| Date: | February 3, 2025 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | Peter Miller, University of Michigan |
| Title: | Universality in the Small-Dispersion Limit of the Benjamin-Ono Equation |
| Abstract: | This talk concerns the Benjamin-Ono (BO) equation of internal wave theory, and properties of the solution of the Cauchy initial-value problem in the situation that the initial data is fixed but the coefficient of the nonlocal dispersive term in the equation is allowed to tend to zero (i.e., the zero-dispersion limit). It is well-known that existence of a limit requires the weak topology because high-frequency oscillations appear even though they are not present in the initial data. Physically, this phenomenon corresponds to the generation of a dispersive shock wave. In the setting of the Korteweg-de Vries (KdV) equation, it has been shown that dispersive shock waves exhibit a universal form independent of initial data near the two edges of the dispersive shock wave, and also near the gradient catastrophe point for the inviscid Burgers equation from which the shock wave forms. In this talk, we will present corresponding universality results for the BO equation. These have quite a different character than in the KdV case; while for KdV one has universal wave profiles expressed in terms of solutions of Painlevé-type equations, for BO one instead has expressions in terms of classical Airy functions and Pearcey integrals. These results are proved for general rational initial data using a new approach based on an explicit formula for the solution of the Cauchy problem for BO. This is joint work with Elliot Blackstone, Louise Gassot, Patrick Gérard, and Matthew Mitchell. |
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| Date: | February 17, 2025 |
| Time: | 4:00pm |
| Location: | Zoom: https://ta |
| Speaker: | Andrea Bertozzi, UCLA |
| Title: | High-throughput optimization of DNA-aptamer secondary structure for classification and machine learning intepretability |
| Abstract: | We consider the secondary structures for aptamers, single
stranded DNA sequences that often fold on themselves and can be designed
to bind to small molecules. Given a specific aptamer sequence, there are
well-established computational tools to identify the lowest energy
secondary structure. However there is a need for a high-throughput process
whereby thousands of DNA structures can be calculated in real time for use
in an interactive setting, in particular when combined with aptamer
selection processes in which thousands of candidate molecules are screened
in the lab. We present a new method called GMfold, which algorithmically
uses subgraph matching ideas, in which the DNA chain is a graph with
nucleotides as graph nodes and adjacency along the chain to define edges
in the primary DNA structure. This allows us to cluster thousands of DNA
strands using modern machine learning algorithms. We present examples
using data from in vitro systematic evolution of ligands by exponential
enrichment (SELEX). This work is intended to serve as a building block for
future machine-learning informed DNA-aptamer selection processes for
target binding and medical therapeutics.
The Seminar will be via Zoom: https://tamu.zoom.us/j/94220070032 |
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| Date: | March 3, 2025 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | Michael Siegel, New Jersey Institute of Technology, Newark |
| Title: | A fast mesh-free boundary integral method for two-phase flow with soluble surfactant |
| Abstract: | We present an accurate and efficient boundary integral (BI) method to simulate the deformation of drops and bubbles in Stokes flow with soluble surfactant. Soluble surfactant advects and diffuses in bulk fluids while adsorbing and desorbing from interfaces. Since the fluid velocity depends on the bulk surfactant concentration C, the advection-diffusion equation governing C is
nonlinear, which precludes the Green’s function formulation necessary for a
BI method. However, in the physically representative large Peclet number
limit an analytical reduction of the surfactant dynamics surprisingly permits
a Green’s function formulation. Despite this, existing fast algorithms for
similar BI formulations, such as those developed for the heat equation, do
not readily apply. To address this challenge, we present a new fast algorithm
for our formulation which gives a mesh-free solution to the fully coupled
moving interface problem, including soluble surfactant effects. The method
extends to other problems involving advection-diffusion in the large Peclet
number limit. This is joint work with Michael Booty (NJIT), Samantha Evans (NJIT), and Johannes Tausch (SMU).
Zoom: https://tamu.zoom.us/j/94220070032 |
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| Date: | March 24, 2025 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | Andre Nachbin, WPI, Worcester, MA |
| Title: | Water waves on graphs |
| Abstract: | We have deduced a weakly nonlinear, weakly dispersive Boussinesq system for water waves on a 1D branching channel, namely on a graph. The model required a new compatibility condition at the graph’s node, where the main reach bifurcates into two reaches. The new nonlinear compatibility condition generalizes the well-known Neumann-Kirchhoff condition and includes forking angles in a systematic fashion. We present numerical simulations comparing solitary waves on the 1D (reduced) graph model with results of the (parent) 2D model, where a compatibility condition is not needed. We will comment on new problems that arise.
Join :https://tamu.zoom.us/j/94220070032 |
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| Date: | March 31, 2025 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | Dimitris Papageorgiou, Imperial College, London |
| Title: | Evolution PDEs arising in multifluid-multiscale flows |
| Abstract: | Viscous multilayer shear flows of immiscible liquids can become unstable due to the presence of interfaces.
The nonlinear patterns that emerge are complex and extreme events such as interface pinching and/or wall touching are possible.
It is of interest, therefore, to develop reduced dimension and reduced parameter nonlinear models that can be used to study the
dynamics and quantify solutions and allied phenomena.
I will describe how asymptotic analysis can be used to derive classes of equations that couple thin and thick regions and produce nonlocal PDEs. The physical origins can be free-stream shear, electric or magnetic fields etc. The particular case of two-layer viscous flows in channels will be considered in detail and comparisons of the nonlocal equations with experiments will be shown that confirm their usefulness. Finally I will discuss models that describe multilayer extrusion processes that in many cases support slip at the interface between viscous fluids. Such slip is the viscous analogue of classical inviscid slip that leads to Kelvin-Helmholtz instability. It will be shown that the viscous case also supports short-wave instabilities, albeit not catastrophic. Nonlinear aspects will also be discussed.
Seminar will be via Zoom: https://tamu.zoom.us/j/94220070032
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| Date: | April 14, 2025 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | John Lowengrub, UC Irvine |
| Title: | Thermodynamically consistent modeling of the chemomechanical regulation of growing tumors |
| Abstract: | Mechanical forces and biochemical signaling networks play a crucial role in cell behavior in growing tissues. The nonlinear dynamics of tissue growth arises from complex interactions between cells and their surroundings. However, the role of chemomechanical regulation which governs the size, shape, and structure of multicellular tissues remains insufficiently understood. To investigate this, we develop a thermodynamically consistent models in the Eulerian framework to simulate nonlinear tissue growth in confined geometries. Our formulation integrates both elastic and chemical energies through an energy variational approach. We consider both sharp and diffuse interface approaches. Additionally, we implement efficient numerical methods with a special boundary treatment to incorporate applied forces. We compare the model results to experimental data on tumor spheroid growth under varying mechanical conditions and find good agreement between the two. We confirm the validity of our diffuse interface (phase-field) model by demonstrating its convergence to the sharp-interface model. Furthermore, we examine how elastic forces, variations in tumor and host stiffness, and external forces influence the evolution of single and multifocal tumors in confined geometries.
Zoom: https://tamu.zoom.us/j/94220070032
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| Date: | April 21, 2025 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Todd Arbogast, University of Texas at Austin, Austin |
| Title: | Self Adaptive Theta (SATh) Schemes for Solving Hyperbolic Conservation Laws |
| Abstract: | We present a discontinuity aware quadrature (DAQ) rule, and use it to develop implicit self-adaptive theta (SATh) schemes for the approximation of scalar hyperbolic conservation laws. Our SATh schemes require the solution of a system of two equations, one controlling the cell averages of the solution at the time levels, and the other controlling the space-time averages of the solution. These quantities are used within the DAQ rule to approximate the time integral of the hyperbolic flux function accurately, even when the solution may be discontinuous somewhere over the time interval. We prove that DAQ is accurate to second order when there is a discontinuity in the solution and third order when it is smooth. The resulting scheme is a finite volume theta time stepping method, with theta defined implicitly (or self-adaptively). When upstream weighted, SATh-up is unconditionally stable, satisfies the maximum principle, and is total variation diminishing under appropriate monotonicity and boundary conditions, provided that theta is restricted to be at least 1/2, or even 0 with additional restrictions. We present numerical results showing the performance of the SATh schemes, sometimes using the more general Lax-Friedrichs numerical flux. Compared to solutions of finite volume schemes using Crank-Nicolson and backward Euler time stepping, SATh solutions often approach the accuracy of the former but without oscillation, and they are numerically less diffuse than the later.
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| Date: | April 28, 2025 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | Christoph Borgers, Tufts University, Medford, MA |
| Title: | The yard-sale convergence theorem |
| Abstract: | Suppose n identical agents engage in a sequence of trades. Each trade involves a random pair of agents, and as a result of the trade, one agent gains some amount of wealth, and the other loses the same amount. The total amount of wealth is conserved, call it W. The amount of wealth transferred is a small fraction of the poorer trading partner’s pre-trade wealth, so nobody ever goes bankrupt. The direction of wealth transfer is random with both possibilities equally likely. The yard-sale convergence theorem states that with probability 1, the wealth of one agent will converge to W, and the wealth of all others will therefore converge to 0. For short, randomness that is fair in expectation inescapably leads to total oligarchy. This was first observed by Anirban Chakraborti 25 years ago. It is an immediate consequence of the martingale convergence theorem. However, I will give a more elementary proof, using nothing heavier than the Borel-Cantelli lemma, of a stronger result. This is joint work with Claude Greengard. |
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