Faculty and graduate students in the Department of Applied and Computational Mathematics and Statistics (ACMS) are always involved in dynamic research projects that enable connections on and off campus.

Integrating Multiscale Modeling and in vivo Experiments for Studying Blood Clot Development
 

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The overall goal of this project is to develop three-dimensional multi-scale mathematical models and a computational toolkit for simulating thrombus formation. These models will be validated with specifically designed experiments to test predictions of thrombus development, structure, and stability. Moreover, the development of reasonable models will serve as a generator of new hypotheses that can be tested in experiments in vivo.

 

Study of the Interplay of Motility Mechanisms During Bacterial Swarming
 

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Many bacteria use motility described as swarming to colonize surfaces in groups that allows them to survive external stresses, including exposure to antibiotics. The main goal of this interdisciplinary project is to combine simulations using new, three-dimensional multiscale modeling environments and specifically designed experiments to study the basic coordination events of bacterial swarming, which is essential to understanding how millions of bacteria function in real environments.

 

Cellular Organization, Computational and Experimental Studies of Microtubule Dynamics and Regulation by Binding Proteins
 

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The long-term goal of this project is to develop a predictive and quantitative understanding of the MT cytoskeleton and its regulation by MTBPs, which will impact fields ranging from systems biology to nanotechnology. The flexible model and tutorials developed through this project will allow researchers to develop and test specific hypotheses about the mechanisms of dynamic instability and MTBP action, which will in turn help design and direct future experiments.

 

 

Longitudal Analysis and Modeling of Large-Scale Social Networks
 

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The data generated by digital communication technologies will be used to:

  1. Test/validate existing social network theories about the mechanism's underlying network dynamics by developing quantitative high-fidelity temporal stochastic models of human behavior within social networks.
  2. Produce a data-driven, dynamic network modeling suite with prediction capabilities.
 

 

Clinical Prognostic Test for Metastasis in Breast Cancer
 

Many breast cancer patients remain free of distant metastasis even without adjuvant chemotherapy. While standard histopathological tests fail to identify these good prognosis patients with adequate precision, analyses of gene expression patterns in primary tumors have resulted in successful diagnostic tests. The accelerated progression relapse test, developed by Buechler (ACMS) using whole-genome microarrays, is one such test, however it requires frozen or fresh-preserved tissue samples. The project includes development of a version of this test that can utilize the tissue source (formalin-fixed, paraffin-embedded) standard in clinical use.

 

 

Wound Healing Modeling
 

The overall goal is to study and understand the behavior of the wound healing process using dynamical system and PDE model through analytical and numerical tools. This research is featured here on SIAM.org.


Collaborators

  • Dr. Avner Friedman (Mathematical Biosciences Institute, Ohio State University)
  • Dr. Bei Hu (ACMS, Notre Dame)
  • Xue Chuan (Postdoc, Mathematical Biosciences Institute, Ohio State University)

 

Development of Bertini
 

Bertini is a software package for computation and manipulation of the solution sets of polynomial systems.


Collaborators

  • Dr. Daniel Bates (Mathematics, Colorado State University)
  • Dr. Jonathan Hauenstein (ACMS, Notre Dame)
  • Dr. Andrew Sommese (ACMS, Notre Dame)
  • Charles Wampler (General Motors R&D, Warren, Michigan)

 

Solution of Systems of Differential Equations Arising in Pattern Formation and Tumor Growth Models
 

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Collaborators

  • Wenrui Hao, Yuan Liu, and Timothy McCoy (Former ACMS Graduate Students, Notre Dame)
  • Dr. Jonathan Hauenstein (ACMS, Notre Dame)
  • Drs. Bei Hu, Andrew Sommese, and Yongtao Zhang (ACMS, Notre Dame)
 

 

Application of Numerical Algebraic Geometry to Problems From the Theory of Mechanisms
 

Collaborators

  • Dr. Jonathan Hauenstein (ACMS, Notre Dame)
  • Dr. Andrew Sommese (ACMS, Notre Dame)
  • Charles Wampler (General Motors R&D, Warren, Michigan)

 

Combination and Comparison of Numerical and Symbolic Methods for Algebraic Computation
 

Collaborators

  • Dr. Daniel Bates (Mathematics, Colorado State University)
  • Dr. Wolfram Decker (Mathematics, University of Kaiserslautern)
  • Dr. Jonathan Hauenstein (ACMS, Notre Dame)
  • Chris Peterson (Mathematics, Colorado State University)
  • Dr. Gerhard Pfister (Mathematics, University of Kaiserslautern)
  • Dr. Andrew Sommese (ACMS, Notre Dame)
  • Dr. Frank-Olaf Schreyer (Computer Science, University of Saarlandes)
 
 

Application of Numerical Methods to Carry Out Algebraic Geometry Calculations
 

Collaborators

  • Dr. Daniel Bates (Mathematics, Colorado State University)
  • Timothy McCoy (Former ACMS Graduate Student, Notre Dame)
  • Chris Peterson (Mathematics, Colorado State University)
  • Dr. Andrew Sommese (ACMS, Notre Dame)

 

Numerical Algorithms to Do Real Algebraic Computations
 

Collaborators

  • Sandra Di Rocco and David Eklund (Royal Institute of Technology, Stockholm)
  • Dr. Jonathan Hauenstein (ACMS, Notre Dame)
  • Dr. Andrew Sommese (ACMS, Notre Dame)
  • Charles Wampler (General Motors R&D, Warren, Michigan)

 

Multiscale Stochastic Model of Swarming Pattern Formation of Pseudomonas Aeruginosa
 

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Pseudomonas aeruginosa is an opportunistic pathogenic bacterium that often, but not always, forms branched tendril patterns during swarming. This physical phenomena occurs only when bacteria produce rhamnolipid, which is regulated by intercellular quorum sensing signaling. Here we report upon our experimental findings of a new behavior of the bacterium P. aeruginosa, an ability to alter its local physical environment by forming and propagating high density waves of cells and rhamnolipid.

 

 

Multiscale Stochastic Model of Bruising
 

This project involves the development of modeling and experimental approaches to detect the nature and aging of a bruise on children, especially in abusive situations.


Collaborators

  • Drs. Mark Alber, Zhiliang Xu, and Gregory Crawford (ACMS and College of Science Dean, Notre Dame)

 

Development of High-Order Accurate, Efficient Discontinuous Galerkin and Limiting Methods for Solving Nonlinear Hyperbolic Conservation Laws (Euler Equations)
 

Collaborators

  • Dr. Zhiliang Xu (ACMS, Notre Dame)
  • Dr. Yingjie Liu (Mathematics, Georgia Institute of Technology)
  • Dr. Chi-Wang Shu (Applied Mathematics, Brown University)

 

Development of High-Order Accurate, Efficient Finite Volume/Element Methods for Magnetohydrodynamics
 

The focus is to develop efficient high-order time-stepping methods, limiting methods, and divergence reconstruction methods.


Collaborators

  • Dr. Dinshaw Balsara (Physics, Notre Dame)
  • Dr. Zhiliang Xu (ACMS, Notre Dame)
  • Guang Lin (Fundamental & Computational Sciences, Pacific Northwest National Laboratory)
  • Huijing Du (Former ACMS graduate student, Notre Dame)

 

High Order Numerical Methods for PDEs on Complex Domains
 

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The goal of this project is to develop efficient and robust high order accuracy numerical methods for solving hyperbolic conservation laws, Hamilton-Jacobi equations and stiff advection-reaction-diffusion equations defined on complex domains. The methods include Weighted ENO methods, discontinuous Galerkin methods, fast sweeping methods and high order time-stepping implicit integration factor methods.

 

 

Computational Analysis of Morphogenesis
 

The goal of this project is to develop computational models to study the tissue patterning during Drosophila (fruit fly) embryo development. Specifically, we study the robustness of multi-protein networks which regulate the morphogen gradient formation.


Collaborators

  • Dr. Arthur Lander (Developmental and Cell Biology and Biomedical Engineering, UC Irvine)
  • Dr. Qing Nie (Mathematics and Biomedical Engineering, UC Irvine)
  • Dr. Frederic Wan (Mathematics and Mechanical and Aerospace Engineering, UC Irvine)
  • Dr. Yongtao Zhang (ACMS, Notre Dame)