Math Colloquium - Spring 2023

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January 20

Raghav Venkatraman, NYU

Host: Travis Askham

The Robustness of ENZ Device

"ENZ" devices are a class of electromagnetic devices that operate at a frequency at which one of their components has dielectric permittivity close to zero. Such devices have curious properties that have made them valuable in creating entirely new kinds of waveguides and resonators. While their analytical study in the physics literature so far has been limited to the idealized "epsilon =0" limit, the robustness of the associated effects to epsilon merely small have only been explored numerically, and an analytical understanding is lacking. 

We will discuss a few different examples in which we can get complete analytical information in the setting where epsilon is merely small. 

This is joint work with Bob Kohn, based on conversations with Nader Engheta. 

January 27

Jonathan Jaquette, Boston University

Host: Michael Siegel

Exploring Global Dynamics and Blowup in Some Nonlinear PDEs

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.

February 3

Ehud Yariv, Department of Mathematics, Technion - Israel Institute of Technology

Host: Michael Siegel

Flows About Superhydrophobic Surfaces

Superhydrophobic surfaces, formed by air entrapment within the cavities of hydrophobic solid substrates, offer a promising potential for hydrodynamic drag reduction. In several of the prototypical surface geometries the flows are two-dimensional, governed by Laplace’s equation in the longitudinal problem and the biharmonic equation in the transverse problem. Moreover, low-drag configurations are typically associated with singular limits. Thus, the analysis of liquid slippage past superhydrophobic surfaces naturally invites the use of both singular-perturbation methods and conformal-mapping techniques. I will discuss the combined application of these methodologies to several emerging problems in the field. 

February 10

Sophie Ramananarivo, LadHyX, Ecole Polytechnique

Host: Anand Oza

Can We Tailor the Behavior of Flexible Sheets in Flows by Adding Cuts or Folds?

Lightweight compliant surfaces are commonly used as roofs (awnings), filtration systems or propulsive appendages, that operate in a fluid environment. Their flexibility allows for shape to change in fluid flows, to better endure harsh or fluctuating conditions, or enhance flight performance of insect wings for example. The way the structure deforms is however key to fulfill its function, prompting the need for control levers. In this talk, we will consider two ways to tailor the deformation of surfaces in a flow, making use of the properties of origami (folded sheet) and kirigami (sheet with a network of cuts). Previous literature showed that the substructure of folds or cuts allows for sophisticated shape morphing, and produces tunable mechanical properties. We will discuss how those original features impact the way the structure interacts with a flow, through combined experiments and theory. We will notably show that a sheet with a symmetric cutting pattern can produce an asymmetric deformation, and study the underlying fluid-structure couplings to further program shape morphing through the cuts arrangement. We will also show that extreme shape reconfiguration through origami folding can cap fluid drag.

February 17

Paul Milewski, Department of Mathematical Sciences, University of Bath

Host: Wooyoung Choi

Embedded Solitary Internal Waves

The ocean and atmosphere are density stratified fluids. This means that a wide variety of gravity waves propagate in their interior, redistributing energy and mixing the fluid, affecting global climate balances. We shall focus on a particular wave-type (the second baroclinic mode) where both observations and modelling have only recently started. Stratified fluids with narrow regions of rapid density variation with respect to depth (pycnoclines) are often modelled as layered flows. We shall adopt this model and examine horizontally propagating internal waves within a three-layer fluid, with a focus on mode-2 waves which have oscillatory vertical structure. Mode-2 waves (typically) occur within the linear spectrum of mode-1 waves (i.e. they travel at lower speeds than mode-1 waves), and thus mode-2 solitary waves are  generically associated with an unphysical resonant mode-1 infinite oscillatory tail. We will present evidence that these tail oscillations can be found to have zero amplitude, thus resulting in families of localised solutions (so called embedded solitary waves) in the Euler equations. This is the first example we know of embedded solitary waves in the Euler equations, and would imply that these waves are longer lived that previously thought.

February 24

Arik Yochelis, Ben-Gurion University of the Negev

Host: Yuan-Nan Young

From Intracellular Actin Waves to Mechanism and Back: How Pattern Formation Theory Aids Biological Understanding and Applications

Self-organized patterns in the actin cytoskeleton are essential for eukaryotic cellular life. They are the building blocks of many functional structures that often operate simultaneously to facilitate, for example, nutrient uptake and movement of cells. However, identifying how qualitatively distinct actin patterns can coexist remains a challenge. In my talk, I will first overview the different aspects of modeling and show how intracellular actin waves are different from many other biological systems and why they are mathematically challenging. In the second part, I will show how using mathematical methods taken from pattern formation theory, it is possible not only to experimentally uncover the mechanism of micropinocytosis (nutrient uptake) but also demonstrate that distinct actin wave patterns can indeed coexist in the cortex of living cells. We believe that the latter is central to different cell functions as well as to synthetic biology, where minimal, yet robust systems are required to reconstitute the essential features of self-organization in the cell cortex.

Selected references:

Bernitt, Döbereiner, Gov, and Yochelis, Fronts and waves of actin polymerization in a bistable-based mechanism of circular dorsal ruffles, Nature Communications 8, 15863 (2017)

Beta, Gov, and Yochelis, Why a large scale mode can be essential for understanding intracellular actin waves, Cells 9, 1533 (2020)

Yochelis, Flemming, and Beta, Versatile patterns in the actin cortex of motile cells: Self-organized pulses can coexist with macropinocytic ring-shaped waves, Physical Review Letters 129, 088101 (2022)

March 3

Speaker and Affiliation: TBA

Tite and Abstract: TBA

March 10

Speaker and Affiliation: TBA

Tite and Abstract: TBA

March 17

Spring Recess - No Colloquium

March 24

Malena Espanol, Arizona State University

Host: Zoi-Heleni Michalopoulou

*This talk will be held virtually via WebEx*

Computational Methods for Solving Inverse Problems in Imaging

Discrete linear and nonlinear inverse problems arise from many different imaging systems. These problems are ill-posed, which means, in most cases, that the solution is very sensitive to the data. Because the data usually contain errors produced by different imaging system parts (e.g., cameras, sensors, etc.), robust and reliable regularization methods need to be developed for computing meaningful solutions. In some imaging systems, massive amounts of data are produced making the data storage and computational cost of the inversion process intractable. In this talk, we will see different imaging systems, we will formulate the corresponding mathematical models, we will introduce regularization methods, and we will show some numerical results.

March 31

Dejan Slepcev, Department of Mathematics, Carnegie Mellon University

Host: Brittany Hamfeldt

Tite and Abstract: TBA

April 7

Good Friday - No Colloquium

April 14

Jeffrey Sachs, Merck

Host: Casey Diekman

Tite and Abstract: TBA

April 21

Gwynn Elfring, Department of Mechanical Engineering, University of British Columbia

Host: Enkeleida Lushi

Tite and Abstract: TBA

April 28

Kasso Okoudjou, Tufts University

Host: Christina Frederick

Tite and Abstract: TBA

Updated: January 23, 2023