For a full description of my research progress, goals and future directions, see my research statement here.

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Research Projects

• Snaking bifurcations in the Brusselator
  
  I study the existence of defect solutions (1-dimensional spiral and target waves) in reaction diffusion equations. Using geometric methods I have shown that defect solutions exist under broad assumptions and that these solutions undergo a version of homoclinic snaking bifurcation. Moreover, I have qualitative and quantitative descriptions of the resulting bifurcation diagrams. 
  
  Keywords: spatial dynamical systems, pattern formation, reaction diffusion equations.
   

• Self-assembly of amphiphilic copolymers
  
  I study the existence of radially symmetric patterns in the formation of amphiphilic copolymers. Amphiphilic copolymers have properties of both hydrophobic and hydrophilic chemicals, and are self-assembling (ie. relatively easy to produce). As such, they have many industrial and medical uses. Importantly, their properties depend on the arrangement of their constituent parts and so there is interest in predicting their geometry from system parameters at onset of assembly.
  
  I have used geometric singular perturbation theory (GSPT) to prove the existence of radially symmetric structures and to obtain leading order expressions for the pattern radius in terms of system parameters.
  
  Keywords: GSPT, Cahn-Hilliard, pattern formation
   

• Spectral stability of fronts via the Riccati-Evans function 
  
  Spectral stability of traveling fronts in reaction-advection-diffusion equations using new geometrically inspired tools. We cast the eigenvalue problem of traveling fronts as a related nonlinear problem on the Grassmannian manifold. Through this, I have implemented a simple and efficient method for numerically computing the location of point spectra for traveling waves.
  
  Keywords: Spectral problem, Riccati methods
   

• Thermoregulation in a model for PO/AH neurons
  
  The pre optic area and anterior hypothalamus (PO/AH) is thought to be an integrative center of thermoregulation in the body, ie. it is responsible for responding to changes in body temperature and environment to stop the body overheating or overcooling. I studied a Hodgkin-Huxley style model for PO/AH neurons to determine how their behavior changes with temperature. I used numerical bifurcation techniques and explained the results GSPT.
  
  Keywords: GSPT, mathematical physiology

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Preprints

1. T Roberts, Bjorn Sandstone. Homoclinic snaking of contact defects in reaction-diffusion equations. [archive]

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​Publications

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1. Kristen E. Harley, Peter van Heijster, Robert Marangell, Graeme J. Pettet, Timothy V. Roberts,and Martin Wechselberger. (In)stability of Travelling Waves in a Model of Haptotaxis. SIAM Journal on Applied Mathematics, 80(4):1629–1653, January 2020.