Israel Michael Sigal

Information for Students

Presently I am working in the following areas:

1. Quantum Field Theory
   Consider a quantum system, such as a hydrogen atom, which is originally in an excited state. We expect such a system to emit photons and descend into its ground state. This is the process of radiation which in particular produces the light we see. We would like to develop mathematical understanding of the dynamics of this and similar processes.
2. Nonlinear Partial Differential Equations of Quantum Physics
   Consider (non-integrable) evolution equations which have single soliton solutions. Examples of such equations are nonlinear Schrödinger, Ginzburg-Landau, Landau-Lifshitz and Heisenberg ferromagnet equations to name just few. One would like to understand solutions of such equations in a presence of inhomogeneities and/or describing several (interacting) solitons.
3. Mathematical Biology
   Here we are interested in aggregation phenomena for living organisms such as cells and bacteria (e.g. Chemotaxis) and for proteins as well as in mathematical models of functions of a cell.
4. Pattern Analysis
   Given an image function, one of the key problems is a decomposition of the image domain into segments corresponding to separate objects or homogenious parts of objects. This is the segmentation problem. Further, one would like to understand which objects are up front and which in the back, and so on. An eye performs such tasks extremely efficiently. One would like to design mathematical models and develop related algorithms allowing computers to perform such tasks as well.
5. Non-equilibrium Statistical Mechanics
   Quantum systems are never isolated but they interact with their environments. The latter can be assumed to be near their states of (local) equilibrium. The problem here is to describe the dynamics of such a coupled system. One would like to see in mathematically rigorous way how dynamics of isolated quantum systems is affected by interaction with its environment.

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