Ravishankar Sundararaman (Rensselaer Polytechnic Institute) Functional approximations for Density-Functional Theory Sponsored by the Meyer Fund
Wed, May. 1 3pm (MATH 350)
Topology
Ravishankar Sundararaman (Rensselaer Polytechnic Institute) Functional approximations for Density-Functional Theory Sponsored by the Meyer Fund
Wed, May. 1 4pm (MATH 350)
Slow Pitch
Yu Wang (CU Boulder) From symplectic manifolds to Kahler manifolds
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Density-Functional Theory (DFT) is the workhorse for computational chemistry and materials science. It relies on exactly mapping intractable many-body problems, which involve computing energies in terms of (classical or quantum) configurations of several particles, on to a problem that involves only the much lower-dimensional density of particles. Practical DFT requires a computable approximate functional that maps density to energy with sufficient accuracy.
In this talk, I will review the general mathematical formalism of DFT, covering both the ubiquitous (quantum) electronic DFT and the less well-known classical DFT of liquids. I will outline the hierarchy of functional approximations that have proven successful in both these fields and discuss the challenge of capturing non-local correlations accurately. I will end with a hypothesis: an old technique in classical DFT named the Percus rank-2 approximation that quickly became analytically intractable, combined with modern machine learning methods, may hold the key to accurate non-local functionals for both quantum and classical DFT.
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Density-Functional Theory (DFT) is the workhorse for computational chemistry and materials science. It relies on exactly mapping intractable many-body problems, which involve computing energies in terms of (classical or quantum) configurations of several particles, on to a problem that involves only the much lower-dimensional density of particles. Practical DFT requires a computable approximate functional that maps density to energy with sufficient accuracy.
In this talk, I will review the general mathematical formalism of DFT, covering both the ubiquitous (quantum) electronic DFT and the less well-known classical DFT of liquids. I will outline the hierarchy of functional approximations that have proven successful in both these fields and discuss the challenge of capturing non-local correlations accurately. I will end with a hypothesis: an old technique in classical DFT named the Percus rank-2 approximation that quickly became analytically intractable, combined with modern machine learning methods, may hold the key to accurate non-local functionals for both quantum and classical DFT.
X
Symplectic manifolds are a special kind of manifolds which have rich applications in physics and mathematics. Roughly speaking, a symplectic manifold is a smooth manifold equipped with a symplectic form, which makes mathematicians only need to study global properties of the manifold. Combining with a compatible almost complex structure, it will give a math version of mirror symmetry conjecture from string theory. Most modern geometry are built up on Kahler manifolds and Calabi-Yau manifolds that can be gotten from giving more analytic structures to a symplectic manifold. In the talk, I am going to roughly talk about basic properties of symplectic manifolds, complex manifolds and Kahler manifolds. Eventually, I am gonna mention a very famous theorem called Hodge decomposition of Kahler manifolds and a famous conjecture named “Homological Mirror Symmetry”.