# Faculty Research Talks - Spring 2023

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

**February 13**

Location: CULM 611

**Micro-swimmers in Complex Environments**

Interactions between micro-swimmers and solid boundaries play an important role in many biological and technological processes. I will discuss our group's ongoing work in modeling and simulations that aim to understand the motion of micro-swimmers such as bacteria, micro-algae, spermatozoa or active colloids in various confinements or in structured environments. We will discuss (1) how to design appropriate models for each type of swimmer, (2) how to build the appropriate computational method with fast algorithmic implementations to trace the collective behavior of thousands of interacting swimmers, and (3) how to compare the simulations with relevant experiments. Our results highlight a complex interplay of the fluidic and contact interactions of the individuals with each-other and the boundaries to give rise to non-trivial individual and collective behavior.

**February 27**

Location: CULM 611

**New Insights into the Leapfrogging Vortex Problem**

We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as 'leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices were known to Grobli (1877) and Love (1883), and can be parameterized by a dimensionless parameter related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophøj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when $\alpha=\phi^{-2}$, where $\phi$ is the golden ratio. These numerical studies indicated a sequence of behaviors that emerge as this parameter is further decreased, leading to the disintegration of the leapfrogging orbit into a pair of dipoles that escape to infinity along transverse rays.

This study has two objectives. The first is to rigorously explain the origin of this remarkable bifurcation value and to generalize this analysis to the leapfrogging of unequal vortex pairs. The second is to understand the sequence of transitions in the phase space of the system that allows for the emergence of the various behaviors. While the first objective is essentially linear, finding the answer requires applying several tricks from the classical mechanics toolkit. The second objective is inherently nonlinear, and our approach involves both analysis and numerics. In particular, we make use of the recently developed technique of Lagrangian descriptors to visualize the phase space structures, including invariant manifolds.

**March 27**

Location: CULM 611

**Emergent Behaviour of Dynamical Systems on Large Random Networks**

I study high-dimensional dynamical systems on an all-to-all network. The connections are assigned a random weight, and each node is subject to white noise. This model has widespread applications, including in neuroscience (sometimes referred to as networks of balanced excitation and inhibition), data science (think of gradient descent for high dimensional datasets) and condensed matter physics (used to model slow ‘glassy dynamics’ in certain metallic alloys). I determine a novel set of ordinary differential equations that indicate the average level of network-wide activity. They are very accurate in the large size limit, and permit the use of dynamical systems techniques to understand macroscopic phenomena such as phase transitions and oscillations.

**April 10**

Location: CULM 611

**An Invitation to Gabor Analysis and the Frame Set Problem**

After a brief introduction to frame theory, I’ll introduce a challenging, longstanding problem in Gabor analysis that is deceptively simple to state.

**April 24**

Location: CULM 611

**Asymptotic Thermal Modeling of Droplet Assembly in Nanoscale Molten Metal Films**

We consider a thin metal film on a thermally conductive substrate exposed to an external heat source in a setup where the heat absorption depends on the local film thickness. Our focus is on modeling film evolution while the film is molten. The film geometry modifies local heat flow, which in turn may influence the film surface evolution through thermal variation of material properties. We use asymptotic analysis to develop a thermal model that is accurate, computationally efficient, and that accounts for the heat flow in both the in-plane and out-of-plane directions. We apply this model to describe metal films of nanoscale thickness exposed to heating and melting by laser pulses, a setup commonly used for self and directed assembly of various metal geometries via dewetting while the films are in the liquid phase. We find that thermal effects play an important role, and in particular that the inclusion of temperature dependence in the metal viscosity modifies the time scale of the evolution significantly. The thickness, thermal conductivity, and rate of heat loss of the underlying substrate are shown to be crucial in accurately modeling film temperatures and subsequent phase changes in the film. Since in many cases the substrate cools the film, modifications to the substrate temperature may induce different dewetting speeds via temperature dependent viscosity of the film. We show via 3D GPU simulations that this may result in various frozen film patterns since full dewetting may not occur while the film is in the liquid phase. This research was supported by NSF CBET-1604351, NSF-DMS-1815613 and by CNMS2020-A-00110.

*Updated: April 20, 2023*