CMSA

Homology, higher derived limits, and set theory
2023-12-11 16:30 - 17:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Justin Moore (Cornell University)
Title: Homology, higher derived limits, and set theory
Abstract: Singular homology has a number of well-known defects when used to study spaces such as the Hawaiian earring and solenoids. It may not reflect the “shape” of the space and can give counterintuitive information about its dimension. One remedy of this is to develop a homology theory based on approximating spaces by polyhedra, computing their homologies, and then taking a limit. This is the approach taken by Steenrod-Sitnikov homology and Lisica and Mardesic’s strong homology. Even within the class of locally compact second countable spaces though, the properties of these homology theories — and the higher derived limits which underly them — are dependent on axioms of set theory beyond ZFC. Recently it was shown that it is consistent with (and therefore independent of) ZFC that strong homology and Steenrod Sitnikov homology coincide in the class of locally compact second countable spaces — and therefore each of these homology theories enjoys the desirable properties of the other. These results also point to how we might develop variants of these homology theories which enjoy their desirable properties, but which are less sensitive to set theory. This is joint work with Nathaniel Bannister, Jeff Bergfalk, and Stevo Todorcevic.

M-theory on nodal Calabi-Yau 3-folds and torsion refined GV-invariants
2023-12-11 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Thorsten Schimannek (Utrecht University)

Title: M-theory on nodal Calabi-Yau 3-folds and torsion refined GV-invariants
Abstract: The physics of M-theory and Type IIA strings on a projective nodal CY 3-folds is determined by the geometry of a small resolution, even if the latter is not Kähler. We will demonstrate this explicitly in the context of a family of Calabi-Yau double covers of P^3. Using conifold transitions, we prove that the exceptional curves in any small resolution are torsion while M-theory develops a discrete gauge symmetry.This leads to a torsion refinement of the ordinary Gopakumar-Vafa invariants, that is associated to the singular Calabi-Yau and captures the enumerative geometry of the non-Kähler resolutions. We further argue that twisted circle compactifications of the 5d theory are dual to IIA compactifications on the nodal CY 3-fold with a flat but topologically non-trivial B-field. As a result, the torsion refined invariants are encoded in the topological string partition functions with different choices for the global topology of a flat B-field.
The talk is based on 2108.09311, 2212.08655 (with S. Katz, A. Klemm, and E. Sharpe) and 2307.00047 (with S. Katz).

Fermi surface symmetric mass generation and its application in nickelate superconductor
2023-12-08 14:30 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint Quantum Matter in Mathematics and Physics & Topological Quantum Matter Seminar
Speaker: Da-Chuan Lu (UCSD)
Title: Fermi surface symmetric mass generation and its application in nickelate superconductor
Abstract: Symmetric mass generation (SMG) is a novel interaction-driven mechanism that generates fermion mass without breaking symmetry, unlike the standard Anderson-Higgs mechanism. SMG can occur in the fermion system without quantum anomalies. In this talk, I will focus on the SMG for the systems with finite fermion density, i.e., the Fermi surface. I will discuss the Fermi surface anomaly and Fermi surface SMG. Lastly, I will talk about its application in the newly found nickelate superconductors, where the superconductivity emerges without a nearby spontaneous symmetry-breaking phase.

Open Problems in Physics and Novel Solutions from Math
2023-12-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Juven Wang
Title: Open Problems in Physics and Novel Solutions from Math

Abstract: (1) How do neutrinos obtain tiny mass and oscillate between e, mu, tau three flavors? What consists of Dark Matter? What causes leptogenesis and baryogenesis?
(2) Why the Parity Symmetry is violated in the weak interaction? How do we quantum mechanically regularize and numerically simulate the Standard Model (chiral fermion/gauge theory) on the lattice in 3+1 spacetime dimensions?
(3) Strong CP problem: Why is the neutron electric dipole moment nearly zero and QCD respects the CP = T symmetry?
(4) Family/Generation problem: Why are there three families/generations of quarks and leptons?
In my talk, I will sketch and propose novel solutions to these challenging open problems in physics based on my own work (available on arXiv, some jointly with my collaborators).
I will show that the eclectic cross-disciplinary fertilization of ideas between the Math-Physics geometry and topology, Quantum Condensed Matter, and High-Energy Physics (theory, pheno, lattice) can help solving these big problems.
The (1) answer relates to Ultra Unification: a Z_{16} class Atiyah-Patodi-Singer eta invariant (Baryon – Lepton)-protected topological superconductor and a symmetry-extended anomalous topological quantum field theory.
The (2) and (3) answers relate to a trivial cobordism class and Symmetric Mass Generation, and disordering and removing the Pecci-Quinn axions.
The (4) answer has to do with the Hirzebruch signature and Atiyah-Singer index theorem, gravitational Chern-Simons theory, and 3 E8 quantum Hall states.

Active structures and flows in living cells
2023-12-07 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Michael Shelley (Flatiron)
Title: Active structures and flows in living cells
Abstract: Flows in the fluidic interior of living cells can serve biological function or act as signatures of how intracellular forces are exerted. I’ll discuss examples of each. One is understanding the emergence of cell-spanning vortical flows in large developing egg cells, while the other arises in studying the nature of force transduction in single cell embryos moving towards their first cell division. Both involve the cytoskeleton, that set of polymers, cross-linkers, and molecular motors that underlie much of the active mechanics within cells, and has led to the development of new coarse-grained active matter models and novel instabilities.

CMSA Q and A Seminar 12/5/2023
2023-12-05 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Mike Douglas (Harvard CMSA) and Anurag Anshu (Harvard Computer Science)
Topics:
Mike Douglas: What are scaling laws in deep learning?
Anurag Anshu: What’s the difference between classical information theory and quantum information theory (or classical computation vs quantum computation)?

The Feynman propagator and self-adjointness
2023-12-05 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Andras Vasy (Stanford)
Title: The Feynman propagator and self-adjointness
Abstract: In this talk I will discuss the Feynman and anti-Feynman inverses for wave operators on certain Lorentzian manifolds; these are two inverses which from a microlocal analysis perspective are more natural than the standard causal (advanced/retarded) ones. For instance, for the spectral family of the wave operator, these are the natural inverses when the spectral parameter is non-real. Indeed, I will explain that these connect to the self-adjointness of the wave operator, and the positivity properties that follow.

Analysis of ALH* gravitational instantons
2023-12-04 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Xuwen Zhu (Northeastern)
Title: Analysis of ALH* gravitational instantons
Abstract: Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will discuss one such type called ALH* metrics which has a non-compact end modelled by the Calabi ansatz with inhomogeneous collapsing near infinity. Such metrics appeared recently in the works on SYZ conjecture, as well as the scaling bubble limits for codimension-3 collapsing of K3 surfaces, where the study of its Laplacian played a central role. In this talk I will talk about the Fredholm mapping property and L^2 cohomology of such metrics. This is ongoing work joint with Rafe Mazzeo.

The Fractional Quantum Hall Effect at ν=5/2: Past, Recent, and Future
2023-12-04 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Ken K. W. Ma (Northeastern University)
Title: The Fractional Quantum Hall Effect at ν=5/2: Past, Recent, and Future
Abstract: The discovery of fractional quantum Hall (FQH) states started a new chapter in modern physics. Nowadays, more than 70 FQH states at different filling factors have been observed. Among them, the FQH state at the filling factor ν=5/2 in GaAs (or the 5/2 state) remains one of the most special and attractive states. Since its discovery in 1987, different possible topological orders have been proposed to describe the 5/2 state. Some of them can host an exotic type of particles, known as non-Abelian anyons. Recent experiments have provided more insights into the understanding of the 5/2 state, but its underlying nature is still under debate.
In this talk, I will review the basics of the 5/2 state [1] and a more unified theoretical description of different possible topological orders of the 5/2 state that we have recently proposed [2]. I will also review the groundbreaking thermal Hall conductance experiment [3] and the follow-up quantum Hall interface experiments by the Weizmann Institute group [4, 5], and discuss what possible lessons that we can learn from the experimental results. Lastly, I will talk about some possible directions and related topics for future investigations.
References:
[1] K. K. W. Ma, M. R. Peterson, V. W. Scarola, and K. Yang, “Fractional quantum Hall effect at the filling factor ν = 5/2” in Encyclopedia of Condensed Matter Physics (Second Edition), edited by T. Chakraborty, Academic Press (2024).
[2] K. K. W. Ma and D. E. Feldman, “The sixteenfold way and the quantum Hall effect at half-integer filling factors”, Phys. Rev. B 100, 035302 (2019).
[3] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, “Observation of half-integer thermal Hall conductance”, Nature (London) 559, 205 (2018).
[4] B. Dutta, W. Yang, R. Melcer, H. K. Kundu, M. Heiblum, V. Umansky, Y. Oreg, A. Stern, D. Mross, “Distinguishing between non-Abelian topological orders in a quantum Hall system”, Science 375, 193 (2021).
[5] B. Dutta, V. Umansky, M. Banerjee, and M. Heiblum, “Isolated ballistic non-Abelian interface channel”, Science 377, 1198 (2022).

CM-minimizers and standard models of Fano fibrations over curves
2023-12-04 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Speaker: Maksym Fedorchuk (Boston College)
Title: CM-minimizers and standard models of Fano fibrations over curves
Abstract: A recent achievement in K-stability of Fano varieties is an algebro-geometric construction of a projective moduli space of K-polystable Fanos. The ample line bundle on this moduli space is the CM line bundle of Tian. One of the consequences of the general theory is that given a family of K-stable Fanos over a punctured curve, the polystable filling is the one that minimizes the degree of the CM line bundle after every finite base change. A natural question is to ask what are the CM-minimizers without base change. In answering this question, we arrive at a theory of Koll’ar stability for fibrations over one-dimensional bases, and standard models of Fano fibrations. I will explain the joint work with Hamid Abban and Igor Krylov in which we show that the CM-minimizers for del Pezzo fibrations are Corti’s standard models and related work in progress on quartic threefold hypersurfaces.

Quantum information: the interplay of mathematics and physics
2023-12-01 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Kaifeng Bu (Harvard University)
Title: Quantum information: the interplay of mathematics and physics
Abstract: I will provide an introduction to quantum information, which points to a new connection with experiment on the one hand, and a potential new area of mathematical analysis on the other. I will introduce two recent results about the application of this new area in quantum information: (1) a novel Quantum Central Limit Theorem (QCLT), and (2) a physically realizable protocol for testing and measuring quantum advantage. This talk is based on a collaboration with Arthur Jaffe, and Weichen Gu (PNAS120(25)2023, arXiv: 2302.08423, arXiv:2306.09292).

A Plane Defect in the 3d O(N) Model
2023-12-01 16:30 - 17:30 | Hybrid
Quantum Matter Seminar
Speaker: Abijith Krishnan (MIT)
Title: A Plane Defect in the 3d O(N) Model
Abstract: It was recently found that the classical 3d O(N) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as (log x)^(-q). This universality class exists for a range 2≤N<Nc and Monte-Carlo simulations and conformal bootstrap indicate Nc>3. In this talk, I’ll extend this result to the 3d O(N) model in an infinite geometry with a plane defect. I’ll explain using the renormalization group (RG) that the extraordinary-log universality class is present for any finite N≥2, and that a line of defect fixed points is present at N=∞. This line of defect fixed points is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N corrections. I’ll show that the line of defect fixed points and the 1/N corrections agree with an a-theorem by Jensen and O’Bannon for 3d CFTs with a boundary. Finally, I’ll conclude by noting some physical systems where the extraordinary-log universality class can be observed.

An introduction to mixture of experts in deep learning
2023-12-01 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Samy Jelassi
Title: An introduction to mixture of experts in deep learning
Abstract: Scale has opened new frontiers in natural language processing – but at a high cost. Mixture-of-Experts (MoE) have been proposed as a path to even larger and more capable language models. They select different parameters for each incoming example. By doing so, the parameter count is decoupled from the compute per example leading to very large, but efficient models. In this talk, I will review the concept of mixture of experts, provide a basic description of the Switch Transformers model, characterize some of their behaviors and conclude by highlighting some open problems in the field. This talk is mainly based on the following papers: https://arxiv.org/pdf/2101.03961.pdf, https://arxiv.org/pdf/2209.01667.pdf .

A Gaussian convexity for logarithmic moment generating function
2023-11-30 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Wei-Kuo Chen (University of Minnesota)
Title: A Gaussian convexity for logarithmic moment generating function
Abstract: Convex functions of Gaussian vectors are prominent objectives in many fields of mathematical studies. In this talk, I will establish a new convexity for the logarithmic moment generating function for this object and draw two consequences. The first leads to the Paouris-Valettas small deviation inequality that arises from the study of convex geometry. The second provides a quantitative bound for the Dotsenko-Franz-Mezard conjecture in the Sherrington-Kirkpatrick mean-field spin glass model, which states that the logarithmic anneal partition function of negative replica is asymptotically equal to the free energy.

Anomalous Landau level analog in solids: search and implications for 2d heterostructures
2023-11-29 17:00 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Valentin Crepel, Flatiron Institute
Title: Anomalous Landau level analog in solids: search and implications for 2d heterostructures
Abstract: The recent experimental observation of fractional Chen insulators — the analog of fractional quantum Hall states realized in absence of any applied magnetic field — calls for more detailed theoretical investigations of the special conditions allowing their emergence. To gain some insight on this problem, we can rely on (i) our extensive knowledge of the physical properties of Landau levels, and (ii) the identification of their exact solid state analogs.
In this talk, (i-ii) will be discussed in the case of ”anomalous Landau levels”, which are protected by an index theorem, and whose most famous representative are the zero-th Landau levels of graphene. In particular, I will derive the generic form of Bloch wave functions realizing the analog of such anomalous Landau levels, and argue that twisted transition metal dichalcogenides homobilayers (in which fractional Chern insulators were first discovered) almost realize such anomalous bands.

A random matrix model towards the quantum chaos transition conjecture
2023-11-28 12:00 - 13:00 | MA
Probability Seminar
Speaker: Jun Yin (UCLA)
Title: A random matrix model towards the quantum chaos transition conjecture
Abstract: The Quantum Chaos Conjecture has long fascinated researchers, postulating a critical spectrum phase transition that separates integrable systems from chaotic systems in quantum mechanics. In the realm of integrable systems, eigenvectors remain localized, and local eigenvalue statistics follow the Poisson distribution. Conversely, chaotic systems exhibit delocalized eigenvectors, with local eigenvalue statistics mirroring the Sine kernel distribution, akin to the standard random matrix ensembles GOE/GUE.
This talk delves into the heart of the Quantum Chaos Conjecture, presenting a novel approach through the lens of random matrix models. By utilizing these models, we aim to provide a clear and intuitive demonstration of the same phenomenon, shedding light on the intricacies of this long-standing conjecture.

Remarkable symmetries of rotating black holes
2023-11-28 11:00 - 12:00 | MA
General Relativity Seminar
Speaker: David Kubiznak (Charles University)
Title: Remarkable symmetries of rotating black holes
Abstract: It is well known that the Kerr geometry admits a non-trivial Killing tensor and its ‘square root’ known as the Killing-Yano tensor. These two objects stand behind Carter’s constant of geodesic motion as well as allow for separability of test field equations in this background. The situation is even more remarkable in higher dimensions, where a single object — the principal Killing-Yano tensor — generates a tower of explicit and hidden symmetries responsible for integrability of geodesics and separability of test fields around higher-dimensional rotating black holes. Interestingly, similar yet different structure is already present for the slowly rotating black holes described by the `magic square’ version of the Lense-Thirring solution, giving rise to a geometrically preferred spacetime that can be cast in the Painleve-Gullstrand form and admits a tower of exact rank-2 and higher rank Killing tensors whose number rapidly grows with the number of spacetime dimensions.

What do topological dynamics, combinatorics, and model theory have in common?
2023-11-27 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Dana Bartosova (University of Florida)
Title: What do topological dynamics, combinatorics, and model theory have in common?
Abstract: A striking correspondence between dynamics of automorphism groups of countable first order structures and Ramsey theory of finitary approximation of the structures was established in 2005 by Kechris, Pestov, and Todocevic. Since then, their work has been generalized and applied in many directions. It also struck a fresh wave of interest in finite Ramsey theory. Many classes of finite structures are shown to have the Ramsey property by encoding their problem in a known Ramsey class and translating a solution back. This is often a case-by-case approach and naturally there is a great need for abstracting the process. There has been much success on this front, however, none of the tools captures every situation. We will discuss one such encoding via a model-theoretic notion of semi-retraction introduced by Lynn Scow in 2012. In a joint work, we showed that a semi-retraction transfers the Ramsey property from one class of structures to another under quite general conditions. We compare semi-retractions to a category-theoretical notion of pre-adjunction revived by Mašulović in 2016. If time permits, I will mention a transfer theorem of the Ramsey property from a class of finite structures to their uncountable ultraproducts, which is an AIMSQuaRE project with Džamonja, Patel, and Scow.

A p-adic Laplacian on the Tate curve
2023-11-27 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: An Huang (Brandeis)
Pre-talk Speaker: TBA: 10:00-10:30 am

Title: A p-adic Laplacian on the Tate curve
Abstract: We shall first explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate’s thesis. We then propose a genus one p-adic string worldsheet action. The key is the definition of a p-adic Laplacian operator on the Tate curve. We show that the genus one p-adic Green’s function exists, is unique under some obvious constraints, is locally constant off diagonal, and has a reflection symmetry. It can also be numerically computed exactly off the diagonal, thanks to some simplifications due to the p-adic setup. Numerics suggest that at least in some special cases, the asymptotic behavior of the Green’s function near the diagonal is a direct p-adic counterpart of the familiar Archimedean case, although the p-adic Laplacian is not a local operator. Joint work in progress with Rebecca Rohrlich.

Modulated Gauge Theories and Fracton Behavior in 2D
2023-11-22 16:00 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Guilherme Delfino Silva (Boston University)
Title: Modulated Gauge Theories and Fracton Behavior in 2D
Abstract: In this talk we investigate deconfined phases of two-dimensional ZN lattice gauge theories associated to spatially modulated symmetries. In order to study the low-energy physics of such modulated gauge theories we propose and explore exactly solvable gapped Hamiltonians, which allow us to fully characterize their low-energy properties at zero temperature. We provide explicit examples and discuss how non-polynomial symmetries, as exponential symmetries, are powerful enough to fully constrain the mobility of isolated excitations.

The analytical challenges of connectomics
2023-11-20 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Jeff W. Lichtman (Harvard University)
Title: The analytical challenges of connectomics
Abstract: Recent progress in generating synapse-level maps of brains, a field known as connectomics, brings both opportunities and challenges. The upside is that the biophysical instantiation of memories, behaviors, and knowledge will soon be before us. The downside is that no one knows exactly how to make sense of this data. I will show what connectomics data sets are and attempt to explain why it is so difficult to unravel their meaning.

Emergent composite-fermion Luttinger liquid at the half-filled Landau level
2023-11-20 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Prashant Kumar, University of Chicago
Title: Emergent composite-fermion Luttinger liquid at the half-filled Landau level
Abstract: The half-filled Landau level is a fascinating point in the phase diagram of quantum Hall effect that has been hypothesized to be a non-Fermi liquid where composite-fermions (CFs) emerge at low energies. In this talk, I will present our numerical calculations of the excitation spectrum of this state on infinite cylinders. By comparing our results with a quasi-1D theory of CFs, we show compelling quantitative evidence for the emergence of a Luttinger liquid of composite-fermions at low energies, distinct from that of electrons. I will comment on some future directions.

CMSA/Tsinghua Math-Science Literature Lecture: Scott Kominers
2023-11-20 09:00 - 10:30 | Virtual
CMSA/Tsinghua Math-Science Literature Lecture

Prof. Scott Kominers will present a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Monday, November 20, 2023
Time: 9:00 – 10:30 am ET
Location: Via Zoom Webinar
Title: 60 Years of Matching: From Gale and Shapley to Trading Networks
Abstract: Gale and Shapley’s 1962 American Mathematical Monthly paper, “College Admissions and the Stability of Marriage,” is by now one of the most cited articles in the journal’s history, having served as the foundation for an entire branch of the field of market design. This success owes in large part to the beautiful, applicable, and surprisingly general theory of matching mechanisms uncovered in Gale and Shapley’s work. This talk traces the history and evolution of matching theory from that paper forward to the present day, along the way touching on real-world applications to everything from medical residency matching to electricity markets.
Moderator: Sergiy Verstyuk

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

A quasi-local mass in general relativity
2023-11-17 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Aghil Alaee
Title: A quasi-local mass in general relativity
Abstract: One of the fundamental open problems in geometric analysis and mathematical relativity is constructing a (universal) energy/mass quantity to compute the energy/mass of a finite region of spacetime, which can include a star, black hole, or galaxy. We define a new gauge-independent quasi-local mass and energy with respect to the Minkowski spacetime, and show its relation to the Brown-York Hamilton-Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Rigidity is also established in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely, the mass vanishes for any such surface. This is joint work with M. Khuri and S.T. Yau.

Thresholds
2023-11-15 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Jinyoung Park (NYU)
Title: Thresholds
Abstract: For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a “threshold.” Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). I will present recent progress on this topic. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

On the Power of Forward pass through Transformer Architectures
2023-11-15 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Abhishek Panigrahi, Dept. of Computer Science, Princeton University
Title: On the Power of Forward pass through Transformer Architectures
Abstract: Highly trained transformers are capable of interesting computations as they infer for an input. The exact mechanism that these models use during forward passes is an interesting area of study. This talk studies two interesting phenomena.
In the first half, we explore how and why pre-trained language models, specifically BERT of moderate sizes, can effectively learn linguistic structures like parse trees during pre-training. Specifically, using synthetic data through PCFGs, we show how moderate-sized transformers can perform forward-backward parsing, also known as the inside-outside algorithm, during inference. We further understand the role of the pre-training loss for the model to learn to parse during pre-training.
In the second half, we consider in-context learning of large language models, where they learn to reason on the fly. An ongoing hypothesis is that transformers simulate gradient descent at inference to perform in-context learning. We propose the Transformer in Transformer (TinT) framework, which creates explicit transformer architectures that can simulate and fine-tune a small pre-trained transformer model during inference. E.g. a 1.3B parameter TINT model can simulate and fine-tune a 125 million parameter model in a single forward pass. This framework suggests that large transformers might execute intricate sub-routines during inference, and provides insights for enhancing their capabilities through intelligent design considerations.

A bulk gap in the presence of edge states for a truncated Haldane pseudopotential
2023-11-15 10:30 - 11:30 | Virtual
Topological Quantum Matter Seminar
Speaker: Amanda Young, UIUC
Title: A bulk gap in the presence of edge states for a truncated Haldane pseudopotential
Abstract: Haldane pseudopotentials were first introduced as Hamiltonian models for the fractional quantum Hall effect, and it has been long expected that they should exhibit the characteristic properties of this exotic phase of matter, including a spectral gap above the ground state energy. We will discuss recent work that verified this gap conjecture for a truncated version of the 1/3-filled Haldane pseudopotential in the cylinder geometry. Numerical evidence suggested that for open boundary conditions the gap of the truncated model closes as the cylinder radius converges to zero and that this closure is due to the presence of edge modes; in contrast, for periodic boundary conditions, the gap remains robustly order one in the same radius limit. The standard scheme for applying spectral gap estimating techniques to the model with periodic boundary conditions, though, produces a lower bound on the bulk gap that still reflects the energy of the edge modes. To obtain an estimate on the bulk gap that reflects its true behavior, a new gap estimating strategy was developed. By customizing the spectral gap method to key invariant subspaces of the Hamiltonian, we are able to successfully avoid the edge states and produce a more accurate lower bound on the bulk gap. In this talk, we discuss this invariant subspace strategy for proving bulk gaps in the presence of edge states. This is based off joint work with S. Warzel.

Quasi-Local Mass in a Binary Black Hole Merger
2023-11-14 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Daniel Kolb (Max Planck Institute)
Title: Quasi-Local Mass in a Binary Black Hole Merger
Abstract: One of the major open problems in classical general relativity is how one should define the mass of a finite region of space. In this talk, we will investigate a promising definition proposed by Wang and Yau in 2009. A closed 2-surface bounding the region of interest is embedded isometrically into Minkowski space. The mass is then calculated by comparing the extrinsic geometries. The Wang-Yau mass has many desirable properties, but it has previously not been calculated for surfaces in dynamical spacetimes. To remedy this, we will discuss how the Wang-Yau mass can be computed in practice and extend the definition to surfaces important in black hole dynamics: their quasi-local horizons. Finally, we look at how this mass behaves in a merger of black holes.

Koszul duality in QFT
2023-11-13 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Brian Williams (Boston University)
Title: Koszul duality in QFT
Abstract: We will describe appearances of the algebraic phenomena of Koszul duality in the context of boundary conditions and defects in quantum field theory. Primarily motivated by topological string theory, this point of view was pioneered by Costello and Li in their proposal for a twisted version of the AdS/CFT correspondence. Since then, many important examples of (twisted) holographic dualities in string and M-theory have been studied in work of Costello, Gaiotto, Paquette and many others. I will survey some of these examples and some current work with Raghavendran and Saberi which uses this formalism to predict exceptional symmetries present in M-theory.

Stacky small resolutions of determinantal octic double solids and noncommutative Gopakumar-Vafa invariants
2023-11-13 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Speaker: Sheldon Katz, UIUC
Title: Stacky small resolutions of determinantal octic double solids and noncommutative Gopakumar-Vafa invariants
Abstract: A determinantal octic double solid is the double cover X of P^3 branched along the degree 8 determinant of a symmetric matrix of homogeneous forms on P^3. These X are nodal CY threefolds which do not admit a projective small resolution. B-model techniques can be applied to compute GV invariants up to g le 32. This raises the question: what is the geometric meaning of these invariants?
Evidence suggests that these enumerative invariants are associated with moduli stacks of coherent sheaves of modules over a sheaf B of noncommutative algebras on X constructed by Kuznetsov. One of these moduli stacks is a stacky small resolution X’ of X itself. This leads to another geometric interpretation of the invariants as being associated with moduli of sheaves on X’ twisted by a Brauer class. Geometric computations based on these working definitions always agree with the B-model computations.
This talk is based on joint work with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe.
