Mathematics Home / Research Seminars: Applied and Computational Mathematics

Time & Location: Typically talks will be in Gibson Hall 325 at 3:00 PM on a Friday.

Organizers: Punshon-Smith, Samuel and Buvoli, Tommaso

**January 20**

**Title:**** **Modeling microswimmers: the effects of cell shape and complex environments

**Rudi Shuech | Tulane | Tulane University**

**Abstract: **

In this two-part talk, first I will summarize my previous work on the effects of curved-rod bacterial shapes on swimming performance and other ecologically important tasks. We used a regularized Stokeslet boundary element method to compute the motion of curved-rod microswimmers propelled by rotating helical flagella. We then showed that Pareto-optimal tradeoffs between efficient swimming, chemotaxis, and cell construction cost can explain the morphological diversity of extant curved bacterial species.

In the second part, I will transition to thinking about the complex environments that microorganisms swim through, which are often composed of a viscous fluid with suspended microstructures such as elastic polymers and filamentous networks. These microstructures can have similar length scales to the microorganisms, leading to complex swimming dynamics. Some microorganisms are also known to remodel the viscoelastic networks they move through. To gain insight into the coupling between the dynamics of the swimmer and the network, we combined our computational framework for microswimmer motion with a model of a discrete viscoelastic network. The network is represented by a cloud of points with virtual Maxwell element links, whose properties (i.e., stiffness, relaxation time) can have non-obvious effects on the swimmer dynamics. We model enzymatic dissolution of the network by bacteria or microrobots by breaking links based on their distance to the microswimmer. We investigate how swimming performance is affected by properties of the network and swimmer.

If time allows, I will also introduce our new work on microswimmers penetrating thin, membrane-like interfaces.

**Time**: 3:00

**Location**: Stanley Thomas 316

**January 27**

**Title:**** **The hydrodynamics of dinoflagellate motility

**Rudi Shuech | Tulane | Tulane University**

**Abstract: **

Flagella are crucial to the interactions of many microorganisms with their surrounding fluid environment. The single-celled dinoflagellates have a unique but remarkably conserved flagellation morphology: a trailing longitudinal flagellum and an exquisitely complex transverse flagellum that encircles the cell. What are the selective advantages offered by this arrangement? We investigated the dinoflagellate design in silico using a high-performance regularized Stokeslet boundary element method, comparing to µPIV observations of swimming cells and quantifying how the morphology affects swimming performance. We found that the helical transverse flagellum provides most forward thrust and, despite its near-cell position, is more hydrodynamically efficient than the trailing flagellum; however, the latter is nonetheless required to enable steering. Flagellar hairs and the sheet-like structure of the transverse flagellum allow dinoflagellates to exert strong propulsive forces and maintain high clearance rates without extending a long conventional flagellum far into the surroundings. This unique morphology has thus been essential to the evolution of the generally large, fast-swimming dinoflagellates.

**Time**: 3:00

**Location**: Stanley Thomas 316

**February 3**

**Title:**** Analysis of solitonic interactions, and an initial connection to random matrix theory**

**Ken McLaughlin | Tulane University**

**Abstract: **I will describe the interaction between a single soliton and a gas of solitons, providing for the first time a mathematical justification for the kinetic theory as posited by Zakharov in the 1970s. Then, if time permits, I will explain an initial connection to random matrix theory, in order to introduce randomness into a large collection of solitons. This is joint work with Manuela Girotti, Tamara Grava, Robert Jenkins, and Alexander Minakov.

Time: 3:00

Location: Gibson Hall 325

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**February 10**

**Title:**** **Analysis of Solitonic interactions including a little more background, and hopefully an initial connection to random matrix theory.

**Ken McLaughlin | Tulane University**

**Abstract: **I will provide more information, including background information, about solitons and their interactions, some sort of a definition of a gas of solitons, and the interaction between a single soliton and a gas of solitons. I will finish with an explanation of an initial connection to random matrix theory, to introduce randomness into a large collection of solitons. This is joint work with Manuela Girotti, Tamara Grava, Robert Jenkins, and Alexander Minakov.

~~Time: 3:00~~

Location: Gibson Hall 325

**February 24**

**Title:**** Phase transitions and log Sobolev inequalities**

**Matias Delgadino | University Texas Austin**

**Abstract: **In this talk, we will study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N-particle system, and the presence or absence of phase transitions for the mean field limit. The non-degeneracy of the LSI constant will be shown to have far reaching consequences, especially in the context of uniform-in-time propagation of chaos and the behaviour of equilibrium fluctuations. This will be done by employing techniques from the theory of gradient flows in the 2-Wasserstein distance, specifically the Riemannian calculus on the space of probability measures.

Time: 3:30 ** Time Change**

Location: Gibson Hall 325

**March 3**

**Title:**** A functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero.**

**David Herzog | Iowa State University**

**Abstract: **We discuss the method of stochastic characteristics to help motivate laws of the iterated logarithm. This method is often used to solve second-order linear boundary-valued PDEs, such as the Dirichlet or Poisson Problem in a bounded domain in R^n. If the operator L defining the equation is uniformly elliptic in a neighborhood of the domain and the boundary is smooth, then we can always use this method with ease to find an expression for the unique classical solution, provided the data is smooth enough. However, if we drop the ellipticity assumption, say L is only hypoelliptic or "weakly" hypoelliptic (i.e. the randomness is degenerate), then the candidate expression typically makes sense, but it may no longer satisfy the equation. The problem moving from the elliptic to hypoelliptic settings is that the natural elliptic approximation of L may not approximate the defining stochastic characteristics near the boundary, even though the processes defined by these operators are close in a very strong sense. In order to determine when it does approximate L near the boundary, we establish laws of the iterated logarithm for the diffusion at time zero.

Time: 3:00

Location: Gibson Hall 325

**March 10**

**Title:**** **Insights from biofluidmechanics: A tale of tails

**Lisa Fauci | Tulane University**

**Abstract: **

The motion of actuated elastic structures in a fluid environment is a common element in many biological and engineered systems. I will present recent work on two very different systems at very different scales, The first is a caricature of the helical flagellar bundle of a bacterium, whose swimming performance improves when confined to a narrow tube. The second model organism I will discuss is the lamprey, the most primitive vertebrate. Using a closed-loop model that couples neural signaling, muscle mechanics, fluid dynamics and sensory feedback, we examine the hypothesis that amplified proprioceptive feedback could restore effective locomotion in lampreys with spinal injuries.

Time: 3:00

Location: ***Hebert Hall, Room 201*** Room change

**March 17**

**Title:**** **Optimizing Reconstruction of Under-sampled Magnetic Resonance Imaging (MRI) for Signal Detection

**Angel Pineda | Manhattan College**

**Abstract: ** Magnetic resonance imaging (MRI) is a versatile imaging modality that suffers from slow acquisition times. Accelerating MRI would benefit patients and improve public health both by reducing the time they need to be in the scanner and the cost of healthcare. Under-sampling the acquired data reduces the scan time but creates challenges for creating clinically useful images. Two recent methods for reconstructing images from under-sampled data are compressed sensing with constrained reconstruction and neural networks. Most of the current research focuses on new neural network architectures or training schemes while using mean squared error (MSE) or structural similarity (SSIM) as loss functions. The goal of this project is to optimize the performance of constrained reconstruction and deep learning based on a model for the clinical task of detecting subtle lesions instead of MSE or SSIM. We developed and experimentally validated observer models for estimating ideal and human observer performance. We have found that commonly used metrics like MSE and SSIM over-estimate the benefits of regularization in constrained reconstruction. In neural network reconstructions, we have also seen hallucination artifacts which are captured by MSE and SSIM but do not affect human observer performance in a signal-known-exactly task with varying backgrounds.

Time: 3:00

Location: Gibson Hall 325

**March 24**

**Title:**** **Slowing Down at Small Scales: Microscale Viscosity Gradients in the Ocean

**Stuart Humphries | University of Lincoln, UK**

**Abstract: ** The need to understand and predict the effect of micro-scale (<1mm) processes in the oceans is a pressing challenge, requiring the integration of several disciplines and across spatial and temporal scales. Interactions between marine microbes drive nutrient cycling and food webs in our oceans, and ultimately influence biogeochemistry on a global scale. Understanding these microscale processes is essential if we are to understand the dynamics of the oceanic system as a whole. At the microscopic scales at which the life of marine microbes unfolds, the physics is dominated by viscosity. Increasing viscosity slows down both the passive transport of solutes and particles and the swimming of motile microorganisms, and thus directly or indirectly affects all aspects of microbial life. Here I will show how we can reveal spatial heterogeneity of viscosity in planktonic systems by using microrheological techniques that allow measurement of viscosity at length scales relevant to microorganisms. I will show the viscous nature and the spatial extent of the phycosphere, the region surrounding phytoplankton, and discuss the implications of this variation for a number of areas, including how we might consider different diffusive situations in the oceans.

Time: 3:00

Location: Gibson Hall 325

**March 31**

**Title:**** TBA**

**Speaker | Tulane University**

**Abstract: **TBA

Time: TBA

Location: Gibson Hall 325

**April 7**

**Title:**** TBA**

**Speaker | Tulane University**

**Abstract: **TBA

Time: 3:00

Location: Gibson Hall 325

**April 14**

**Title:**** **Applied Mathematics for Multiphysics Simulations with Transport

**Ben Southworth | Los Alamos National Labs**

**Abstract: **Simulating multiphyics phenomena on the computer is a complex task, pulling from many branches of physics and mathematics. Broadly, the goal is to construct robust numerical methods with high physics fidelity, that can be run in parallel environments with thousands of CPUs or GPUs. Some of the challenges include high dimensionality of problems (>>3), large changes in scale of the physical behavior (space and time), and stiff nonlinear coupling between different variables or physics. In this talk I will discuss the numerical solution and evolution of transport equations and the coupling to hydrodynamics, with a particular focus on the time evolution and efficient implicit solution. The main objective is developing computationally feasible ways to approximate the physically stiff behavior. I will review linear algebra theory we have developed as well as physical insight that guides our approximations. We then apply our methods to radiative shocks and a hohlraum problem motivated by inertial confinement fusion.

Time: 3:00

Location: Gibson Hall 325

**April 21**

**Title:**** TBA**

**Greg Lyng | Optum Labs**

**Abstract: **TBA

~~Time: 3:00~~

Location: Gibson Hall 325

**April 28**

**Title:**** Long-time Asymptotics of the KdV Steplike Solutions**

**Iryna Egorova | ILTPE, Kharkiv**

**Abstract: **The long-time asymptotics of steplike solutions of the Korteweg – de Vries equation on constant backgrounds has long been well studied at the physical level of rigor. This talk will present some recent mathematically rigorous results that refine and justify these asymptotics. We will start with an introduction to the classical scattering theory for the Schrodinger operator with fast decaying potential. We then briefly discuss the basics of two most common methods of long-time asymptotic analysis of the integrable systems: the Inverse Scattering Transform and the Nonlinear Steepest Descent, and their applicability to the analysis of the KdV steplike solutions in soliton regions. In addition, we will get acquainted with rigorous asymptotics of the KdV rarefaction and shock waves.

Time: 3:00

Location: Gibson Hall 325

**May 3**

Last day of Class