CMSA

Swampland program, extra dimensions and supersymmetry breaking
2023-01-30 11:00 - 12:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Swampland Seminar
Speaker: Ignatios Antoniadis (LPTHE Paris)

Title: Swampland program, extra dimensions and supersymmetry breaking

Abstract: I will argue on the possibility that the smallness of some physical parameters signal a universe corresponding to a large distance corner in the string landscape of vacua. Such parameters can be the scales of dark energy and supersymmetry breaking, leading to a generalisation of the dark dimension proposal. I will discuss the theoretical framework and some of its main physical implications to particle physics and cosmology.

AI and Theorem Proving
2023-01-31 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Mike Douglas
Title: AI and Theorem Proving
Abstract: We survey interactive theorem proving and the Lean theorem prover, and the use of AI and large language models to improve this technology. We hope to start a discussion on projects we can do at the CMSA.

Near extremal de Sitter black holes and JT gravity
2023-02-02 09:30 - 10:30 | Virtual
General Relativity Seminar
Speaker: Chiara Toldo (Harvard)
Title: Near extremal de Sitter black holes and JT gravity
Abstract: In this talk I will explore the thermodynamic response near extremality of charged black holes in four-dimensional Einstein-Maxwell theory with a positive cosmological constant. The latter exhibit three different extremal limits, dubbed cold, Nariai and ultracold configurations, with different near-horizon geometries. For each of these three cases I will analyze small deformations away from extremality, and construct the effective two-dimensional theory, obtained by dimensional reduction, that captures these features. The ultracold case in particular shows an interesting interplay between the entropy variation and charge variation, realizing a different symmetry breaking with respect to the other two near-extremal limits.

Neural Optimal Stopping Boundary
2023-02-02 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Max Reppen (Boston University)
Title: Neural Optimal Stopping Boundary
Abstract: A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.

Interacting Active Matter
2023-02-02 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Amin Doostmohammadi, Niels Bohr Institute, University of Copenhagen
Title: Interacting Active Matter
Abstract: I will focus on the interaction between different active matter systems. In particular, I will describe recent experimental and modeling results that reveal how interaction forces between adhesive cells generate activity in the cell layer and lead to a potentially new mode of phase segregation. I will then discuss mechanics of how cells use finger-like protrusions, known as filopodia, to interact with their surrounding medium. First, I will present experimental and theoretical results of active mirror-symmetry breaking in subcellular skeleton of filopodia that allows for rotation, helicity, and buckling of these cellular fingers in a wide variety of cells ranging from epithelial, mesenchymal, cancerous and stem cells. I will then describe in-vivo experiments together with theoretical modeling showing how during embryo development specialized active cells probe and modify other cell layers and integrate within an active epithelium.

Third Annual Yip Lecture
2023-02-02 19:00 - 20:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Andrew Strominger will give the Third Annual Yip Lecture on February 2, 2023.
Time: 7:00-8:00 pm ET
Location: Harvard Science Center Hall A

Title: Black Holes: The Most Mysterious Objects in the Universe
Abstract: In the last decade black holes have come to center stage in both theoretical and observational science. Theoretically, they were shown a half-century ago by Stephen Hawking and others to obey a precise but still-mysterious set of laws which imply they are paradoxically both the simplest and most complex objects in the universe. Compelling progress on this paradox has occurred recently. Observationally, they have finally and dramatically been seen in the sky, including at LIGO and the Event Horizon Telescope. Future prospects for progress on both fronts hinge on emergent symmetries occurring near the black holes. An elementary presentation of aspects of these topics and their interplay will be given.
Andrew Strominger is the Gwill E. York Professor of Physics and a senior faculty member at the Black Hole Initiative at Harvard University.
Introduction: Peter Galison (Harvard Physics & Black Hole Initiative)
Moderator: Daniel Kapec (Harvard CMSA)
The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

The previous Yip Lecture featured Avi Loeb (Harvard), who spoke on Extraterrestrial Life.

Fracton orders in hyperbolic space and its excitations with fractal mobility
2023-02-03 10:30 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: Han Yan (Rice U)
Title: Fracton orders in hyperbolic space and its excitations with fractal mobility
Abstract: Unlike ordinary topological quantum phases, fracton orders are intimately dependent on the underlying lattice geometry. In this work, we study a generalization of the X-cube model, on lattices embedded in a stack of hyperbolic planes. We demonstrate that for certain hyperbolic lattice tesselations, this model hosts a new kind of subdimensional particle, treeons, which can only move on a fractal-shaped subset of the lattice. Such an excitation only appears on hyperbolic geometries; on flat spaces, treeons become either a lineon or a planeon. Additionally, we find intriguingly that for certain hyperbolic tessellations, a fracton can be created by a membrane operator (as in the X-cube model) or by a fractal-shaped operator within the hyperbolic plane. Our work shows that there are still plenty of exotic behaviors from fracton order to be explored, especially when the embedding geometry is curved.
Reference: H. Yan, K. Slage, A. H. Nevidomskyy, arXiv:2211.15829

Motivic Geometry of Two-Loop Feynman Integrals
2023-02-07 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Chuck Doran
Title: Motivic Geometry of Two-Loop Feynman Integrals
Abstract: We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into a mixed Tate piece and a variation of Hodge structure from families of hyperelliptic curves, elliptic curves, or rational curves depending on the space-time dimension. We give more precise results for two-loop graphs with a small number of edges. In particular, we recover a result of Spencer Bloch that in the well-known double box example there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the “non-planar” two-loop tardigrade graph is that of a family of K3 surfaces of generic Picard number 11. Lastly, we show that generic members of the multi-scoop ice cream cone family of graph hypersurfaces correspond to pairs of multi-loop sunset Calabi-Yau varieties. Our geometric realization of these motives permits us in many cases to derive in full the homogeneous differential operators for the corresponding Feynman integrals. This is joint work with Andrew Harder and Pierre Vanhove.

From spin glasses to Boolean circuits lower bounds - Algorithmic barriers from the overlap gap property
2023-02-08 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: David Gamarnik (MIT)
Title: From spin glasses to Boolean circuits lower bounds. Algorithmic barriers from the overlap gap property
Abstract: Many decision and optimization problems over random structures exhibit an apparent gap between the existentially optimal values and algorithmically achievable values. Examples include the problem of finding a largest independent set in a random graph, the problem of finding a near ground state in a spin glass model, the problem of finding a satisfying assignment in a random constraint satisfaction problem, and many many more. Unfortunately, at the same time no formal computational hardness results exist which explains this persistent algorithmic gap.
In the talk we will describe a new approach for establishing an algorithmic intractability for these problems called the overlap gap property. Originating in statistical physics theory of spin glasses, this is a simple to describe property which a) emerges in most models known to exhibit an apparent algorithmic hardness; b) is consistent with the hardness/tractability phase transition for many models analyzed to the day; and, importantly, c) allows to mathematically rigorously rule out a large class of algorithms as potential contenders, specifically the algorithms which exhibit a form of stability/noise insensitivity.
We will specifically show how to use this property to obtain stronger (stretched exponential) than the state of the art (quasi-polynomial) lower bounds on the size of constant depth Boolean circuits for solving the two of the aforementioned problems: the problem of finding a large independent set in a sparse random graph, and the problem of finding a near ground state of a p-spin model.
Joint work with Aukosh Jagannath and Alex Wein

Bakry-Emery theory and renormalisation
2023-02-08 15:30 - 16:30 | Hybrid
Probability Seminar
Speaker: Roland Bauerschmidt (Cambridge)

Title: Bakry-Emery theory and renormalisation
Abstract: I will discuss an approach to log-Sobolev inequalities that
combines the Bakry-Emery theory with renormalisation and present several
applications. These include log-Sobolev inequalities with polynomial
dependence for critical Ising models on Z^d when d>4 and singular SPDEs
with uniform dependence of the log-Sobolev constant on both the
regularisation and the volume. The talk is based on joint works with
Thierry Bodineau and Benoit Dagallier.

Quasinormal modes and Ruelle resonances: mathematician's perspective
2023-02-09 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Maciej Zworski, UC Berkeley
Title: Quasinormal modes and Ruelle resonances: mathematician’s perspective
Abstract: Quasinormal modes of gravitational waves and Ruelle resonances in hyperbolic classical dynamics share many general properties and can be considered “scattering resonances”: they appear in expansions of correlations, as poles of Green functions and are associated to trapping of trajectories (and are both notoriously hard to observe in nature, unlike, say, quantum resonances in chemistry or scattering poles in acoustical scattering). I will present a mathematical perspective that also includes zeros of the Riemann zeta function (scattering resonances for the Hamiltonian given by the Laplacian on the modular surface) and stresses the importance of different kinds of trapping phenomena, resulting, for instance, in fractal counting laws for resonances.

Special Lectures on Machine Learning and Protein Folding
2023-02-09 15:30 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The CMSA hosted a series of three 90-minute lectures on the subject of machine learning for protein folding.
Thursday Feb. 9, Thursday Feb. 16, & Thursday March 9, 2023, 3:30-5:00 pm ET
Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138 & via Zoom

Speaker: Nazim Bouatta, Harvard Medical School
Abstract: AlphaFold2, a neural network-based model which predicts protein structures from amino acid sequences, is revolutionizing the field of structural biology. This lecture series, given by a leader of the OpenFold project which created an open-source version of AlphaFold2, will explain the protein structure problem and the detailed workings of these models, along with many new results and directions for future research.
Thursday, Feb. 9, 2023

Thursday, Feb. 9, 2023
3:30–5:00 pm ET
Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm space
A brief intro to protein biology. AlphaFold2 impacts on experimental structural biology. Co-evolutionary approaches. Space of ‘algorithms’ for protein structure prediction. Proteins as images (CNNs for protein structure prediction). End-to-end differentiable approaches. Attention and long-range dependencies. AlphaFold2 in a nutshell.

Thursday, Feb. 16, 2023
3:30–5:00 pm ET
Lecture 2: Machine learning for protein structure prediction, Part 2: AlphaFold2 architecture
Turning the co-evolutionary principle into an algorithm: EvoFormer. Structure module and symmetry principles (equivariance and invariance). OpenFold: retraining AlphaFold2 and insights into its learning mechanisms and capacity for generalization. Applications of variants of AlphaFold2 beyond protein structure prediction: AlphaFold Multimer for protein complexes, RNA structure prediction.

Thursday, March 9, 2023
3:30–5:00 pm ET
Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFold
Limitations of AlphaFold2 and evolutionary ML pipelines. OpenFold: retraining AlphaFold2 yields new insights into its capacity for generalization.

Biography: Nazim Bouatta received his doctoral training in high-energy theoretical physics, and transitioned to systems biology at Harvard Medical School, where he received training in cellular and molecular biology in the group of Prof. Judy Lieberman. He is currently a Senior Research Fellow in the Laboratory of Systems Pharmacology led by Prof. Peter Sorger at Harvard Medical School, and an affiliate of the Department of Systems Biology at Columbia, in the group of Prof. Mohammed AlQuraishi. He is interested in applying machine learning, physics, and mathematics to biology at multiple scales. He recently co-supervised the OpenFold project, an optimized, trainable, and completely open-source version of AlphaFold2. OpenFold has paved the way for many breakthroughs in biology, including the release of the ESM Metagenomic Atlas containing over 600 million predicted protein structures.

Chair: Michael Douglas (Harvard CMSA)
Moderators: Farzan Vafa & Sergiy Verstyuk (Harvard CMSA)

Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm space

Lecture 2: Machine learning for protein structure prediction, Part 2: AlphaFold2 architecture

Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFold

Non-invertible Symmetry Enforced Gaplessness
2023-02-10 10:30 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: Ho Tat Lam (MIT)
Title: Non-invertible Symmetry Enforced Gaplessness
Abstract: Quantum systems in 3+1-dimensions that are invariant under gauging a one-form symmetry enjoy novel non-invertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain infrared dynamics. We show that such non-invertible symmetries often forbid a symmetry-preserving vacuum state with a gapped spectrum, leaving only two possibilities for the infrared dynamics: a gapless state or spontaneous breaking of the non-invertible symmetries. These non-invertible symmetries are realized in lattice gauge theories, which serve to illustrate our results.

Parity and Cobordism
2023-02-13 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Swampland Seminar
Speaker: Jake McNamara (Caltech)

Title: Parity and Cobordism

Abstract: The swampland cobordism conjecture provides a convenient way to discuss conserved charges associated with the topology of spacetime. However, much of the power of the cobordism conjecture comes from a mathematical black box: the Adams spectral sequence. In this talk, I will give physical meaning to this black box through a concrete example: domain walls arising from the spontaneously breaking of parity symmetry, which arise in particle physics in Nelson-Barr models. I will argue that parity domain walls are exactly stable, and interpret this stability as the result of an unusual type of gauge symmetry that can only occur in gravitational theories.

Complete Calabi-Yau metrics: Recent progress and open problems
2023-02-13 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Tristan Collins, MIT
Title: Complete Calabi-Yau metrics: Recent progress and open problems
Abstract: Complete Calabi-Yau metrics are fundamental objects in Kahler geometry arising as singularity models or “bubbles” in degenerations of compact Calabi-Yau manifolds. The existence of these metrics and their relationship with algebraic geometry are the subjects of several long standing conjectures due to Yau and Tian-Yau. I will describe some recent progress towards the question of existence, and explain some future directions, highlighting connections with notions of algebro-geometric stability.

Dynamics of active nematic defects on cones
2023-02-14 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Farzan Vafa
Title: Dynamics of active nematic defects on cones
Abstract: In the first part of the talk, we investigate the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on cones. The cone apex develops an effective topological charge, which in analogy to electrostatics, leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. We find three types of ground-state configurations as a function of cone angle, which is determined by charged defects screening the effective apex charge: (i) for sharp cones, all of the +1/p defects are absorbed by the apex; (ii) at intermediate cone angles, some of the +1/p defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (iii) for nearly flat cones, all of the +1/p defects lie equally spaced along a concentric ring on the flank. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement. In the second part of the talk, we investigate the dynamics of an active nematic on a cone, and via simulations find long-time circular orbits of either one or two flank defects, with transitions between these states mediated by the apex via defect absorption, emission, or defect pair creation.

Manifold Fitting: An Invitation to Statistics
2023-02-15 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Zhigang Yao (Harvard CMSA/National University of Singapore)

Title: Manifold Fitting: An Invitation to Statistics
Abstract: This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set $Y subset mathbb{R}^D$. When can we construct a smooth $d$-dimensional submanifold $widehat{M} subset mathbb{R}^D$ to approximate $Y$, and how well can $widehat{M}$ estimate $Y$ in terms of distance and smoothness? To address these problems, various mathematical approaches have been proposed (see Fefferman et al. (2016, 2018, 2021b)). However, many of these methods rely on restrictive assumptions, making extending them to efficient and workable algorithms challenging. As the manifold hypothesis (non-Euclidean structure exploration) continues to be a foundational element in statistics, the manifold fitting Problem, merits further exploration and discussion within the modern statistical community. The talk will be partially based on a recent work Yao and Xia (2019) along with some on-going progress. Relevant reference: https://arxiv.org/abs/1909.10228

Towards programmable living materials and quantitative models of active matter
2023-02-16 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Jörn Dunkel, MIT
Title: Towards programmable living materials and quantitative models of active matter
Abstract: Over the last two decades, major progress has been made in understanding the self-organization principles of active matter. A wide variety of experimental model systems, from self-driven colloids to active elastic materials, has been established, and an extensive theoretical framework has been developed to explain many of the experimentally observed non-equilibrium pattern formation phenomena. Two key challenges for the coming years will be to translate this foundational knowledge into functional active materials, and to identify quantitative mathematical models that can inform and guide the design and production of such materials. Here, I will describe joint efforts with our experimental collaborators to realize self-growing bacterial materials [1], and to implement computational model inference schemes for active and living systems dynamics [2,3].
[1] Nature 608: 324, 2022
[2] PNAS 120: e2206994120, 2023
[3] eLife 10: e68679, 2021

Quasinormal Modes from Penrose Limits
2023-02-16 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Kwinten Fransen (UC Santa Barbara)
Title: Quasinormal Modes from Penrose Limits
Abstract: In this talk, I will explain how to describe quasinormal modes with large real frequencies using Penrose limits. To do so, I first recall relevant aspects of the Penrose limit, and its resulting plane wave spacetimes, as well as quasinormal modes to subsequently tie these together. Having established the main principle, I will illustrate the usefulness of this point of view with the geometric realization of the emergent symmetry algebra underlying the quasinormal modes in the large real frequency limit and present its application to the astrophysically important example of Kerr black holes. Based on arXiv:2301.06999.

Quantum Spin Lakes: NISQ-Era Spin Liquids from Non-Equilibrium Dynamics
2023-02-17 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: Rahul Sahay (Harvard)
Title: Quantum Spin Lakes: NISQ-Era Spin Liquids from Non-Equilibrium Dynamics
Abstract: While many-body quantum systems can in principle host exotic quantum spin liquid (QSL) states, realizing them as ground states in experiments can be prohibitively difficult. In this talk, we show how non-equilibrium dynamics can provide a streamlined route toward creating QSLs. In particular, we show how a simple Hamiltonian parameter sweep can dynamically project out condensed anyons from a family of initial product states (e.g. dynamically “un-Higgs”), yielding a QSL-like state. We christen such states “quantum spin lakes” which, while not thermodynamically large QSLs, enable their study in NISQ-era quantum simulators. Indeed, we show that this mechanism sheds light on recent experimental and numerical observations of the dynamical state preparation of the ruby lattice spin liquid in Rydberg atom arrays. Time permitting, we will discuss how our theory motivates a tree tensor network-based numerical tool—reliant on our theory—that quantitatively reproduces the experimental data two orders of magnitude faster than conventional brute-force simulation methods. Finally, we will highlight that even spin liquid states that are unstable in equilibrium—namely, 2 + 1D U(1) spin liquid states—can be robustly prepared by non-equilibrium dynamics.

Hints of Flat Space Holography
2023-02-21 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Dan Kapec
Title: Hints of Flat Space Holography
Abstract: Despite our detailed understanding of holography in Anti-de Sitter space, flat space holography remains somewhat mysterious. “Celestial CFT” is a formalism which attempts to recast quantum gravity in (d+2)-dimensional asymptotically flat spacetimes in terms of a d-dimensional Euclidean conformal field theory residing at the conformal boundary. I will discuss certain universal aspects of this correspondence. As in Anti-de Sitter space, bulk gravitons produce a boundary stress tensor, and bulk gluons furnish boundary-conserved currents. I will also show that continuous spaces of vacua in the bulk map directly onto the conformal manifold of the boundary CFT. This correspondence provides a new perspective on the role of the BMS group in flat space holography and offers a new interpretation of the antisymmetric double-soft gluon theorem in terms of the curvature of an infinite-dimensional vacuum manifold.

The Black Hole Information Paradox: A Resolution on the Horizon?
2023-02-22 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Netta Engelhardt (MIT)
Title: The Black Hole Information Paradox: A Resolution on the Horizon?
Abstract: The black hole information paradox — whether information escapes an evaporating black hole or not — remains one of the most longstanding mysteries of theoretical physics. The apparent conflict between validity of semiclassical gravity at low energies and unitarity of quantum mechanics has long been expected to find its resolution in a complete quantum theory of gravity. Recent developments in the holographic dictionary, and in particular its application to entanglement and complexity, however, have shown that a semiclassical analysis of gravitational physics can reproduce a hallmark feature of unitary evolution. I will describe this recent progress and discuss some promising indications of a full resolution of the information paradox.

Thresholds for edge colorings
2023-02-22 15:30 - 16:30 | Virtual
Probability Seminar
Speaker: Vishesh Jain (University of Illinois Chicago)

Title: Thresholds for edge colorings

Abstract: We show that if each edge of the complete bipartite graph K_{n,n} is given a random list of C(log n) colors from [n], then with high probability, there is a proper edge coloring where the color of each edge comes from the corresponding list. We also prove analogous results for Latin squares and Steiner triple systems. This resolves several related conjectures of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus. I will discuss some of the main ingredients which go into the proof: the Kahn-Kalai conjecture, absorption, and the Lovasz Local Lemma distribution. Based on joint work with Huy Tuan Pham.

Formation of trapped surfaces in the Einstein-Yang-Mills system
2023-02-23 09:30 - 10:30 | Virtual
General Relativity Seminar
Speaker: Nikolaos Athanasiou (University of Crete, Greece)
Title: Formation of trapped surfaces in the Einstein-Yang-Mills system
Abstract: The purpose of this talk is to give an overview of a semi-global existence result and a trapped surface formation results in the context of the Einstein-Yang-Mills system. Adopting a “signature for decay rates” approach first introduced by An, we develop a novel gauge (and scale) invariant hierarchy of non-linear estimates for the Yang-Mills curvature which, together with the estimates for the gravitational degrees of freedom, yield the desired semi-global existence result. Once semi-global existence has been established, we will explain how the formation of a trapped surface follows from a standard ODE argument. This is joint work with Puskar Mondal and Shing-Tung Yau.

On the convexity of general inverse $sigma_k$ equations and some applications
2023-02-24 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Chao-Ming Lin (University of California, Irvine)
Title: On the convexity of general inverse $sigma_k$ equations and some applications
Abstract: In this talk, I will show my recent work on general inverse $sigma_k$ equations and the deformed Hermitian-Yang-Mills equation (hereinafter the dHYM equation). First, I will show my recent results. This result states that if a level set of a general inverse $sigma_k$ equation (after translation if needed) is contained in the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the Monge-Ampère equation, the J-equation, the dHYM equation, the special Lagrangian equation, etc. Second, I will introduce some semialgebraic sets and a special class of univariate polynomials and give a Positivstellensatz type result. These give a numerical criterion to verify whether the level set will be contained in the positive orthant. Last, as an application, I will prove one of the conjectures by Collins-Jacob-Yau when the dimension equals four. This conjecture states that under the supercritical phase assumption, if there exists a C-subsolution to the dHYM equation, then the dHYM equation is solvable.

Conference on Geometry and Statistics
2023-02-27 09:00 - 2023-03-01 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On Feb 27-March 1, 2023 the CMSA will host a Conference on Geometry and Statistics.
Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138
Organizing Committee:
Stephan Huckemann (Georg-August-Universität Göttingen)
Ezra Miller (Duke University)
Zhigang Yao (Harvard CMSA and Committee Chair)
Scientific Advisors:
Horng-Tzer Yau (Harvard CMSA)
Shing-Tung Yau (Harvard CMSA)
Speakers:

Tamara Broderick (MIT)
David Donoho (Stanford)
Ian Dryden (Florida International University in Miami)
David Dunson (Duke)
Charles Fefferman (Princeton)
Stefanie Jegelka (MIT)
Sebastian Kurtek (OSU)
Lizhen Lin (Notre Dame)
Steve Marron (U North Carolina)
Ezra Miller (Duke)
Hans-Georg Mueller (UC Davis)
Nicolai Reshetikhin (UC Berkeley)
Wolfgang Polonik (UC Davis)
Amit Singer (Princeton)
Zhigang Yao (Harvard CMSA)
Bin Yu (Berkeley)

Moderator: Michael Simkin (Harvard CMSA)

SCHEDULE
Monday, Feb. 27, 2023 (Eastern Time)

8:30 am
Breakfast

8:45–8:55 am
Zhigang Yao
Welcome Remarks

8:55–9:00 am
Shing-Tung Yau*
Remarks

Morning Session Chair: Zhigang Yao

9:00–10:00 am
David Donoho
Title: ScreeNOT: Exact MSE-Optimal Singular Value Thresholding in Correlated Noise
Abstract: Truncation of the singular value decomposition is a true scientific workhorse. But where to Truncate?
For 55 years the answer, for many scientists, has been to eyeball the scree plot, an approach which still generates hundreds of papers per year.
I will describe ScreeNOT, a mathematically solid alternative deriving from the many advances in Random Matrix Theory over those 55 years. Assuming a model of low-rank signal plus possibly correlated noise, and adopting an asymptotic viewpoint with number of rows proportional to the number of columns, we show that ScreeNOT has a surprising oracle property.
It typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance – i.e. the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy dataset and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure.
The talk is based on joint work with Matan Gavish and Elad Romanov, Hebrew University.

10:00–10:10 am
Break

10:10–11:10 am
Steve Marron
Title: Modes of Variation in Non-Euclidean Spaces
Abstract: Modes of Variation provide an intuitive means of understanding variation in populations, especially in the case of data objects that naturally lie in non-Euclidean spaces. A variety of useful approaches to finding useful modes of variation are considered in several non-Euclidean contexts, including shapes as data objects, vectors of directional data, amplitude and phase variation and compositional data.

11:10–11:20 am
Break

11:20 am–12:20 pm
Zhigang Yao
Title: Manifold fitting: an invitation to statistics
Abstract: While classical statistics has dealt with observations which are real numbers or elements of a real vector space, nowadays many statistical problems of high interest in the sciences deal with the analysis of data which consist of more complex objects, taking values in spaces which are naturally not (Euclidean) vector spaces but which still feature some geometric structure. This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set $Y subset mathbb{R}^D$. When can we construct a smooth $d$-dimensional submanifold $widehat{M} subset mathbb{R}^D$ to approximate $Y$, and how well can $widehat{M}$ estimate $Y$ in terms of distance and smoothness? To address these problems, various mathematical approaches have been proposed (see Fefferman et al. (2016, 2018, 2021b)). However, many of these methods rely on restrictive assumptions, making extending them to efficient and workable algorithms challenging. As the manifold hypothesis (non-Euclidean structure exploration) continues to be a foundational element in statistics, the manifold fitting Problem, merits further exploration and discussion within the modern statistical community. The talk will be partially based on a recent work Yao and Xia (2019) along with some on-going progress. Relevant reference:https://arxiv.org/abs/1909.10228

12:20–1:50 pm
12:20 pm Group Photo
followed by Lunch

Afternoon Session Chair: Stephan Huckemann

1:50–2:50 pm
Bin Yu*
Title: Interpreting Deep Neural Networks towards Trustworthiness
Abstract: Recent deep learning models have achieved impressive predictive performance by learning complex functions of many variables, often at the cost of interpretability. This lecture first defines interpretable machine learning in general and introduces the agglomerative contextual decomposition (ACD) method to interpret neural networks. Extending ACD to the scientifically meaningful frequency domain, an adaptive wavelet distillation (AWD) interpretation method is developed. AWD is shown to be both outperforming deep neural networks and interpretable in two prediction problems from cosmology and cell biology. Finally, a quality-controlled data science life cycle is advocated for building any model for trustworthy interpretation and introduce a Predictability Computability Stability (PCS) framework for such a data science life cycle.

2:50–3:00 pm
Break

3:00-4:00 pm
Hans-Georg Mueller
Title: Exploration of Random Objects with Depth Profiles and Fréchet Regression
Abstract: Random objects, i.e., random variables that take values in a separable metric space, pose many challenges for statistical analysis, as vector operations are not available in general metric spaces. Examples include random variables that take values in the space of distributions, covariance matrices or surfaces, graph Laplacians to represent networks, trees and in other spaces. The increasing prevalence of samples of random objects has stimulated the development of metric statistics, an emerging collection of statistical tools to characterize, infer and relate samples of random objects. Recent developments include depth profiles, which are useful for the exploration of random objects. The depth profile for any given object is the distribution of distances to all other objects (with P. Dubey, Y. Chen 2022).
These distributions can then be subjected to statistical analysis. Their mutual transports lead to notions of transport ranks, quantiles and centrality. Another useful tool is global or local Fréchet regression (with A. Petersen 2019) where random objects are responses and scalars or vectors are predictors and one aims at modeling conditional Fréchet means. Recent theoretical advances for local Fréchet regression provide a basis for object time warping (with Y. Chen 2022). These approaches are illustrated with distributional and other data.

4:00-4:10 pm
Break

4:10-5:10 pm
Stefanie Jegelka
Title: Some benefits of machine learning with invariances
Abstract: In many applications, especially in the sciences, data and tasks have known invariances. Encoding such invariances directly into a machine learning model can improve learning outcomes, while it also poses challenges on efficient model design. In the first part of the talk, we will focus on the invariances relevant to eigenvectors and eigenspaces being inputs to a neural network. Such inputs are important, for instance, for graph representation learning. We will discuss targeted architectures that can universally express functions with the relevant invariances – sign flips and changes of basis – and their theoretical and empirical benefits.
Second, we will take a broader, theoretical perspective. Empirically, it is known that encoding invariances into the machine learning model can reduce sample complexity. For the simplified setting of kernel ridge regression or random features, we will discuss new bounds that illustrate two ways in which invariances can reduce sample complexity. Our results hold for learning on manifolds and for invariances to (almost) any group action, and use tools from differential geometry.
This is joint work with Derek Lim, Joshua Robinson, Behrooz Tahmasebi, Lingxiao Zhao, Tess Smidt, Suvrit Sra, and Haggai Maron.

Tuesday, Feb. 28, 2023 (Eastern Time)

8:30-9:00 am
Breakfast

Morning Session Chair: Zhigang Yao

9:00-10:00 am
Charles Fefferman*
Title: Lipschitz Selection on Metric Spaces
Abstract: The talk concerns the problem of finding a Lipschitz map F from a given metric space X into R^D, subject to the constraint that F(x) must lie in a given compact convex “target” K(x) for each point x in X. Joint work with Pavel Shvartsman and with Bernat Guillen Pegueroles.

10:00-10:10 am
Break

10:10-11:10 am
David Dunson
Title: Inferring manifolds from noisy data using Gaussian processes
Abstract: In analyzing complex datasets, it is often of interest to infer lower dimensional structure underlying the higher dimensional observations. As a flexible class of nonlinear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the original data with lower dimensional coordinates without providing an estimate of the manifold in the observation space or using the manifold to denoise the original data. This article proposes a new methodology for addressing these problems, allowing interpolation of the estimated manifold between fitted data points. The proposed approach is motivated by novel theoretical properties of local covariance matrices constructed from noisy samples on a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing application of Gaussian processes for probabilistic manifold reconstruction. In addition to theory justifying the algorithm, we provide simulated and real data examples to illustrate the performance. Joint work with Nan Wu – see https://arxiv.org/abs/2110.07478

11:10-11:20 am
Break

11:20 am-12:20 pm
Wolfgang Polonik
Title: Inference in topological data analysis
Abstract: Topological data analysis has seen a huge increase in popularity finding applications in numerous scientific fields. This motivates the importance of developing a deeper understanding of benefits and limitations of such methods. Using this angle, we will present and discuss some recent results on large sample inference in topological data analysis, including bootstrap for Betti numbers and the Euler characteristics process.

12:20–1:50 pm
Lunch

Afternoon Session Chair: Stephan Huckemann

1:50-2:50 pm
Ezra Miller
Title: Geometric central limit theorems on non-smooth spaces
Abstract: The central limit theorem (CLT) is commonly thought of as occurring on the real line, or in multivariate form on a real vector space. Motivated by statistical applications involving nonlinear data, such as angles or phylogenetic trees, the past twenty years have seen CLTs proved for Fréchet means on manifolds and on certain examples of singular spaces built from flat pieces glued together in combinatorial ways. These CLTs reduce to the linear case by tangent space approximation or by gluing. What should a CLT look like on general non-smooth spaces, where tangent spaces are not linear and no combinatorial gluing or flat pieces are available? Answering this question involves figuring out appropriate classes of spaces and measures, correct analogues of Gaussian random variables, and how the geometry of the space (think “curvature”) is reflected in the limiting distribution. This talk provides an overview of these answers, starting with a review of the usual linear CLT and its generalization to smooth manifolds, viewed through a lens that casts the singular CLT as a natural outgrowth, and concluding with how this investigation opens gateways to further advances in geometric probability, topology, and statistics. Joint work with Jonathan Mattingly and Do Tran.

2:50-3:00 pm
Break

3:00-4:00 pm
Lizhen Lin
Title: Statistical foundations of deep generative models
Abstract: Deep generative models are probabilistic generative models where the generator is parameterized by a deep neural network. They are popular models for modeling high-dimensional data such as texts, images and speeches, and have achieved impressive empirical success. Despite demonstrated success in empirical performance, theoretical understanding of such models is largely lacking. We investigate statistical properties of deep generative models from a nonparametric distribution estimation viewpoint. In the considered model, data are assumed to be observed in some high-dimensional ambient space but concentrate around some low-dimensional structure such as a lower-dimensional manifold structure. Estimating the distribution supported on this low-dimensional structure is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. We obtain convergence rates with respect to the Wasserstein metric of distribution estimators based on two methods: a sieve MLE based on the perturbed data and a GAN type estimator. Such an analysis provides insights into i) how deep generative models can avoid the curse of dimensionality and outperform classical nonparametric estimates, and ii) how likelihood approaches work for singular distribution estimation, especially in adapting to the intrinsic geometry of the data.

4:00-4:10 pm
Break

4:10-5:10 pm
Conversation session

Wednesday, March 1, 2023 (Eastern Time)

8:30-9:00 am
Breakfast

Morning Session Chair: Ezra Miller

9:00-10:00 am
Amit Singer*
Title: Heterogeneity analysis in cryo-EM by covariance estimation and manifold learning
Abstract: In cryo-EM, the 3-D molecular structure needs to be determined from many noisy 2-D tomographic projection images of randomly oriented and positioned molecules. A key assumption in classical reconstruction procedures for cryo-EM is that the sample consists of identical molecules. However, many molecules of interest exist in more than one conformational state. These structural variations are of great interest to biologists, as they provide insight into the functioning of the molecule. Determining the structural variability from a set of cryo-EM images is known as the heterogeneity problem, widely recognized as one of the most challenging and important computational problem in the field. Due to high level of noise in cryo-EM images, heterogeneity studies typically involve hundreds of thousands of images, sometimes even a few millions. Covariance estimation is one of the earliest methods proposed for heterogeneity analysis in cryo-EM. It relies on computing the covariance of the conformations directly from projection images and extracting the optimal linear subspace of conformations through an eigendecomposition. Unfortunately, the standard formulation is plagued by the exorbitant cost of computing the N^3 x N^3 covariance matrix. In the first part of the talk, we present a new low-rank estimation method that requires computing only a small subset of the columns of the covariance while still providing an approximation for the entire matrix. This scheme allows us to estimate tens of principal components of real datasets in a few minutes at medium resolutions and under 30 minutes at high resolutions. In the second part of the talk, we discuss a manifold learning approach based on the graph Laplacian and the diffusion maps framework for learning the manifold of conformations. If time permits, we will also discuss the potential application of optimal transportation to heterogeneity analysis. Based on joint works with Joakim Andén, Marc Gilles, Amit Halevi, Eugene Katsevich, Joe Kileel, Amit Moscovich, and Nathan Zelesko.

10:00-10:10 am
Break

10:10-11:10 am
Ian Dryden
Title: Statistical shape analysis of molecule data
Abstract: Molecular shape data arise in many applications, for example high dimension low sample size cryo-electron microscopy (cryo-EM) data and large temporal sequences of peptides from molecular dynamics simulations. In both applications it is of interest to summarize the shape evolution of the molecules in a succinct, low-dimensional representation. However, Euclidean techniques such as principal components analysis (PCA) can be problematic as the data may lie far from in a flat manifold. Principal nested spheres gives a fundamentally different decomposition of data from the usual Euclidean subspace based PCA. Subspaces of successively lower dimension are fitted to the data in a backwards manner with the aim of retaining signal and dispensing with noise at each stage. We adapt the methodology to 3D sub-shape spaces and provide some practical fitting algorithms. The methodology is applied to cryo-EM data of a large sliding clamp multi-protein complex and to cluster analysis of peptides, where different states of the molecules can be identified. Further molecular modeling tasks include resolution matching, where coarse resolution models are back-mapped into high resolution (atomistic) structures. This is joint work with Kwang-Rae Kim, Charles Laughton and Huiling Le.

11:10-11:20 am
Break

11:20 am-12:20 pm
Tamara Broderick
Title: An Automatic Finite-Sample Robustness Metric: Can Dropping a Little Data Change Conclusions?
Abstract: One hopes that data analyses will be used to make beneficial decisions regarding people’s health, finances, and well-being. But the data fed to an analysis may systematically differ from the data where these decisions are ultimately applied. For instance, suppose we analyze data in one country and conclude that microcredit is effective at alleviating poverty; based on this analysis, we decide to distribute microcredit in other locations and in future years. We might then ask: can we trust our conclusion to apply under new conditions? If we found that a very small percentage of the original data was instrumental in determining the original conclusion, we might not be confident in the stability of the conclusion under new conditions. So we propose a method to assess the sensitivity of data analyses to the removal of a very small fraction of the data set. Analyzing all possible data subsets of a certain size is computationally prohibitive, so we provide an approximation. We call our resulting method the Approximate Maximum Influence Perturbation. Our approximation is automatically computable, theoretically supported, and works for common estimators. We show that any non-robustness our method finds is conclusive. Empirics demonstrate that while some applications are robust, in others the sign of a treatment effect can be changed by dropping less than 0.1% of the data — even in simple models and even when standard errors are small.

12:20-1:50 pm
Lunch

Afternoon Session Chair: Ezra Miller

1:50-2:50 pm
Nicolai Reshetikhin*
Title: Random surfaces in exactly solvable models in statistical mechanics.
Abstract: In the first part of the talk I will be an overview of a few models in statistical mechanics where a random variable is a geometric object such as a random surface or a random curve. The second part will be focused on the behavior of such random surfaces in the thermodynamic limit and on the formation of the so-called “limit shapes”.

2:50-3:00 pm
Break

3:00-4:00 pm
Sebastian Kurtek
Title: Robust Persistent Homology Using Elastic Functional Data Analysis
Abstract: Persistence landscapes are functional summaries of persistence diagrams designed to enable analysis of the diagrams using tools from functional data analysis. They comprise a collection of scalar functions such that birth and death times of topological features in persistence diagrams map to extrema of functions and intervals where they are non-zero. As a consequence, variation in persistence diagrams is encoded in both amplitude and phase components of persistence landscapes. Through functional data analysis of persistence landscapes, under an elastic Riemannian metric, we show how meaningful statistical summaries of persistence landscapes (e.g., mean, dominant directions of variation) can be obtained by decoupling their amplitude and phase variations. This decoupling is achieved via optimal alignment, with respect to the elastic metric, of the persistence landscapes. The estimated phase functions are tied to the resolution parameter that determines the filtration of simplicial complexes used to construct persistence diagrams. For a dataset obtained under geometric, scale and sampling variabilities, the phase function prescribes an optimal rate of increase of the resolution parameter for enhancing the topological signal in a persistence diagram. The proposed approach adds to the statistical analysis of data objects with rich structure compared to past studies. In particular, we focus on two sets of data that have been analyzed in the past, brain artery trees and images of prostate cancer cells, and show that separation of amplitude and phase of persistence landscapes is beneficial in both settings. This is joint work with Dr. James Matuk (Duke University) and Dr. Karthik Bharath (University of Nottingham).

4:00-4:10 pm
Break

4:10-5:10 pm
Conversation session

5:10-5:20 pm
Stephan Huckemann, Ezra Miller, Zhigang Yao
Closing Remarks

* Virtual Presentation

New bounds on lattice covering volumes, and nearly uniform covers
2023-03-02 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Barak Weiss
Title: New bounds on lattice covering volumes, and nearly uniform covers
Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.

The string/black hole transition in anti de Sitter space
2023-03-02 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Erez Urbach, Weizmann Institute of Science
Title: The string/black hole transition in anti de Sitter space
Abstract: String stars, or Horowitz-Polchinski solutions, are string theory saddles with normalizable condensates of thermal-winding strings. In the past, string stars were offered as a possible description of stringy (Euclidean) black holes in asymptotically flat spacetime, close to the Hagedorn temperature. I will discuss the thermodynamic properties of string stars in asymptotically (thermal) anti-de Sitter background (including AdS3 with NS-NS flux), their possible connection to small black holes in AdS, and their implications for holography. I will also present new “winding-string gas” saddles for confining holographic backgrounds such as the Witten model, and their relation to the deconfined phase of 3+1 pure Yang-Mills.

Strongly coupled ultraviolet fixed point and symmetric mass generation in four dimensions with 8 Kähler-Dirac fermions
2023-03-03 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: Anna Hasenfratz (University of Colorado)
Title: Strongly coupled ultraviolet fixed point and symmetric mass generation in four dimensions with 8 Kähler-Dirac fermions

Abstract: 4-dimensional gauge-fermion systems exhibit a quantum phase transition from a confining, chirally broken phase to a conformal phase as the number of fermions is increased. While the existence of the conformal phase is well established, very little is known about the nature of the phase transition or the strong coupling phase.

Lattice QCD methods can predict the RG $beta$ function, but the calculations are often limited by non-physical bulk phase transition that prevent exploring the strong coupling region of the phase diagram. Even the critical flavor number is controversial, estimates vary between $N_f=8$ and 14 for fundamental fermions.

Using an improved lattice actions that include heavy Pauli-Villars (PV) type bosons to reduce ultraviolet fluctuations, I was able to simulate an SU(3) system with 8 fundamental flavors at much stronger renormalized coupling than previously possibly. The numerical results indicate a smooth phase transition from weak coupling to a strongly coupled phase.
I investigate the critical behavior of the transition using finite size scaling. The result of the scaling analysis is not consistent with a first order phase transition, but it is well described by Berezinsky-Kosterlitz-Thouless or BKT scaling. BKT scaling could imply that the 8-flavor system is the opening of the conformal window, an exciting possibility that warrants further investigations.

The strongly coupled phase appear to be chirally symmetric but gapped, suggesting symmetric mass generation (SMG). This could be the consequence of the lattice fermions used in this study. Staggered fermions in the massless limit are known to be anomaly free, allowing an SMG phase in the continuum limit.

References:
Phys.Rev.D 106 (2022) 1, 014513 • e-Print: 2204.04801
Phys.Rev.D 104 (2021) 7, 074509 • e-Print: 2109.02790
For anomalies and staggered fermion, see
Phys.Rev.D 104 (2021) 9, 094504 • e-Print: 2101.01026

https://www.youtube.com/watch?v=3jtNsFGszjE&list=PL0NRmB0fnLJQAnYwkpt9PN2PBKx4rvdup&index=14

Categorical Symmetry of the Standard Model from Gravitational Anomaly
2023-03-07 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Juven Wang
Title: Categorical Symmetry of the Standard Model from Gravitational Anomaly
Abstract: In the Standard Model, the total “sterile right-handed” neutrino number n_{νR} is not equal to the family number Nf. The anomaly index (-Nf+n_{νR}) had been advocated to play an important role in our previous work on Cobordism and Deformation Class of the Standard Model [2112.14765, 2204.08393] and Ultra Unification [2012.15860] in order to predict new highly entangled sectors beyond the Standard Model. Moreover, the invertible baryon minus lepton number B−L symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical counterpart of the B−L symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by ℤ and ℤ16 respectively. For this construction, we utilize the anomaly inflow concept and the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate chiral central charge, or the 3d boundary topological order of 4d ℤTF4-time-reversal symmetric topological superconductor [2302.14862].
