CMSA

Anomalous Landau level analog in solids: search and implications for 2d heterostructures
2023-11-29 17:00 - 18:00 (America/New_York) | 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.

Uniqueness of Landau levels and their analogs with higher Chern numbers
2023-11-09 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Bruno Mera, Instituto Superior Tecnico
Title: Uniqueness of Landau levels and their analogs with higher Chern numbers
Abstract: Lowest Landau level wavefunctions are eigenstates of the Hamiltonian of a charged par- ticle in two dimensions under a uniform magnetic field. They are known to be holomorphic both in real and momentum spaces, and also exhibit uniform, translationally invariant, geometrical properties in momentum space. In this talk, using the Stone-von Neumann the- orem, we show that lowest Landau level wavefunctions are indeed the only possible states with unit Chern number satisfying these conditions. We also prove the uniqueness of their direct analogs with higher Chern numbers and provide their expressions.
Ref: Bruno Mera and Tomoki Ozawa. Uniqueness of Landau levels and their analogs with higher Chern numbers. arXiv:2304.00866, 2023. arXiv:2304.00866.

Nuclear chromodynamics: non-equilibrium phase transition in the nucleus of a living cell
2023-11-09 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Alexander Grosberg (NYU)

Title: Nuclear chromodynamics: non-equilibrium phase transition in the nucleus of a living cell
Abstract: Nucleus of a living cell houses a cell genome – a polymer called chromatin, which is a functional form of DNA. It is very long, e.g., 2 meters long for every human cell. Nucleus is also an arena of incessant energy-driven activity. Experiments show that chromatin undergoes large scale motions sustained over long times of order seconds. In the talk, after reviewing the phenomenology, I will show how these flows may arise due to a phase transition in which chromatin-driving motors, such as RNA polymerase, form a polar (“ferromagnetic”) order controlled by hydrodynamic interactions. The talk is based on the joint work with I.Eshghi and A.Zidovska.

Fitting ellipsoids to random points
2023-11-08 15:30 - 16:30 | Virtual
Probability Seminar
Speaker: Antoine Maillard (ETH Zürich)
Title: Fitting ellipsoids to random points
Abstract: We consider the problem of exactly fitting an ellipsoid (centered at 0) to n standard Gaussian random vectors in dimension d, for very large n and d. This problem has connections to questions in statistical learning and theoretical computer science, and is conjectured to undergo a sharp transition: with high probability, it has a solution if n < d^2/4, while it is not satisfiable if n > d^2/4. In this talk we will discuss the origin of this conjecture, and highlight some recent progress, in three different directions:

A proof that the problem is feasible for n < d^2 / C, for some (large) constant C, significantly improving over previously-known bounds.
A non-rigorous characterization of the conjecture, as well as significant generalizations, using analytical methods of statistical physics.
A rigorous proof of a satisfiability transition exactly at n = d^2 / 4 in a slightly relaxed version of the problem, the first rigorous result characterizing the expected phase transition in ellipsoid fitting. The proof is inspired by the non-rigorous characterization discussed above.

This talk is based on the three manuscripts: arXiv:2307.01181, arXiv:2310.01169, arXiv:2310.05787, which are joint works with A. Bandeira, Tim Kunisky, Shahar Mendelson and Elliot Paquette.

Peano: Learning Formal Mathematical Reasoning Without Human Data
2023-11-08 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Gabriel Poesia, Dept. of Computer Science, Stanford University
Title: Peano: Learning Formal Mathematical Reasoning Without Human Data
Abstract: Peano is a theorem proving environment in which a computational agent can start tabula rasa in a new domain, learn to solve problems through curiosity-driven exploration, and create its own higher level actions. Gabriel will describe the system, present case studies on learning to solve simple algebra problems from the Khan Academy platform, and describe work on progress on learning the Natural Number Game, a popular introduction to theorem proving in Lean for mathematicians.

A Long Exact Sequence in Symmetry Breaking
2023-11-07 16:30 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Cameron Krulewski (MIT) and Leon Liu (Harvard)
Title: A Long Exact Sequence in Symmetry Breaking
Abstract: We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations that sometimes occur at the cores of these objects. These are topologically protected by an ’t Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the “symmetry breaking long exact sequence” (SBLES).
The mathematical backbone of the SBLES is the Smith homomorphism, which gives a family of maps between twisted bordism groups. Though many examples have been studied, we give the first completely general account of the Smith homomorphism. We lift it to a map of Thom
spectra and identify the cofiber, producing a long exact sequence of twisted bordism groups; the SBLES is the Anderson dual of that long exact sequence. Our work develops further the theory of higher Berry phase and its bulk-boundary correspondence and serves as a new computational tool for classifying symmetry protected topological phases.

CMSA Q and A Seminar 11/7/2023
2023-11-07 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Michael Hopkins (Harvard Math) and Minhea Popa (Harvard Math)
Topics:
Mike Hopkins: What are topological modular forms?
Mihnea Popa: What is the abundance conjecture and why should it to be true?

Fluid stabilization in slowly expanding cosmological spacetime
2023-11-07 11:00 - 12:00 | Hybrid – G10
General Relativity Seminar
Speaker: David Fajman (Vienna)
Title: Fluid stabilization in slowly expanding cosmological spacetime
Abstract: Relativistic fluids are known to form shocks during their evolution from near-homogeneous initial data. In expanding spacetimes, shock formation is suppressed, if the expansion is sufficiently strong. We refer to this effect as fluid stabilization. The occurrence of this phenomenon depends on features of the fluid and has implications for our understanding of structure formation and cosmological evolution. While the effect is well studied in the regime of accelerated expansion, in recent years it has been shown that fluid stabilization occurs as well in spacetimes with slower expansion rates. In this talk we present different recent results on fluid stabilization in slowly expanding spacetimes and aspects of the methods involved in the respective proofs.

Talk via Zoom, broadcast in G10

Impossibility results in classical dynamical systems
2023-11-06 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Matthew Foreman (UC Irvine)
Title: Impossibility results in classical dynamical systems
Abstract: In 1932, motivated by questions in statistical and celestial mechanics, von Neumann proposed classifying the statistical behavior of dynamical systems. In the 1960’s, motivated by work of Poincaré, Smale proposed classifying the qualitative behavior of dynamical systems. These questions laid the groundwork for enormous amounts of work, but the fundamental questions remain open. This talk shows that they are impossible to answer in a rigorous sense. The talk will discuss various kinds of impossibility results and describe how they apply to von Neumann’s program and Smale’s program.

Deformations of Landau-Ginzburg models and their fibers
2023-11-06 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Andrew Harder, Lehigh University
Title: Deformations of Landau-Ginzburg models and their fibers
Abstract: In mirror symmetry, the dual object to a Fano variety is a Landau-Ginzburg model. Broadly, a Landau-Ginzburg model is quasi-projective variety Y with a superpotential function w, but not all such pairs correspond to Fano varieties under mirror symmetry, so a very natural question to ask is: Which Landau-Ginzburg models are mirror to Fano varieties? In this talk, I will discuss a cohomological characterization of mirrors of (semi-)Fano varieties, focusing on the case of threefolds. I’ll discuss how this characterization relates to the deformation and Hodge theory of (Y,w), and in particular, how the classification of (semi-)Fano threefolds is related to questions about moduli spaces of lattice polarized K3 surfaces.

Deformations of Landau-Ginzburg models and their fibers
2023-11-03 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Zi Yang Kang
Title: Deformations of Landau-Ginzburg models and their fibers
Abstract: In mirror symmetry, the dual object to a Fano variety is a Landau-Ginzburg model. Broadly, a Landau-Ginzburg model is quasi-projective variety Y with a superpotential function w, but not all such pairs correspond to Fano varieties under mirror symmetry, so a very natural question to ask is: Which Landau-Ginzburg models are mirror to Fano varieties? In this talk, I will discuss a cohomological characterization of mirrors of (semi-)Fano varieties, focusing on the case of threefolds. I’ll discuss how this characterization relates to the deformation and Hodge theory of (Y,w), and in particular, how the classification of (semi-)Fano threefolds is related to questions about moduli spaces of lattice polarized K3 surfaces.

Landscape of quantum phases in quantum materials
2023-11-02 16:30 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint Quantum Matter in Mathematics and Physics and Topological Quantum Matter Seminar
Speaker: Liujun Zou (Perimeter Institute)
Title: Landscape of quantum phases in quantum materials

Abstract: A central goal of condensed matter physics is to understand which quantum phases of matter can emerge in a quantum material. For this purpose, one should be able to not only describe the quantum phases using some effective field theories, but also capture the important microscopic information of the material via mathematical formulation. In this talk, I will present a framework to classify quantum phases in quantum materials, where the microscopic information of a material is encoded in its quantum anomaly. I will talk about the application of this framework to classify various exotic quantum phases of matter in different lattice systems. Using our framework, we have obtained many results unexpected from the previous literature.

Solving spin systems, the Babylonian way
2023-11-02 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Nicola Kistler (Johann Wolfgang Goethe-Universität Frankfurt am Main)
Title: Solving spin systems, the Babylonian way
Abstract: The replica method, together with Parisi’s symmetry breaking mechanism, is an extremely powerful tool to compute the limiting free energy of virtually any mean field disordered system. Unfortunately, the tool is dramatically flawed from a mathematical point of view. I will discuss a truly elementary procedure which allows to rigorously implement two (out of three) steps of the replica method, and conclude with some remarks on the relation between this new point of view and old work by Mezard and Virasoro on the microstructure of ultrametricity, the latter being the fundamental yet unjustified Ansatz in the celebrated Parisi solution. We are still far from a clear understanding of the issues, but quite astonishingly, evidence is mounting that Parisi’s ultrametricity assumption, the onset of scales and the universal hierarchical self-organisation of random systems in the infinite volume limit, is intimately linked to hidden geometrical properties of large random matrices which satisfy rules reminiscent of the popular SUDOKU game.

Universality of max-margin classifiers
2023-11-01 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Youngtak Sohn (MIT)
Title: Universality of max-margin classifiers
Abstract: Many modern learning methods, such as deep neural networks, are so complex that they perfectly fit the training data. Despite this, they generalize well to the unseen data. Motivated by this phenomenon, we consider high-dimensional binary classification with linearly separable data. First, we consider Gaussian covariates and characterize linear classification problems for which the minimum norm interpolating prediction rule, namely the max-margin classification, has near-optimal prediction accuracy. Then, we discuss universality of max-margin classification. In particular, we characterize the prediction accuracy of the non-linear random features model, a two-layer neural network with random first layer weights. The spectrum of the kernel random matrices plays a crucial role in the analysis. Finally, we consider the wide-network limit, where the number of neurons tends to infinity, and show how non-linear max-margin classification with random features collapse to a linear classifier with a soft-margin objective.
Joint work with Andrea Montanari, Feng Ruan, Jun Yan, and Basil Saeed.

Unveiling Correlated Topological Insulators through Fermionic Tensor Network States
2023-11-01 10:30 - 11:30 | Virtual
Topological Quantum Matter Seminar
Speaker: Shenghan Jiang, Kavli Institute for Theoretical Sciences UCAS
Title: Unveiling Correlated Topological Insulators through Fermionic Tensor Network States
Abstract: The study of topological band insulators has revealed fascinating phases characterized by band topology indices, harboring extraordinary boundary modes protected by anomalous symmetry actions. In strongly correlated systems, it has been established that topological insulator phases persist as stable phases. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wavefunctions for numerical simulations is highly desirable.
In this talk, we tackle this challenge by developing a comprehensive framework with fermionic tensor network states. Starting from simple assumptions, we write down tensor equations, construct edge theories and extract quantum anomaly data for topological insulators. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for corresponding topological insulator phases. Our methodology provides a crucial first step towards simulating topological insulators in strongly correlated systems.
