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Alexander Bihlo
CRM, U Montreal and University of Vienna
Mon 11 Jun 2012, 3:00pm SPECIAL
Symmetries and Differential Equations Seminar
Math Annex 1118
Invariant turbulence modeling
Math Annex 1118
Mon 11 Jun 2012, 3:00pm-4:00pm


Symmetries are among the most successfully employed concepts in science and mathematics. They form a cornerstone of various branches of physics, such as classical and quantum mechanics, particle physics and relativity. The governing equations of hydrodynamics generally possess wide symmetry
groups and therefore there is a great potential to exploit these symmetries so as to derive similarity solutions, conservation laws and invariants or to study the effects of symmetry breaking due to the
presence of boundaries or additional body-forces. However, to date,symmetries are often used in a non-explicit or indirect way in hydrodynamics and turbulence theory. On the other hand, there exist
powerful and general methods introduced in the field of group analysis of differential equations which, when suitably adapted, can be readily applied to the aforementioned fields. In this talk we will introduce an algorithmic method which allows associating with a given object its invariant counterpart. The object
under consideration can be, e.g., a turbulence closure model or a finite-difference discretization of a differential equation, which can then be invariantized to yield a turbulence model or a finite-difference
discretization that is invariant under the same Lie point symmetry group as admitted by the original governing equations of hydrodynamics. This method can therefore be used to correct artificial symmetry breaking due to non-appropriately designed turbulence models. As an example it is shown that classical hyperdiffusion as used in two-dimensional (decaying) turbulence simulations violates the symmetries of the incompressible Euler equations. Invariantization of these hyperdiffusion terms yields symmetry-preserving but nonlinear diffusion-like terms. Using the notion of differential invariants it is
demonstrated that the invariantized hyperdiffusion models can be modified with quite some flexibility while still preserving their desired invariance characteristics. First numerical tests show that the invariant hyperdiffusion schemes which can be obtained by this method might be able to reproduce the -3 slope of the energy spectrum in the enstrophy inertial range.
David Steinberg
Fri 15 Jun 2012, 12:30pm SPECIAL
Graduate Student Center, Room 203
Doctoral Exam
Graduate Student Center, Room 203
Fri 15 Jun 2012, 12:30pm-3:00pm


Abstract: The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be
equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but Bryan, Cadman, and Young have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold. The crepant resolution conjecture of Bryan, Cadman, and Young gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland.