# Applied Math Colloquium - Fall 2021

Colloquia are held on Fridays at 11:30 a.m. in CULM Lecture Hall 2 or online via Webex. Please note the location listed in the schedule below.

Colloquia are held on Fridays at 11:30 a.m. in CULM Lecture Hall 2 or online via Webex. Please note the location listed in the schedule below.

To join the Applied Mathematics Colloquium mailing list visit https://groups.google.com/a/njit.edu/forum/?hl=en#!forum/math-colloquium/join (Google Profile required). To join the mailing list without a Google Profile, submit the seminar request form.

**Michael Booty, NJIT**

**Location: Webex**

Flows with Soluble Surfactant or Electrokinetic Effects

Surfactants are energetically favored to seek out the interface between immiscible fluids, where they influence the dynamics of the interface by reducing its surface tension. Surfactants can also occur in dissolved form away from the interface and transfer between interfacial and bulk forms. Because surfactant molecules are large relative to solvent molecules they diffuse very slowly relative to other time scales such as the relaxation time associated with surface tension.

Electrokinetic flows, on the other hand, consist of a strong electrolyte solution, meaning that the solute is almost completely disassociated into ions, in the presence of an electric field. The solution is charge-neutral except near an interface between immiscible electrolytes, where the electric field induces separation of charge to occur in thin Debye layers, and the Coulomb force acting on the separated charge induces flow.

These are both examples where a spatially narrow boundary layer or transition layer develops adjacent to a sharp fluid-fluid interface, and where the layer exerts a substantial element of control over the flow’s evolution and dynamics. Asymptotic techniques are introduced that allow reduction of the governing equations to a more tractable form that resolves the layer dynamics but nonetheless needs numerical solution. Various physical examples will be presented. This is collaborative work with Michael Siegel and a number of past and present doctoral students and postdocs in DMS.

**Peter Baddoo, MIT**

**Location: CULM 611 **

New Methods for Data-Driven Modeling of Dynamical Systems

Current techniques for data-driven modeling of dynamical systems struggle to capture high-dimensional nonlinearities and rarely produce models that respect the known physical properties of the system at hand. These challenges are addressed with two novel data-driven frameworks. Firstly, the Linear and Nonlinear Disambiguation Optimization (LANDO) is a kernel learning architecture that uses data measurements to construct efficient representations of high-dimensional nonlinear systems. Furthermore, LANDO enables an accurate disambiguation of the underlying linear and nonlinear mechanisms, which produces a set of physically interpretable linear modes. Secondly, the physics-informed Dynamic Mode Decomposition (piDMD) performs a constrained optimization to learn the best-fit linear operator that respects given physical properties. piDMD incorporates a range of physical structures including symmetries, invariances, causalities, spatial locality and conservation laws. As well as enforcing known physics, piDMD can also be employed to discover unknown physics. These new methods are applied to a sequence of problems from the physical sciences, thus demonstrating their improved diagnostic, predictive and interpretative abilities.

This is a collaborative project with Profs Benjamin Herrmann (U. of Chile), Beverley McKeon (CalTech), Steven Brunton (U. of Washington) and Nathan Kutz (U. of Washington).

**Guido De Philippis, NYU**

**Location: Webex**

(Boundary) Regularity for Mass Minimising Currents

Plateau problem consists in finding a surface of minimal area among the ones spanning a given curve. It is among the oldest problem in the calculus of variations and its study lead to wonderful development in mathematics. Federer and Fleming integral currents provide a suitably weak solution to the Plateau problem in arbitrary Riemannian manifolds, in any dimension and co-dimension. Once this weak solution has been found a natural question consists in understanding whether it is classical one. i.e. a smooth minimal surface. This is the topic of the regularity theory, which naturally splits into interior regularity and boundary regularity. After the monumental work of Almgren, revised by De Lellis and Spadaro, interior regularity is by now well understood. Boundary regularity is instead less clear and some new phenomena appear. Aim of the talk is to give an overview of the problem and to present some boundary regularity results we have obtained in the last years.

**Stephen Shipman, LSU**

**Location: Webex**

Embedded Eigenvalues for the Neumann-Poincaré

The PDEs governing sub-wavelength quasi-static scattering and resonance in small particles are intimately connected to the spectral theory of the Neumann-Poincaré (NP) boundary-integral operator in potential theory. In two and three dimensions, we prove the existence of scatterers for which the NP operator has eigenvalues embedded in its continuous spectrum. These eigenvalues arise due to a combination of a sharp point and symmetry in the scatterer. This is joint work with Wei Li (DePaul) and Karl-Mikael Perfekt (NTNU).

**Jiaoyang Huang, NYU**

**Location: TBA**

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**Jeremy Marzuola, UNC**

**Location: TBA**

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**Eduardo Corona, UC Boulder**

**Location: TBA**

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**Ed Large, U Conn.**

**Location: TBA**

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**Eleni Katifori, U Penn**

**Location: TBA**

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**Shahriar Afkhami, NJIT**

**Location: TBA**

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**Alex Barnett, Flatiron**

**Location: TBA**

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**Simone Maras, NJIT**

**Location: TBA**

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**Eric Keaveny, Imperial**

**Location: TBA**

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Updated: September 20, 2021