# Student/Postdoc Working Geometry Seminar

The seminar meets Mondays at 5pm in Blocker 628.

Date Time |
Location | Speaker | Title – click for abstract | |
---|---|---|---|---|

01/1310:00am |
zoom | V. Batista Yale U. |
Tensor train methods for simulations of quantum dynamics and global optimizationWe introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) algorithm for simulations of multidimensional quantum dynamics [J. Chem. Theory Comput. 13: 4034-4042 (2017)]. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the wavepacket tensor components in arbitrary basis sets, bypassing the need of actually computing the wave function in its full-rank tensor product space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the interconversion dynamics of pyrazine after photoexcitation into an electronically excited state. Furthermore, we introduce the iterative power algorithm (IPA) [J. Chem. Theory Comput. submitted (2021)] for global optimization, including a formal proof of convergence for both discrete and continuous optimization problems. The IPA is essentially the imaginary time propagation method with infinite mass. It is based on the power recurrence relation ρk+1(r) = Uρk/|Uρk(r)| where U = e−V (r) is defined by the scaled potential energy surface V(r), and ρk(r) is the density distribution after the k-th optimization step. We show how to implement the IPA for high-dimensional potential energy surfaces by approximating ρ(r) and V (r) in terms of low-rank quantics tensor trains (QTT) generated by fast adaptive interpolation of multidimensional arrays. The resulting QTT implementation by-passes the curse of dimensionality and the need to evaluate V(r) at all possible values of r. We illustrate the capabilities of IPA as applied to the highly rugged potential energy surface V (r) = mod(N, r) in the space of primes r {2, 3, 5, 7, 11, . . . } folded as a d-dimensional 2_1 × 2_2 × · · · × 2_d tensor. We find that IPA resolves the degenerate global minima corresponding to the prime factors of numbers N with as many as 2,773 digits, enabling the solution of | |

01/2010:00am |
zoom | A. Casarotti Ferrara |
Defectiveness and Identifiability: a geometric point of view on tensor analysisIdentifiability problems arise naturally in many fields of mathematics, from the abstract world of birational geometry to the applied setup of tensor analysis. In this talk we link the identifiability property for a variety X to its secant behavior and the geometry of the tangential contact loci. In the first part, after reviewing the main properties of the tangential contact locus associated to h general points of X, we give a numerical bound under which the non h-secant defectiveness ensures the h-identifiability, with h subgeneric. Note that this result is of birational nature and so does not strictly depend on the geometry of the particular tensor variety we choose. Finally we apply our result to many classes of varieties which play a central role in tensor analysis. In the second part we move on the generic rank case, where a clever use of the infinitesimal Bertini's theorem and an implementation of Noether-Fano's inequalities enable us to link the generic identifiability with the infinitesimal tangential contact locus. This finally shows the non generic identifiability for many partially symmetric tensors satisfying a mild numerical bound on their dimensions and degrees | |

02/0310:00am |
zoom | P. Buergisser TU Berlin |
TBA | |

02/1010:30am |
zoom | Elina Robeva UBC |
Orthogonal decomposition of tensor trainsTensor decomposition has many applications. However, it is often a hard problem. Orthogonally decomposable tensors form are a small subfamily of tensors and retain many of the nice properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. The decomposition of such tensors can be found efficiently, their eigenvectors can be computed efficiently, and the set of orthogonally decomposable tensors of low rank is closed and can be described by a set of quadratic equations. One of the issues with orthogonally decomposable tensors, however, is that they form a very small subset of the set of all tensors. We expand this subset and consider orthogonally decomposable tensor trains. These are formed by placing an orthogonally decomposable tensor at each of the vertices of a tensor train, and then contracting. We give algorithms for decomposing such tensors both in the setting where the tensors at the vertices of the tensor train are symmetric and non-symmetric. This purely theoretical work is based on joint work with Karim Halaseh and Tommi Muller. | |

02/1710:15am |
zoom | Thomas Barthel Duke U. Physics |
Tensor-network simulations of quantum matter and entanglementThe non-locality of quantum many-body systems can be quantified by entanglement
measures. One finds that the entanglement in most states occurring in nature is
far below the theoretical maximum. Hence, it is possible to describe such
systems with a reduced set of effective degrees of freedom. This is exploited
in simulation techniques based on tensor network states. I will discuss
corresponding algorithms employing different types of tensor networks: matrix
product states (MPS), projected entangled-pair states (PEPS), and the
multiscale entanglement renormalization ansatz (MERA). The computation costs
are intimately related to the scaling of entanglement entropies in the
simulated systems, for which I will give a short overview.
| |

02/189:30pm |
zoom | JM Landsberg TAMU |
Gauss maps and secant varieties via differential geometry IThis will be a sequence of 2 expository talks on Gauss maps and secant varieties. After a brief history, I'll give the proofs that (i) projective varieties with degenerate Gauss maps are singular (ii) Zak's theorem on tangencies, (iii) Zak's theorem on linear normailty, (iv) interesting tensors associated to varieties with degenerate secant varieties. The talks will also serve as an introduction to the moving frame. | |

02/2410:15pm |
zoom | JM Landsberg TAMU |
Secant varieties IISlides/recording of part I available upon request. |