**Email:** lastname at berkeley dot edu

**Office:** 1089 Evans Hall

**Background:** I am an assistant professor in the math department at UC Berkeley.
Here is my CV (8/2023).

**Research interests:** I work on computational methods driven by numerical linear algebra, optimization, and randomization, with a special focus on
high-dimensional scientific computing problems. These include quantum many-body
problems arising in quantum chemistry and condensed matter physics, as well as various problems in applied probability. Approaches draw on a wide variety of
techniques including semidefinite relaxation, Monte Carlo sampling, and optimization over parametric function classes such as tensor networks and neural networks.

For an introduction to quantum many-body physics written for an applied math audience, see
my dissertation (8/2019) and in particular the first part.
Also see this talk for a tutorial on many-body perturbation theory and Green's
function methods.

**News: ** I am a 2024 Sloan Research Fellow.

Past semesters: [Fall 2023]

** Background.** For many intractable optimization problems, it is possible to formulate
a convex relaxation by making use of simple structure in the objective function. Examples range from combinatorial optimization problems (such as the famous Max-Cut problem)
to the problem of determining the ground state energy of a quantum many-body system, which is of fundamental importances in
quantum chemistry and condensed matter physics. Relaxations are derived by writing down necessary constraints satisfied by
some reduced variable in the original problem. Often the resulting problem has the structure of
a semidefinite program (e.g., the Goemans-Williamson algorithm for Max-Cut and the 2-RDM theories of quantum chemistry).

*Pictured*:
A partition of a lattice system into disjoint fragments.

** Background.** The fundamental difficulty of quantum
many-body problems is that the complexity of the quantum wavefunction grows exponentially
with the number of particles in the system. Recently, quantum embedding theories have made
progress in addressing this difficulty by dividing systems into smaller fragments, which
can themselves be treated with an accurate 'high-level' theory, perhaps even implemented on
a quantum computer! These local problems are coupled together
self-consistently via some coarse global physical quantities.

*Pictured*:
The Rayleigh-Gauss-Newton (RGN) method exhibits fast
convergence for the 1D transverse-field Ising model, maintained through the quantum phase transition at *h*=1.

** Background.** Many scientific computing problems can be viewed as optimization problems
over functions on high-dimensional state spaces (discrete or continuous). Such problems include the ground-state
eigenvalue problem for quantum many-body problems, as well as the problem of computing committor functions to
characterize reaction pathways in computational chemistry. A well-chosen parametric class of functions
(such as a tensor network or neural network parametrization) can make such an optimization problem
tractable.

*Pictured*:
Mean-field dynamics for an ensemble sampler.

Monte Carlo sampling is the most flexible and widely-used tool for estimating sums and integrals in high dimensions.
It plays a supporting role in Variational Monte Carlo (see __High-dimensional functions__ above), where
efficient sampling is crucial for wavefunction optimization. Conversely, parametric function classes can support the goal of Monte Carlo sampling. Some results in this area
should be coming soon!

I am also interested in generic sampling problems. We introduced an
ensemble MCMC sampler that can circumvent slow
mixing due to metastability.

*Pictured*:
Bold Feynman diagram expansion for the GW approximation to the Luttinger-Ward functional.

** Background.**
Beyond the single-particle picture of density functional theory (DFT), many of the most
widely used computational methods in quantum many-body physics are based on the formalism
of Green's functions, which encode response and excitation properties of a system. Many-body
perturbation theory (MBPT), which expands quantities of interest perturbatively in
the strength of the inter-particle interaction, defines a major category of approaches to computing
Green's functions. The terms in this expansion correspond to the so-called
bare Feynman diagrams. These can in turn be reorganized into a series of bold Feynman diagrams.
These objects can be understood in terms of a construction called the Luttinger-Ward (LW) functional.
But the construction is only formal, and recently the existence
of a single-valued LW functional has been cast into doubt by theoretical and numerical evidence.

**Scalable variational embedding for quantum many-body problems**

SIAM Conference on Mathematical Aspects of Materials Science (May 2021)

[slides]

**Optimization for variational Monte Carlo with neural quantum states**

Modeling and Simulation Group Seminar: Machine Learning in Science at NYU (April 2021)

[slides]

**Optimal transport via a Monge-Ampère optimization problem**

SIAM Conference on the Mathematics of Data Science (May 2020)

[slides]

**Toward sharp error analysis of extended Langrangian molecular dynamics for polarizable force field simulation**

Ki-Net Young Researchers Workshop (October 2019)

[slides]

**Semidefinite relaxation of multi-marginal optimal transport, with application to strictly correlated electrons in second quantization**

ICIAM (July 2019)

[slides]

**A classical statistical mechanics approach to understanding Green's function methods and the Luttinger-Ward formalism**

Oberwolfach (March 2018)

[slides]

**Adaptive compression for Hartree-Fock-like equations**

SIAM Conference on Applied Linear Algebra (May 2018)

[slides]

**Efficient quantum trace estimation with reconfigurable real-time circuits
**

Yizhi Shen, Katherine Klymko, Eran Rabani, Norm M. Tubman, Daan Camps, Roel Van Beeumen, and Michael Lindsey

[arXiv:2401.04176]

**Multiscale interpolative construction of quantized tensor trains
**

Michael Lindsey

[arXiv:2311.12554]

**Multimarginal generative modeling with stochastic interpolants
**

with Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden

[arXiv:2310.03695]

**Nested gausslet basis sets
**

Steven R. White and Michael Lindsey

*J. Chem. Phys.* 159, 234112 (2023)

[journal]
[arXiv:2309.10704]

**Fast randomized entropically regularized semidefinite programming
**

Michael Lindsey

[arXiv:2303.12133]

**Understanding and eliminating spurious modes in variational Monte Carlo using collective variables**

Huan Zhang, Robert J. Webber, Michael Lindsey, Timothy C. Berkelbach, and Jonathan Weare

*Phys. Rev. Research* 5, 023101 (2023)

[journal]
[arXiv:2211.09767]

**Non-Hertz-Millis scaling of the antiferromagnetic quantum critical metal via scalable Hybrid Monte Carlo **

Peter Lunts, Michael S. Albergo, and Michael Lindsey

*Nat. Commun.* 14, 2547 (2023)

[journal]
[arXiv:2204.14241]

**Generative modeling via tensor train sketching**

YoonHaeng Hur, Jeremy G. Hoskins, Michael Lindsey, E.M. Stoudenmire, and Yuehaw Khoo

*Appl. Comput. Harmon. Anal.* 67, 101575 (2023)

[journal]
[arXiv:2202.11788]

**Committor functions via tensor networks**

with Yian Chen, Jeremy Hoskins, and Yuehaw Khoo

*J. Comput. Phys.* 472, 111646 (2023)

[journal]
[arXiv:2106.12515]

**Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo**

Robert J. Webber and Michael Lindsey

*Phys. Rev. Research* 4, 033099 (2022)

[journal]
[arXiv:2106.10558]

**Ensemble Markov chain Monte Carlo with teleporting walkers**

with Jonathan Weare and Anna Zhang

*SIAM/ASA JUQ* 10, 860 (2022)

[journal] [arXiv:2106.02686]

**Scalable semidefinite programming approach to variational embedding for quantum many-body problems**

with Yuehaw Khoo

[arXiv:2106.02682]

**Multiscale semidefinite programming approach to positioning problems with pairwise structure**

with Yian Chen and Yuehaw Khoo

[arXiv:2012.10046]

**Towards sharp error analysis of extended Lagrangian molecular dynamics**

with Dong An and Lin Lin

*J. Comput. Phys.* 466, 111403 (2022)

[journal]
[arXiv:2010.07508]

**Enhancing robustness and efficiency of density matrix embedding theory via semidefinite programming and local correlation potential fitting**

Xiaojie Wu, Michael Lindsey, Tiangang Zhou, Yu Tong, and Lin Lin

*Phys. Rev. B* 102, 085123 (2020)

[journal]
[arXiv:2003.00873]

(Note: Editor's Suggestion.)

**Variational embedding for quantum many-body problems**

with Lin Lin

*Comm. Pure Appl. Math.* 75, 2033 (2022)

[journal] [arXiv:1910.00560]

**Efficient hybridization fitting for dynamical mean-field theory via semi-definite relaxation**

Carlos Mejuto-Zaera, Leonardo Zepeda-Núñez, Michael Lindsey, Norm Tubman, Birgitta Whaley, and Lin Lin

*Phys. Rev. B* 101, 035143 (2020)

[journal]
[arXiv:1907.07191]

**Semidefinite relaxation of multi-marginal optimal transport for strictly correlated electrons in second quantization**

with Yuehaw Khoo, Lin Lin, and Lexing Ying

*SIAM J. Sci. Comput.* 42, B1462 (2020)

[journal]
[arXiv:1905.08322]

**Projected density matrix embedding theory with applications to the two-dimensional Hubbard model**

Xiaojie Wu, Zhi-Hao Cui, Yu Tong, Michael Lindsey, Garnet Kin-Lic Chan, and Lin Lin

*J. Chem. Phys.* 151, 064108 (2019)

[journal]
[arXiv:1905.00886]

**Sparsity pattern of the self-energy for classical and quantum impurity problems**

with Lin Lin

*Ann. Henri Poincaré* 21, 2219 (2020)

[journal]
[arXiv:1902.04796]

**Bold Feynman diagrams and the Luttinger-Ward formalism via Gibbs measures. Part II: Non-perturbative analysis**

with Lin Lin

*Arch. Ration. Mech. Anal.* 242, 527 (2021)

[journal]
[arXiv:1809.02901]

**Bold Feynman diagrams and the Luttinger-Ward formalism via Gibbs measures. Part I: Perturbative approach**

with Lin Lin

*Arch. Ration. Mech. Anal.* 242, 581 (2021)

[journal]
[arXiv:1809.02900]

**Variational structure of Luttinger-Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory**

with Lin Lin

*Proc. Natl. Acad. Sci.* 115, 2282 (2018)

[journal]
[.pdf] [supporting info]

**Convergence of adaptive compression methods for Hartree-Fock-like equations**

with Lin Lin

*Comm. Pure Appl. Math.* 72, 451 (2019)

[journal] [arXiv:1703.05441]

**Optimal transport via a Monge-Ampère optimization problem**

with Yanir A. Rubinstein

*SIAM J. Math. Anal.* 49, 3073 (2017)

[journal] [.pdf]

(Note: recognized with the
SIAM Student Paper Prize.)

**On discontinuity of planar optimal transport maps**

with Otis Chodosh, Vishesh Jain, Lyuboslav Panchev, and Yanir A. Rubinstein

*Journal of Topology and Analysis* 07, 239 (2015)

[journal] [arXiv:1312.2929]

(Note: research carried out during
SURIM 2012.)

**Infrared imagery of streak formation in a breaking wave**

Ivan Savelyev, Robert A. Handler, and Michael Lindsey

*Physics of Fluids* 24, 121701 (2012)

[journal]

**UC Berkeley**

Fall 2023: Math 228A

Spring 2023: Math 128B

Fall 2022: Math 228A

**New York University**

Spring 2022: Linear Algebra I (graduate)

Fall 2021: Mathematical Statistics

Spring 2021: Linear Algebra I (graduate)

Fall 2020: Calculus I

**UC Berkeley (GSI)**

Spring 2018: Math 54, Alexander Paulin [section page]

Spring 2016: Math 53, Denis Auroux [quizzes]

Fall 2015: Math 1B, Ole Hald

Here is a list of what I read in graduate school, which may suggest some interesting resources.

**Two views on optimal transport and its numerical solution** (2015).
My undergraduate thesis (supervised by Yanir Rubinstein
and Rafe Mazzeo),
which presented two new formulations of optimal transport problems leading to two corresponding methods for numerically solving them.
This thesis won the Kennedy Thesis Prize for the top undergraduate thesis in the natural sciences at Stanford.

[.pdf]

**Asymptotics of Hermite polynomials** (2015). A largely expository paper
(for a course on orthogonal polynomials) about asymptotics of Hermite polynomials and the Gaussian Unitary Ensemble (GUE).
Presents a result about the stationary states for the quantum harmonic oscillator, which,
though likely nothing new, I think is fairly cool.

[.pdf]

**Spectral methods for neural computation** (2013/14). A presentation for the
Brains in Silicon
lab outlining some ideas about how favorable Fourier-domain properties of certain neural
tuning curves are naturally suited (in an idealized setting) for the computation of simple functions.

[slides]

**3D Shape Understanding Using Machine Learning** (2013). A presentation about my
work on using a deep learning framework to perform labeled segmentation of discrete
surfaces and extract multiscale learned shape descriptors. To help with training,
I introduced a new set of shape descriptors based on conformal maps.

[slides]

(Note: research carried out during CURIS
2013 in Stanford's Geometric Computation Group.)

**The k-discs algorithm and its kernelization** (2012). Introduces and
analyzes an extension of

[.pdf]

(Note: class project for CS 229 at Stanford.)