__COVID-19 update 03-01-2023: __

The safety of all IPAM participants and staff is our number one priority, and we will follow all guidelines established by UCLA, LA county, the state of California, and the CDC.

All current and future IPAM activities are expected to be held in person at this time.

Please check this page as well as the UCLA campus web page: https://covid-19.ucla.edu/updates/ for the latest COVID-19 updates.

]]>The PRIMES award is a two-year grant that begins with the PI’s participation in programs at the research institute. During the second year of the award, the PI receives support for travel to work with collaborators etc. Faculty interested in partnering with IPAM to submit a proposal must participate in one of two long programs at IPAM during the 2023-2024 academic year. These are “**Mathematical and Computational Challenges in Quantum Computing**” running September 11-December 15, 2023 or “**Geometry, Statistical Mechanics, and Integrability**” running March 11-June 14, 2024.

Please email Associate Director, Selenne Bañuelos, at **sbanuelos@ipam.ucla.edu** if you are interested in submitting a PRIMES proposal and partnering with IPAM. NSF’s full proposal target date is **May 26, 2023**.

We are thrilled to report that Luis Caffarelli, Mathematician at the University of Texas, Austin, has been awarded the 2023 Abel Prize. The Abel Prize honors groundbreaking contributions to mathematical science and is regarded by many as one of the top prizes in mathematics. Dr. Caffarelli received recognition for his work on equations that are crucial for explaining physical phenomena such as the flow of fluids and the melting of ice.

Dr. Caffarelli served as an organizer and speaker for one of IPAM’s workshops, **Nonlocal PDEs, Variational Problems and their Applications**, and also participated as a speaker in other IPAM workshops such as **Random Media: Homogenization and Beyond**, **Aspects of Optimal Transport in Geometry and Calculus of Variations**, and **International Forum on Multiscale Methods and Partial Differential Equations**.

Congratulations, Luis Caffarelli!

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Professor Tao has served on IPAM’s Science Advisory Board as well as organized and spoke at numerous IPAM programs including the workshop on **Machine Assisted Proofs** that took place recently on February 13 – 17, 2023. This highly successful program brought together scientists and mathematicians from all over the world and triggered new and interesting discussions within the field. In fact, the talk he has given at the Grand Medal ceremony is titled “Machine Assisted Proofs”.

Professor Tao has served an organizer for IPAM’s 2018 spring long program **Quantitative Linear Algebra** and as an invited core participant of 2014 spring long program **Algebraic Techniques for Combinatorial and Computational Geometry**. His many engagements with IPAM include: speaking at workshops on **Turbulent Dissipation, Mixing and Predictability** (2017), **Combinatorial Geometry Problems at the Algebraic Interface** (2014), **Tools from Algebraic Geometry **(2014), **The Kakeya Problem, Restriction Problem, and Sum-product Theory** (2014), and a conference titled **Latinos in the Mathematical Sciences Conference 2015**. In addition to this, the has organized a workshop on **Finding Algebraic Structures in Extremal Combinatorial** (2014) and participated in workshop **Zariski-dense Subgroups** (2015).

Known as the “Mozart of Math”, Professor Tao’s valuable contributions have shaped the fields of partial differential equations, combinatorics, harmonic analysis and additive number theory.

Congratulations, Terence Tao!

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This whitepaper summarizes the activities and outcomes of the long program on “Computational Microscopy” at the Institute of Pure and Applied Mathematics (IPAM) from September 12 to December 16, 2022.

For more than three centuries, lens-based microscopy, such as optical, phase-contrast,

fluorescence, confocal, and electron microscopy, has played an important role in the evolution of

modern science and technology. In 1999, a novel form of microscopy, known as coherent

diffractive imaging (CDI), was developed and transformed our traditional view of microscopy, as

the diffraction pattern of a noncrystalline object was first measured and then directly phased to

obtain a high-resolution image. The well-known phase problem—the usually unavoidable loss of

phase information in the diffraction intensity—was solved by a combination of coherent

illumination and computational algorithms. Over the years, various CDI methods including

plane-wave CDI, ptychography (i.e. scanning CDI), Bragg CDI and Fourier ptychography have

been broadly implemented using synchrotron radiation, X-ray free electron lasers, high harmonic

generation, and optical and electron microscopy. Furthermore, the 2017 Nobel Prize in chemistry

was awarded to Richard Henderson, Joachim Frank, and Jacques Dubochet for developing

cryo-electron microscopy (cryo-EM) for the high-resolution structure determination of

biomolecules in solution. All these groundbreaking developments require the use of advanced

computational algorithms and mathematical tools. This IPAM long program brought together

senior and junior applied mathematicians, physicists, chemists, materials scientists, engineers,

and biologists to discuss and debate on the current status and future perspectivesof modern

microscopy using computation, mathematics, and modeling. The program hosted four workshops

focusing on different aspects of computational microscopy:

● Workshop I: “Diffractive Imaging with Phase Retrieval” focused on advanced

computational methods to solve the phase problem using iterative algorithms and deep

learning.

● Workshop II: “Mathematical Advances for Multi-Dimensional Microscopy” focused on

the incorporation of state-of-the-art mathematical and computational methods into

multi-dimensional electron microscopy.

● Workshop III: “Cryo-Electron Microscopy and Beyond” focused on the current

challenges and future perspectives of the cryo-EM field.

● Workshop IV: “Multi-Modal Imaging with Deep Learning and Modeling” focused on the

integration of data acquisition, mathematical modeling, and deep learning in multimodal

microscopy.

In addition to these four workshops, we formed seven working groups, including 1) Simulation

for electron and optical microscopy, 2) Inverse problems in cryo-EM and phase retrieval, 3) AI

& learning theory, 4) Data-driven information extraction from microscopy data, 5) Multimodal

data processing and acquisition, 6) Space-time models, and 7) Geometry in data processing for

microscopy. The working groups met regularly during the program and tackled a number of

outstanding problems in the field. Below we provide the open challenges that we identified in

computational microscopy, the progress that we made at IPAM, and the research directions that

we will continue to investigate in the future rections in the field of electronic structure theory and computational chemistry as well as related fields that were discussed during the program.

IPAM’s 2022 Newsletter highlights, among others, the highly successful **Latinx in the Mathematical Sciences Conference 2022**. The third in the series, the conference brought together Latinx mathematicians and scientists from all stages of their careers including senior and junior faculty, researchers, career professionals, industry leaders, postdocs, as well as graduate, undergraduate, and high school students. Riding the momentum of this success, IPAM is committed to organize the fourth conference in 2025 (page 4). Also worth noting is the Summer School on Algorithmic Fairness that was hosted by IPAM during the 2022 summer. This was followed by a thematically connected workshop on Sex and Gender Bias in Data. These two programs proved to be greatly compelling and drew participants from a wide range of disciplines (pages 1 and 7). You will also find an interesting discussion on the advent of super computers that are ushering in a new era of computation and knowledge (page 2). You are cordially invited to read it and share it with your friends who might be interested in IPAM’s work.

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The white paper summarizes the activities and outcomes of the Long Program “Advancing Quantum Mechanics with Mathematics and Statistics” which was held at the Institute of Pure and Applied Mathematics (IPAM) from March 7 to June 10, 2022. It also briefly explores some of the current open questions and future directions in the field of electronic structure theory and computational chemistry as well as related fields that were discussed during the program.

**Field Theory Approaches: **One challenge in quantum mechanics for large systems is connecting established methods across scales. Quantum Field Theory (QFT) provides a natural framework to derive existing methods from first principles, highlighting common approximations and a pathway to more accurate ones. This framework allows the description of collective and long-range effects in large complexes. In this context, the development of a rigorous effective field theory approach and the analysis of the light-matter interaction beyond the semiclassical approximations in large molecular systems have been identified as key open questions.

**Excited and open quantum systems: **Multiscale approaches to quantum mechanics need to describe collective degrees of freedom arising from large time and length scales. Open quantum system theory offers a productive way to describe excited electronic states, especially in the context of light-matter interactions. This accurate description is a key aspect both for cavity molecular quantum electrodynamics and large biomolecular complexes. The development of rigorous, accurate, and efficient methodologies for multiscale modeling across optical, electronic, and vibrational degrees of freedom has been envisaged as one of the main directions for future research.

**Embedding theories: **Embedding theories provide an efficient way to combine computational methods of different scales to treat large quantum systems. Within this program, a variety of embedding methods were discussed and scrutinized. Particular attention has been given to density matrix embedding theory, for which a numerical remedy for the chemical potential fitting problem has been developed. The implementation thereof is important as it extends the numerical applicability of embedding methods to larger and more complicated systems.

**Response functions in molecules and solids: **The number of materials and molecules grows combinatorially with the elements and building blocks considered. Dominating quantum effects can often be recovered with response functions describing how electronic structure changes under external influences. Participants devised new methods to obtain some of these response functions relevant to interacting systems and many-body interactions. Open challenges include obtaining high-quality response functions and the systematic improvement over current limitations of quantum mechanical calculations.

**Non-covalent Interactions: **Macroscopic properties of matter are chiefly determined by intermolecular interactions. In large systems, non-covalent interactions are highly non-local with van der Waals (vdW) dispersion being of particular importance. Due to its long-range nature and complex quantum many-body character, an accurate treatment of the vdW dispersion is currently challenging for large systems. During the program an interdisciplinary effort addressed the current lack of accurate and efficient methods capturing vdW dispersion interactions. In addition to the development of novel numerical approaches, this multidisciplinary perspective allows for a better understanding of the physical origins of non-covalent interactions.

**Machine Learning for Quantum Mechanics: **Quantum mechanical problems are high-dimensional and often nonlinear. If machine learning methods can yield models of relevant accuracy, more applications and systems become feasible. The program explored multiple complementary methods that aim to describe long-range interactions and improve transferability by introducing physical constraints directly. The participants identified interpretability of models, inclusion of functional derivatives, and treatment of large-scale many-body systems as some of the key open questions.

**Additional Long Program Activity – 2D Moire Materials Workshop: **Since its isolation by Geim, Novoselov, and collaborators in 2004, single layer graphene has drawn intense interest for its remarkable physical and theoretical properties. Graphene has the highest known tensile strength, and conductivity. More recently, theoretical studies have focused on the mathematical modeling of twisted bilayer graphene (TBG), a metamaterial constructed by stacking two layers of graphene and then rotating one layer with respect to the other. This has led to the development of models such as the Bistritzer-Mcdonald model (2011) and the Chiral model proposed by Tarnopolsky and coworkers (2019). The working group on __2D__ materials focused on reading and understanding the most recent mathematical results on the spectral properties of TBG and strained graphene at magic angles.

Dr. Lieb had participated as a speaker in IPAM’s 2013 workshop Semiclassical Origins of Density Functional Approximations which has led to a number of follow up programs at IPAM.

Congratulations, Elliott Lieb!

]]>Congratulations on this great achievement!

]]>Intense interest in 2D materials was sparked by the exfoliation of graphene, a single layer of carbon atoms with hexagonal structure by Geim and Novoselov in 2004. Graphene had previously been theoretically shown by Wallace (1947) to have a linear dispersion near conical Dirac points and later shown to have quasi-particles that propagate by a massless Dirac equation for a two-component wave function (in analogy to Dirac’s relativistic equation with four-component wave functions that propagate at the speed of light). This analogy to Dirac’s relativistic equation has recently motivated the exploration of phenomena such as Klein’s paradox in graphene. A rigorous and general approach by Weinstein and Fefferman to the derivation of the massless Dirac equation for the long wave, low energy wave packets in hexagon structures was presented by Michael Weinstein and an application to the investigation of edge states was presented by Alexis Druhot and Alexander Watson.

Placing a two-dimensional lattice on another with a small rotation gives rise to periodic “moiré” patterns on a superlattice scale much larger than the original lattice analogous to the beating of two waves with slightly different wavelength (see twisted hexagonal lattices below left). This effective large-scale fundamental domain has opened the possibility of discovering new phenomena at the moiré scale that were previously inaccessible at the atomistic scale. An early example was the observation of the fractal Hofstadter butterfly in twisted bilayer graphene by Philip Kim, et al. (2013). Mitchell Luskin and Paul Cazeaux gave presentations describing their general mathematical theory of moiré patterns in relaxed incommensurate 2D heterostructures (see relaxation of twisted bilayer hexagonal lattices below right).

Rafi Bistrizter and Allan MacDonald (2010) developed a low energy continuum model for the electronic structure of twisted bilayer graphene at the moiré scale and showed that the group velocity of electronic quasi-particles is nearly zero at a “magic” angle of approximately 1°, when the interlayer coupling balances the separation of the Dirac cones of the twisted bilayer. This low group velocity and the localization of the electron density led Bistrizter and MacDonald to suggest that correlated electronic phases such as unconventional superconductivity and Mott insulation might be observed in magic angle twisted bilayer graphene. Such an observation (2018) by Pablo Jarillo-Herrero’s group galvanized the scientific world and has led to an intense synergistic development and investigation of new correlated physics models in parallel with experimental investigations.

MacDonald gave an overview of his recent research on the electronic and optical properties of 2D moiré superlattices and Francisco Guinea, another pioneer in the theory of 2D materials, gave a presentation on electron-electron interactions in graphene systems. Theorists and mathematicians gave lectures on Wannier functions for 2D layered materials (Shiang Fang), band-free approaches (Stephen Carr), and momentum space methods for relaxed incommensurate bilayers (Daniel Massatt).

Novel 2D heterostructures beyond twisted bilayer graphene are actively being investigated in the search for new platforms for correlated physics. Zoe Zhu gave theory and computation for twisted trilayer graphene (see figure below) that highlighted the challenges presented by moiré of moiré structures. Experimental results demonstrating correlated phenomena for twisted trilayer graphene were then presented by Ke Wang.

Mathematics and physics at the moiré scale continues to be intensely investigated and has been a major theme in the IPAM long program on Advancing Quantum Mechanics with Mathematics and Statistics. An informal mini workshop will be held during May 18-20, 2022.

]]>UCLA’s Mathematics Department and the Institute for Pure and Applied Mathematics (IPAM) will be hosting a satellite of this year’s workshop at the Luskin Center from 7:30am – 1:00pm. Click **here to Register**!

Prineha Narang has participated as a speaker in IPAM’s 2020 workshop Theory and Computation 2D Materials, a core participant in IPAM’s 2022 spring long program Advancing Quantum Mechanics with Mathematics and Statistics, and an organizer in the Institute’s 2023 fall long program Mathematical and Computational Challenges in Quantum Computing.

Welcome to UCLA, Prineha!

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