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I'm a mathematician with main reasearch in Global Analysis with applications to mathematical physics. My main interest are symmetries of infinite dimensional dynamical systems, i.e. partial differential equations (PDEs) with an infinite number of degrees of freedom. Ex. the equations of hydrodynamics (Euler, KdV, Navier Stokes), of electromagnetism (Maxwell) of plasma physics (Maxwell-Vlasov) , of general relativity (Einstein), of quantum field theories , QED, QCD (Yang-Mills), of string theories (Witten, Chern-Simon) Symmetry is the essential organizing principle in physics, it leads to conserved quantities and a reduction of number of degrees of freedom. eg. invariance under time translation (relativity!) --> conservation of energy invariance under space translation (classical mechanics) --> conservation of linear momentum invariance under space rotation --> conservation of angular momentum momentum gauge invariance of Maxwell's equation --> charge conservation without Lorentz invariance general relativity does not work gauge invariance of Yang-Mills equations --> current conservation internal symmetries in quantum field theories SU(3)xSU(2)xU(1) --> quark flavor, color, spin super symmetry --> Higgs mechanism how elementary particle require mass broken symmetries --> solutions are not invariant although the equations are (how does the universe go from an undifferentiated point to the complex structure of today)
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