Interfacial phenomena can be observed in many fields and across vastly different scales, such invading population of species in a shifting climate, the geographical spread of epidemic infections, the growth of cancer tumors at the cellular level, and the propagation of electric signals in neuroscience. This workshop focuses on interfacial phenomena and emphasizes the role of analysis of partial differential equations in understanding them. We aim at exposing students and researchers to challenging problems in the fields of biology and life sciences and the development of new mathematical frameworks to advance current understanding.
This workshop will include a poster session for early career researchers (including graduate students). In order to propose a poster, you must first register for the workshop, and then submit a proposal using the form that will become available on this page after you register. The registration form should not be used to propose a poster.
The deadline for proposing is November 4, 2024. If your proposal is accepted, you should plan to attend the event in-person.
Pierre-Emmanuel Jabin
Pennsylvania State University
I
K
Inwon Kim
University of California, Los Angeles (UCLA)
J
L
Jian-Guo Liu
Duke University
I
M
Idriss Mazari
Ceremade – Université Paris Dauphine
A
M
Antoine Mellet
University of Maryland
Y
M
Yoichiro Mori
University of Pennsylvania
M
P
Marya Ptashnyk
Heriot-Watt University
L
R
Lenya Ryzhik
Stanford University
R
S
Rachidi Salako
University of Nevada, Las Vegas
D
S
Delphine Salort
Sorbonne University
M
T
Min Tang
Shanghai Jiaotong University
O
T
Olga Turanova
Michigan State University
N
W
Ning Wei
Purdue University
C
W
Chang-Hong Wu
National Yang Ming Chiao Tung University
S
Y
Shugo Yasuda
Univerity of Hyogo
X
Z
Xinyue Zhao
University of Tennessee
Schedule
Monday, December 9, 2024
8:30-9:00 CST
Check-In and Breakfast
9:00-9:30 CST
A model on Tumor growth with nutrients
Speaker: Inwon Kim (University of California, Los Angeles (UCLA))
9:30-9:40 CST
Q&A
9:40-9:45 CST
Tech Break
9:45-10:15 CST
Predicting front invasion speeds via marginal stability
Speaker: Montie Avery (Boston University)
Front propagation into unstable states often mediates state transitions in spatially extended systems, in biological models and across the sciences. Classical examples include the Fisher-KPP equation for population genetics, Lotka-Volterra models for competing species, and Keller-Segel models for bacterial motion in the presence of chemotaxis. A fundamental question is to predict the speed of the propagating front as well as which new state is selected in its wake. The marginal stability conjecture asserts that front invasion speeds are determined by the spectrum of the linearization about traveling wave solutions of the PDE model. We present a formulation and proof of the marginal stability conjecture in general reaction-diffusion systems, and give an outlook towards establishing invasion properties in complex pattern-forming systems.
10:15-10:25 CST
Q&A
10:25-10:55 CST
Coffee Break
10:55-11:25 CST
Reaction rate of the flux-limited chemotaxis system
Speaker: Jing An (Duke University)
11:25-11:35 CST
Q&A
11:35-11:40 CST
Tech Break
11:40-12:10 CST
Optimal control of free boundary models for tumor growth
Speaker: Xinyue Zhao (University of Tennessee Knoxville)
In this talk, we will investigate the optimal control of treatment in free boundary PDE models for tumor growth. The optimal control strategy is designed to inhibit tumor growth while minimizing side effects. In order to characterize it, the optimality system is derived, and a necessary condition is obtained. Numerical simulations will be presented to illustrate the theoretical findings and assess the impact of the optimal control strategy on tumor growth dynamics.
12:10-12:20 CST
Q&A
12:20-13:20 CST
Lunch Break
13:20-13:50 CST
On the coexistence and exclusion of strains of a diffusive epidemic model
Speaker: Salako Rachidi (University of Nevada, Las Vegas)
13:50-14:00 CST
Q&A
14:00-14:30 CST
Social Hour
14:30-15:00 CST
Poster session (with continuation of Social Hour)
Tuesday, December 10, 2024
8:30-9:00 CST
Check-In and Breakfast
9:00-9:30 CST
TBA
Speaker: Yoichiro Mori (University of Pennsylvania)
9:30-9:40 CST
Q&A
9:40-9:45 CST
Tech Break
9:45-10:15 CST
TBA
Speaker: Shugo Yasuda (University of Hyogo)
10:15-10:25 CST
Q&A
10:25-10:55 CST
Coffee Break
10:55-11:25 CST
Singular Limits in Mechanical Models of Tissue Growth
Speaker: Noemi David (University of Lyon, France)
"Based on the mechanical perspective that living tissues exhibit fluid-like behavior, PDE models inspired by fluid dynamics are increasingly applied to describe tissue growth at the macroscopic level. These models link the pressure to the velocity field depending on the type of tissue, using either Brinkman’s law (viscoelastic models) or Darcy’s law (porous-medium equation, PME). The stiffness of the pressure law plays a crucial role in distinguishing between compressible (density-based) models and incompressible (free boundary) problems, where density saturation occurs.
In this talk, I will explore how different mechanical models of living tissues can be related through singular limits. Specifically, I will discuss the inviscid limit leading to the PME and the incompressible limit from the PME to Hele-Shaw free boundary problems. Furthermore, I will present a recent result on their joint limit, which is derived using energy dissipation inequalities, a method reminiscent of the EDI characterization for gradient flows."
11:25-11:35 CST
Q&A
11:35-11:40 CST
Tech Break
11:40-12:10 CST
Well-posedness and front propagation for aerotaxis models
Speaker: Christopher Henderson (University of Arizona)
"The amoeba Dictyostelium discoideum exhibits aerotaxis: in experiments, when a colony is covered, cells quickly consume the available oxygen and then move outwards at a constant speed to find new oxygen. In this talk, we discuss a reaction-diffusion equation, arising from a large population limit of an stochastic differential equation, for this behavior that was proposed by Demircigil and Tomašević. This model is quite degenerate, making even the well-posedness difficult to show.
The main goal of the talk is to discuss the existence, uniqueness, and long-time behavior of solutions. We show a transition in the character of the behavior as the strength of the aerotactic response varies. Our results use crucially the shape defect function, an object that measures the distance from the solution to the traveling wave in a novel way related to Stein's method in probability. This is a joint work with Mete Demircigil."
12:10-12:20 CST
Q&A
12:20-13:50 CST
Lunch Break
13:50-14:20 CST
TBA
Speaker: Antoine Mellet (University of Maryland)
14:20-14:30 CST
Q&A
14:30-15:00 CST
Coffee Break
15:00-15:30 CST
(Bistable and Monostable) Mean Field Game models for fisheries
Speaker: Idriss Mazari (Université Paris Dauphine)
In this talk we will review some recent works with Z. Kobeissi and D. Ruiz-Balet devoted to the understanding of MFG models for the optimal management of fisheries. We will review te bistable case, with a special emphasis on traveling wave solutions, and the monostable case, in which we will focus more on existence, uniqueness and long-time behaviour of the solutions.
15:30-15:40 CST
Q&A
Wednesday, December 11, 2024
8:30-9:00 CST
Check-In and Breakfast
9:00-9:30 CST
Macroscopic Dynamics for Nonequilibrium Biochemical Reactions from a Hamiltonian Perspective
Speaker: Jian-Guo Liu (Duke University)
Most biochemical reactions in living cells are not closed systems; they interact with their surroundings by exchanging energy and materials. At a mesoscopic scale, the quantity of each chemical can be modeled by random time-changed Poisson processes. Understanding macroscopic behaviors is facilitated by a nonlinear reaction rate equation that describes species concentrations. In the thermodynamic limit, the large deviation rate function from the chemical master equation is governed by a Hamilton–Jacobi equation. We decompose the general macroscopic reaction rate equation into an Onsager-type strong gradient flow, supplemented by conservative dynamics. We will also present findings on the large deviation principle and the importance sampling of transition paths that connect metastable states in chemical reactions.
9:30-9:40 CST
Q&A
9:40-9:45 CST
Tech Break
9:45-10:15 CST
Asymptotic dynamic of neural models with partial diffusion
Speaker: Delphine Salort (Sorbonne University)
10:15-10:25 CST
Q&A
10:25-10:55 CST
Coffee Break
10:55-11:25 CST
The mean-field Limit of sparse networks of integrate and fire neurons
Speaker: Pierre-Emmanuel Jabin (Pennsylvania State University)
"We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. Our approach allows to derive a continuous limit for the macroscopic behavior of the system, the 1-particle distribution, for a large number of neurons with no structural assumptions on the connection map outside of a generalized mean-field scaling. We propose a novel notion of observables that naturally extends the notion of marginals to systems with non-identical or non-exchangeable agents. Our new observables satisfy a complex approximate hierarchy, essentially a tree-indexed extension of the classical BBGKY hierarchy. We are able to pass to the limit in this hierarchy as the number of neurons increases through novel quantitative stability estimates in some adapted weak norm. While we require non-vanishing diffusion, this approach notably addresses the challenges of sparse interacting graphs/matrices and singular interactions from Poisson jumps, and requires no additional regularity on the initial distribution.
This is a joint work with D. Zhou."
11:25-11:35 CST
Q&A
11:40-12:10 CST
The Homogeneous Landau Equation
Speaker: Luis Silvestre (University of Chicago)
12:10-12:20 CST
Q&A
12:20-13:20 CST
Lunch Break
Thursday, December 12, 2024
8:30-9:00 CST
Check-In and Breakfast
9:00-9:30 CST
Multiscale modelling, analysis and simulation of plant biomechanics and growth
Speaker: Marya Ptashnyk (Heriot-Watt University)
The interplay between the complex microscopic structures and microscopic (cellular) processes enables many biological tissues to combine the ability to resist high pressure and mechanical forces with the flexibility required for large expansions and growth. To analyse the interactions between the mechanics, microstructure, and the chemistry we derive microscopic models for plant biomechanics, assuming that the elastic/viscoelastic properties of plant cell walls depend on the chemical processes and chemical reactions depend on the mechanical stresses. Multiplicative decomposition of the deformation gradient into elastic and growth parts is used to model the stress or strain based growth. To analyse and simulate the complex multiscale models, the macroscopic effective equations are derived using periodic or stochastic homogenization techniques. Numerical solutions for the macroscopic models demonstrate the impact of the microscopic structure and tissue heterogeneity on deformation and growth.
9:30-9:40 CST
Q&A
9:40-9:45 CST
Tech Break
9:45-10:15 CST
“Particle approximation of diffusion”
Speaker: Olga Turanova (Michigan State University)
Particle approximation of diffusion
I will present recent work on deterministic particle methods for a wide range of diffusion equations, including the heat equation, the porous medium equation, fast diffusion equations, and height constrained transport. The main idea is to introduce a nonlocal version of the relevant equation and prove that solutions of the nonlocal equation converge to those of the original PDE. In this talk, I will highlight the how the convergence arguments take advantage of the gradient flow structures of the PDEs under consideration. Based on joint work with Katy Craig and Matt Jacobs.
10:15-10:25 CST
Q&A
10:25-10:55 CST
Coffee Break
10:55-11:25 CST
KPP fronts in the diffusive Rosenzweig-MacArthur model
Speaker: Anna Ghazaryan (Miami University)
We consider a diffusive Rosenzweig - MacArthur predator-prey model in the situation when the prey diffuses at the rate much smaller than that of the predator. Depending on the parameter regime, phenomenologically different types of fronts exist in this model. In particular, there exists a regime when the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and, according to the Geometric Singular Perturbation Theory, the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. It is of interest whether the stability of the fronts is also governed by the scalar Fisher-KPP equation. The techniques of the analysis include a construction of unstable augmented bundles and their treatment as multiscale topological structures. This is a joint project with Stephane Lafortune, Yuri Latushkin, and Vahagn Manukian.
11:25-11:35 CST
Q&A
11:35-11:40 CST
Tech Break
11:40-12:10 CST
TBD
Speaker: Matt Holzer (George Mason University)
12:10-13:20 CST
Q&A
12:20-13:20 CST
Lunch Break
13:50-14:20 CST
The impact of ephaptic coupling and ionic electrodiffusion on arrhythmogenesis in the heart
Speaker: Ning Wei (Purdue University)
14:20-14:30 CST
Q&A
14:30-15:00 CST
Coffee Break
15:00-15:30 CST
Spreading behavior of a two-species diffusion-competition system with free boundaries
Speaker: Chang-Hong Wu (National Yang Ming Chiao Tung University)
This talk will discuss the dynamics of two invading competitors using a two-species diffusion-competition model with two free boundaries. We aim to understand all possible invasion profiles of the system, where strong and weak competitors invade the spatial environment simultaneously. We show that this system has five distinct types of invasion profiles. This work is based on joint work with Professor Yihong Du (University of New England).
15:30-15:40 CST
Q&A
Friday, December 13, 2024
8:30-9:00 CST
Check-In and Breakfast
9:00-9:30 CST
The lottery regime in diffusion of knowledge models
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