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Events
Excited States in Nonlinear Algebra

Lettering: “Conference - Excited States in Nonlinear Algebra” in front of a green-orange color gradient

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

24.10.25

Excited States in Nonlinear Algebra

conference

24.10.25

Excited States in Nonlinear Algebra

This one-day workshop serves as a follow-up to the event "Nonlinear Algebra Meets Quantum Chemistry". It will feature five recent projects in Nonlinear Algebra, all inspired by challenges in Quantum Chemistry. The presentations are designed for a broad audience, and the final segment of the workshop will be dedicated to discussions about future developments. The primary goal is to foster collaboration between mathematicians and chemists in preparation for the Spring 2027 research semester titled "Numerical Algebraic Geometry and Correlated Electrons: Generalized Grassmannians, Response Functions, and Excited States," to be held at the Institute for Pure and Applied Mathematics (IPAM) in Los Angeles.

There will be a live stream of the lectures. Please register for receiving an email that contains all the information for following the talks via Zoom.

Speakers

Viktoriia Borovik

MPI MiS, Leipzig

Paul Breiding

University Osnabrück

Elke Neuhaus

MPI MiS, Leipzig

Svala Sverrisdóttir

MPI MiS, Leipzig

Julian Weigert

MPI MiS, Leipzig

Program

08:30 - 09:00	Welcome Coffee
09:00 - 10:00	Svala Sverrisdóttir The Fermi-Dirac algebra and Hamiltonians In quantum chemistry, many-electron states are represented as elements of an exterior algebra, the Fock space. The fermionic creation and annihilation operators generate the Fermi-Dirac algebra, which can be realized as a Clifford algebra acting on the Fock space. The elements of the Fermi–Dirac algebra act as endomorphisms of the Fock space. Among these are the two-body electronic Hamiltonian and the cluster operator. Their structure leads to identities that truncate the Baker-Campbell-Hausdorff expansion, thereby allowing us to express the coupled cluster equations as a polynomial system of degree at most four. Video (720p) Video (1080p)
10:00 - 10:30	Coffee Break
10:30 - 11:30	Paul Breiding Homotopy Iterators We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which tracks them along the homotopy. Doing so produces a representation of the target solutions, bypassing the need to hold all solutions in memory. We discuss several applications of this datatype ranging from solution counting to data compression. This is joint work with Taylor Brysiewicz and Hannah Friedman. Video (720p) Video (1080p)
11:30 - 12:30	Elke Neuhaus The Coupled Cluster Doubles Truncation Variety The quantum states arise from the cluster amplitudes via the exponential parametrization, given by the first column of the exponential of a lower triangular matrix T, representing the cluster operator. The coordinates of the cluster amplitudes describe the probability of different electrons leaving the reference state and can therefore be categorized into different levels, based on the number of moved electrons. For the sake of simplifying the electronic Schrödinger equation, one truncates the eigenvalue problem to only allow cluster amplitudes truncated to different levels. The quantum states arising in this way belong to the so-called truncation variety. We study this zero-set and its generators in small cases for truncation at level 2. Video (720p) Video (1080p)
12:30 - 13:30	Lunch
13:30 - 14:30	Julian Weigert Nonlinear Kalman varieties A common problem in quantum chemistry concerns finding eigenvectors of the Hamiltonian H. This matrix is usually very large and has to be approximated by smaller matrices when doing explicit computations. The eigenvectors of the Hamiltonian are constrained, meaning that quantum chemists know explicit equations that are fulfilled by some eigenvector of H. When approximating the matrix H arbitrarily, this additional information is usually lost due to the imprecise nature of the approximation of H. There are two ways to handle this issue: Either we relax the condition of "being an eigenvector the approximation" in trade of the condition that an "almost eigenvector" satisfies the constraints, or we choose the approximation of H in a clever way such that it has an eigenvector that exactly satisfies the constraints. In this talk we focus on the second approach which leads us to study nonlinear Kalman varieties. These are varieties of matrices which admit an eigenvector that satisfies certain nonlinear constraints. Inspired by the linear case, which was studied by Ottaviani and Sturmfels, we discuss the basic questions that an algebraic geometer might ask about these varieties, like dimension, degree and singularities. We also present a construction that produces explicit equations for Kalman varieties based on the minors of a large matrix with polynomial entries. Video (720p) Video (1080p)
14:30 - 15:00	Coffee Break
15:00 - 16:00	Viktoriia Borovik Numerical Algebraic Geometry for Energy Computations We consider an optimization problem arising in quantum chemistry. Given a real symmetric matrix H, called a Hamiltonian, and a projective variety V, we aim to minimize the Rayleigh-Ritz quotient of H over V. This corresponds to computing the ground state energy of a quantum system. The number of complex critical points of this optimization problem for a generic symmetric matrix H is known as the Rayleigh-Ritz degree of the variety V. Using a one-to-one parameterization, we develop numerical homotopy continuation methods to compute all RR-degree many critical points for algebraic varieties of tensors with bounded tensor train rank. We compare our approach to the well-known alternating least squares (ALS) method. This talk is based on joint work in progress with Hannah Friedman, Serkan Hoşten, and Max Pfeffer. Video (720p) Video (1080p)
16:00 - 17:30	Discussion
18:30 -	Conference Dinner at Nuva (self-paid)

Participants

George Balla

MPI MiS Leipzig

Viktoriia Borovik

MPI MiS, Leipzig

Paul Breiding

University Osnabrück

Max Buchholz

University Leipzig

Veronica Calvo Cortes

MPI MiS

Shelby Cox

MPI MiS

Claire de Korte

Max Planck Institute for Mathematics in the Sciences

Angelo El Saliby

MPI MiS

Vincenzo Galgano

MPI-CBG Dresden

Oskar Henriksson

MPI-CBG Dresden

Serkan Hosten

San Francisco State University

Yue Jiao

MPI-CBG

Paul Kalbitz

Universität Leipzig

Daria Kriuchkova

MIPT

Jiayi Li

MPI-CBG

Todd Martinez

Stanford University

Elke Neuhaus

MPI MiS, Leipzig

Niharika Paul

MPI MIS

Irem Portakal

MPI MiS leipzig

Andrea Rosana

MPI MiS, Leipzig

Flavio Salizzoni

MPI-MiS

Anton Schiela

Universität Bayreuth

Otto Theodor Primus Schmidt

MPI MiS Leipzig / BEC Pitaevskii Trento

Marisha Singh

Georg-August-Universität

Luca Sodomaco

Max Planck Institute for Mathematics in the Sciences

Martin Stoehr

Stanford University

Bernd Sturmfels

MPI MiS

Svala Sverrisdóttir

MPI MiS, Leipzig

Philippe Tranchida

Mpi Mis

Julian Weigert

MPI MiS, Leipzig

Laura Weigl

University of Bayreuth

Julia Westermayr

Leipzig University

Teresa Yu

MPI MiS Leipzig

Organizers

Luca Sodomaco

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail