# Faculty Research Talks - Fall 2023

Talks will be held at 2:30PM on every other Monday (M) at 2:30PM in CULM 611 unless otherwise noted.

**September 18**

**Location:** CULM 611

**Inverse Scattering Problems in Locally Perturbed Periodic Layers**

Inverse scattering problems arise from many real-life applications, such as non-destructive testing, medical imaging, geophysical protesting,... An inverse scattering problem typically aims to determine some information about a scattering object or an inhomogeneous medium from some knowledge about the waves that are scattered by that object or medium.

This talk will briefly introduce an inverse scattering problem in general and inverse scattering problems in a locally perturbed periodic domain with an unknown background. Some open questions and projects will be discussed.

**October 2**

**Location:** CULM 611

**Soft Matter Systems: Big-data, Networks, Topology, Machine Learning**

This project focuses on modeling soft matter systems using methods that include modeling, discrete element simulations, network analysis using computational topology methods, and machine learning. The considered setups are typical for big-data problems involving large amounts of dynamic data. The plan is to use various mathematical methods to simplify these data sets, focusing on extracting physical mechanisms governing the behavior of underlying systems. The systems of interest include porous media flow, as well as wet and dry granular systems that are of relevance to a number of soft matter systems, such as suspensions, including active matter ones, among others. The part of the project at NJIT focuses on modeling and simulations and will be carried out in close collaboration with the groups focusing on physical experiments.

The presentation will provide an overview of various approaches being used, as well as of specific problems that have been explored recently. In addition, we will briefly discuss a few other projects that are currently considered by Complex Flow and Soft Matter Group members; more information about current and past projects can be found on the group page, cfsm.njit.edu.

**October 16**

**Location:** CULM 611

**Exploring Global Dynamics and Blowup in Some Nonlinear PDEs with Validated Numerics**

Computational models are exceedingly effective at simulating complex systems in science and engineering, however it is essential to understand their sources of error and bias. For example, when the National Hurricane Center reports to the public on the likely path of a hurricane, they do not present a single trajectory but instead a forecast cone within which the hurricane is expected to stay 66% of the time. Standard numerical analysis can provide local bounds on the error, and show that it goes to zero as the resolution (and computation time!) tends to infinity. However one is only ever able to run a finite computation, and a global error bound requires the cumulative effect from all of the sources of error to be quantified.

In this talk I will present an introduction to validated numerics, which allows one to compute such a global error bound. Then I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow. In a recent series of papers, we have shown (using validated numerics and 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.

*Updated: October 12, 2023*