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Harvard University

1.
Brantner, David Lukas Benjamin.
The Lubin-Tate Theory of *Spectral* *Lie* * Algebras*.

Degree: PhD, 2017, Harvard University

URL: http://nrs.harvard.edu/urn-3:HUL.InstRepos:41140243

We use equivariant discrete Morse theory to establish a general technique in poset topology and demonstrate its applicability by computing various equivariant properties of the partition complex and related posets in a uniform manner. Our technique gives new and purely combinatorial proofs of results on algebraic and topological André-Quillen homology. We then carry out a general study of the relation between monadic Koszul duality and unstable power operations. Finally, we combine our techniques to compute the operations which act on the homotopy groups K(n)-local Lie algebras over Lubin-Tate space.

Mathematics

Subjects/Keywords: Morava E-theory; Lubin-Tate space; spectral Lie algebras; poset topology; discrete Morse theory; Andre-Quillen homology; monoids; Koszul duality

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Brantner, D. L. B. (2017). The Lubin-Tate Theory of Spectral Lie Algebras. (Doctoral Dissertation). Harvard University. Retrieved from http://nrs.harvard.edu/urn-3:HUL.InstRepos:41140243

Chicago Manual of Style (16^{th} Edition):

Brantner, David Lukas Benjamin. “The Lubin-Tate Theory of Spectral Lie Algebras.” 2017. Doctoral Dissertation, Harvard University. Accessed January 20, 2020. http://nrs.harvard.edu/urn-3:HUL.InstRepos:41140243.

MLA Handbook (7^{th} Edition):

Brantner, David Lukas Benjamin. “The Lubin-Tate Theory of Spectral Lie Algebras.” 2017. Web. 20 Jan 2020.

Vancouver:

Brantner DLB. The Lubin-Tate Theory of Spectral Lie Algebras. [Internet] [Doctoral dissertation]. Harvard University; 2017. [cited 2020 Jan 20]. Available from: http://nrs.harvard.edu/urn-3:HUL.InstRepos:41140243.

Council of Science Editors:

Brantner DLB. The Lubin-Tate Theory of Spectral Lie Algebras. [Doctoral Dissertation]. Harvard University; 2017. Available from: http://nrs.harvard.edu/urn-3:HUL.InstRepos:41140243

University of Cambridge

2. Singh, Pranav. High accuracy computational methods for the semiclassical Schrödinger equation.

Degree: PhD, 2018, University of Cambridge

URL: https://www.repository.cam.ac.uk/handle/1810/274913

The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings.
In this thesis we follow an alternative strategy{-}we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator.
This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the ℤ_{2}-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability.
The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of it{asymptotic splitting:} exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy.
Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus – Zassenhaus schemes{-}one where the integrals are discretised using Gauss – Legendre quadrature at the outset and another where integrals are preserved throughout.
These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus – Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices.
An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

Subjects/Keywords: Semiclassical Schrödinger equations; time-dependent potentials; exponential splittings; Zassenhaus splitting; Magnus expansions; Lanczos iterations; Magnus – Zassenhaus schemes; commutator free; high-order methods; asymptotic analysis; Lie algebras; Jordan polynomials; symmetrised differential operators; spectral collocation

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Singh, P. (2018). High accuracy computational methods for the semiclassical Schrödinger equation. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/274913

Chicago Manual of Style (16^{th} Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Doctoral Dissertation, University of Cambridge. Accessed January 20, 2020. https://www.repository.cam.ac.uk/handle/1810/274913.

MLA Handbook (7^{th} Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Web. 20 Jan 2020.

Vancouver:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Internet] [Doctoral dissertation]. University of Cambridge; 2018. [cited 2020 Jan 20]. Available from: https://www.repository.cam.ac.uk/handle/1810/274913.

Council of Science Editors:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Doctoral Dissertation]. University of Cambridge; 2018. Available from: https://www.repository.cam.ac.uk/handle/1810/274913

University of Cambridge

3. Singh, Pranav. High accuracy computational methods for the semiclassical Schrödinger equation.

Degree: PhD, 2018, University of Cambridge

URL: https://www.repository.cam.ac.uk/handle/1810/274913 ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691

The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy{-}we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the ℤ_{2}-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of it{asymptotic splitting:} exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus – Zassenhaus schemes{-}one where the integrals are discretised using Gauss – Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus – Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

Subjects/Keywords: Semiclassical Schro¨dinger equations; time-dependent potentials; exponential splittings; Zassenhaus splitting; Magnus expansions; Lanczos iterations; Magnus – Zassenhaus schemes; commutator free; high-order methods; asymptotic analysis; Lie algebras; Jordan polynomials; symmetrised differential operators; spectral collocation

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Singh, P. (2018). High accuracy computational methods for the semiclassical Schrödinger equation. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/274913 ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691

Chicago Manual of Style (16^{th} Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Doctoral Dissertation, University of Cambridge. Accessed January 20, 2020. https://www.repository.cam.ac.uk/handle/1810/274913 ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691.

MLA Handbook (7^{th} Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Web. 20 Jan 2020.

Vancouver:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Internet] [Doctoral dissertation]. University of Cambridge; 2018. [cited 2020 Jan 20]. Available from: https://www.repository.cam.ac.uk/handle/1810/274913 ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691.

Council of Science Editors:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Doctoral Dissertation]. University of Cambridge; 2018. Available from: https://www.repository.cam.ac.uk/handle/1810/274913 ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691