DA generalizes the variational, Bayesian, and minimum variance formulation of the Kalman filter. Here, I will show how multiple model ensembles can be combined for both DA and forecasting in a flow-dependent manner using a multi-model ensemble Kalman filter (MM-EnKF). This methodology is applied to multiscale chaotic models and results in significant error reductions compared to the best model and to an unweighted multi-model ensemble. Lastly, I will discuss the prospects of using the MM-EnKF for hybrid forecasting.
October 13 Zoom (Recording)
David Newstein (Statistical and Epidemiologicial Consultant, Independent)
Insights on Bertrand's paradox from a statistician's perspective
I consider Bertrand's Paradox (1889) from the perspective of the concept of The Principle of Indifference and introduce this concept with some familiar examples. Then I give an example illustrating one of the pitfalls of applying this concept recklessly (The Hidden Cube Example). I state and give a proof of the Law of The Unconscious Statistician, which clarifies and gives a valid solution to this example. I then introduce Bertrand's Paradox and mention its historical relation to The Principle of Indifference. I proceed to derive the Probability Density Functions (PDFs) for the three cases of the Paradox utilizing another theorem (The Change of Variables Formula) which is borrowed from analysis. A proof is provided for this stochastic version of the theorem. I conduct simulations which generate the Simulated Empirical Probability Density Functions for the three cases of the Paradox, and compare them to the analytically derived PDFs, and then check their goodness of fit. I finish with a brief mention of a physicist's (E.T. Jaynes) considerations on this subject, from his paper of 1973, and examine the validity of his assertions regarding the Paradox and its relation to The Principle of Indifference.
Hiatus
October 27 Thomas Hunter Hall, Room 412, Zoom (Recording)
Andres Contreras Murcillo (New Mexico State University, Department of Mathematical Sciences)
Orbital stability of domain walls in coupled Gross-Pitaevskii systems
In joint work with Pelinovsky and Plum, we establish an improved form of orbital stability of domain walls for a class of coupled Gross-Pitaevskii systems. We work in a suitable weighted H1-space, adapted to the domain walls to overcome the degeneracy of the linearized operator and lack of coercivity. Our proof does not make use of any integrability assumption and it is thus quite flexible. Also, our approach is strong enough to allow for controlling the modulation parameters in the time evolution of the modulation equations.
November 3 Thomas Hunter Hall, Room 412, (Recording)
Dana Ferranti (Tulane University, Department of Mathematics)
Computational Modeling of Bodies Immersed in Viscous Fluids
The Stokes equations describe fluid flows where viscous forces dominate inertial forces. These flows are relevant in the modeling of microorganism swimming, like bacteria or spermatozoa. We will give an accessible introduction to the Stokes equations, including the scaling analysis that leads to the equations and the properties which distinguish them from the broader Navier-Stokes equations. Additionally, we will discuss the method of regularized Stokeslets (MRS), a popular computational method for simulating flows generated by forces on the surfaces of bodies immersed in the fluid. The accuracy of the method relies on the choice of a blob parameter, which depends on the discretization parameter for the surface. In some applications, this dependence requires surface discretizations that are unnecessarily fine, which reduces the efficiency of the method. A modification of the MRS will be introduced which alleviates the coupling between these two parameters.
November 10 Thomas Hunter Hall, Room 412, Zoom (Slides)
Mirjeta Pasha (Tufts University, Department of Mathematics)
Modern Challenges in Large-Scale and High Dimensional Data Analysis
Rapidly growing fields such as data science, uncertainty quantification, and machine learning rely on fast and accurate methods for inverse problems. Three emerging challenges on obtaining relevant solutions to large-scale and data-intensive inverse problems are ill-posedness of the problem, large dimensionality of the parameters, and the complexity of the model constraints. Tackling the immediate challenges that arise from growing model complexities (spatiotemporal measurements) and data-intensive studies (large-scale and high-dimensional measurements collected as time-series), state-of-the-art methods can easily exceed their limits of applicability. In this talk we discuss efficient methods for computing solutions to dynamic inverse problems, where both the quantities of interest and the forward operator may change at different time instances. We consider large-scale ill-posed problems that are made more challenging by their dynamic nature and, possibly, by the limited amount of available data per measurement step. In the first part of the talk, to remedy these difficulties, we apply efficient regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each time instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy. In the remainder of the talk, we focus on designing spatio-temporal Bayesian Besov priors for computing the MAP estimate in large-scale and dynamic inverse problems. Numerical examples from a wide range of applications, such as biomedical applications, tomographic reconstruction, image deblurring, and multichannel dynamic tomography are used to illustrate the effectiveness of the described approaches.
November 17 Thomas Hunter Hall, Room 412, Zoom (Recording)
David Goluskin (University of Victoria, Department of Mathematics and Statistics)
Studying nonlinear dynamics using computational polynomial optimization
For nonlinear ODEs and PDEs that cannot be solved exactly, various properties can be inferred by constructing functions that satisfy suitable inequalities. Although the most familiar example is proving nonlinear stability of an equilibrium by constructing Lyapunov functions, similar approaches can produce many other types of mathematical statements, including for systems with chaotic behavior. Such statements include bounds on attractor properties or on transient behavior, estimates of basins of attraction, and design of nonlinear controls. Analytical results of these types often give overly conservative results in order to remain tractable. Much stronger results can be achieved by using computational methods of polynomial optimization to construct functions that satisfy the desired inequalities. This talk will provide an overview of the different ways in which polynomial optimization can be used to study dynamics. I will show various examples in which polynomial optimization produces arbitrarily sharp results while other methods do not. I will focus on the ODE case, where theory and computational methods are more complete.
Thanksgiving break
December 1 Thomas Hunter Hall, Room 412, Zoom (Recording)
Joshua Hudson (Sandia National Laboratories, Combustion Research Facility)
Data assimilation and inverse problems for fluid dynamics using nudging.
Nudging is a data assimilation technique where a damping term based on observational data is added to a dynamical system and acts as a corrective force on the system state. When successful, the algorithm results in convergence of the data assimilation approximation to the true solution at a much finer scale than that of the observations. In 2014, a rigorous mathematical proof was given by Azouani Olson and Titi giving conditions for the success of nudging for the 2D Navier-Stokes equations. Subsequently, similar results were obtained for several related equations. We will present some theoretical and computational results of nudging applied to the Magnetohydrodynamic equations. Then, we will discuss how nudging can be used to solve the inverse problem of inferring model parameters from the coarse state observations. We will focus on the Navier-Stokes equations with the viscosity as an unknown parameter, and discuss how and when the viscosity can be recovered. We will end with a discussion about the determining-map (the mapping of data and viscosity to a solution on the attractor), and discuss how our results extend the concept of determining modes to include the viscosity as a finitely determined quantity - this can be interpreted as a limitation on how different attractors of the Navier--Stokes (parameterized by the viscosity) can intersect.
December 8 Thomas Hunter Hall, Room 412, Zoom (Recording)
Daniel Ginsberg (Princeton University, Department of Mathematics)
Flexibility and rigidity of steady fluid motion and the distribution of heat in a fibered magnetic field
Motivated by problems in plasma physics and a conjecture of Grad, we consider some questions related to flexibility and rigidity of steady states of fluid equations. We also discuss how these problems are related to the problem of determining the distribution of heat in a strongly magnetized plasma. This addresses a recent physical conjecture of Helander, Hudson, and Paul about how the dynamical and geometric properties of the magnetic field influence heat transport. This is based on joint works with Peter Constantin, Theodore D. Drivas, and Hezekiah Grayer II.
December 15 Thomas Hunter Hall, Room 412, Zoom (Recording)
Thomas Joy and Michael Pallante (CUNY Hunter College, Department of Mathematics and Statistics)
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