CMSA

General-relativistic viscous fluids
2022-09-29 10:30 - 11:30 (America/New_York) | Virtual
General Relativity Seminar
Speaker: Marcelo Disconzi, Vanderbilt University
Title: General–relativistic viscous fluids

Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical relativity simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and comprehensive theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

GLSM, Homological projective duality and nc resolutions
2022-09-30 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Mauricio Romo, Tsinghua University
Title: GLSM, Homological projective duality and nc resolutions

Abstract: Kuznetsov’s Homological projective duality (HPD) in algebraic geometry is a powerful theorem that allows to extract information about semiorthogonal decompositions of derived categories of certain varieties. I will give a GLSMs perspective based on categories of B-branes. I will focus mostly on the case of Fano (hypersurfaces) manifolds. In general, for such cases the HPD can be interpreted as a non-commutative (nc) resolution of a compact variety. I will give a physical interpretation of this fact and present some conjectures.

Kahler geometry in twisted materials
2022-09-30 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Jie Wang
Title: Kahler geometry in twisted materials
Abstract: Flatbands are versatile platform for realizing exotic quantum phases due to the enhanced interactions. The canonical example is Landau level where fractional quantum Hall physics exists. Although interaction is strong, the fractional quantum Hall effect is relatively well understood thanks to its model wavefunction, exact parent Hamiltonian, conformal field theory analogous and other exact aspects. In generic flatbands, the interacting physics is controlled by the interplay between the interaction scale and intrinsic quantum geometries, in particular the Berry curvature and the Fubini-Study metric, which are in general spatially non-uniform. It is commonly believed that the non-uniform geometries destroy these exact properties of fractional quantum Hall physics, making many-body states less stable in flatbands.
In this talk, I will disprove this common belief by showing a large family of flatbands (ideal flatbands) where quantum geometries can be highly non-uniform, but still exhibit exact properties such as model wavefunctions, density algebra, exact parent Hamiltonians. I will discuss both the theory of ideal flatband, its experimental realization in Dirac materials as well as implications.

Holomorphic Twists and Confinement in N=1 SYM
2022-10-04 09:30 - 11:00 | Virtual
Quantum Matter Seminar
Speaker: Justin Kulp (Perimeter Institute)
Title: Holomorphic Twists and Confinement in N=1 SYM
Abstract: Supersymmetric QFT’s are of long-standing interest for their high degree of solvability, phenomenological implications, and rich connections to mathematics. In my talk, I will describe how the holomorphic twist isolates the protected quantities which give SUSY QFTs their potency by restricting to the cohomology of one supercharge. I will briefly introduce infinite dimensional symmetry algebras, generalizing Virasoro and Kac-Moody symmetries, which emerge. Finally, I will explain a potential novel UV manifestation of confinement, dubbed “holomorphic confinement,” in the example of pure SU(N) super Yang-Mills. Based on arXiv:2207.14321 and 2 forthcoming works with Kasia Budzik, Davide Gaiotto, Brian Williams, Jingxiang Wu, and Matthew Yu.

Minerva: Solving Quantitative Reasoning Problems with Language Models
2022-10-05 14:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Guy Gur-Ari, Google Research
Title: Minerva: Solving Quantitative Reasoning Problems with Language Models
Abstract: Quantitative reasoning tasks which can involve mathematics, science, and programming are often challenging for machine learning models in general and for language models in particular. We show that transformer-based language models obtain significantly better performance on math and science questions when trained in an unsupervised way on a large, math-focused dataset. Performance can be further improved using prompting and sampling techniques including chain-of-thought and majority voting. Minerva, a model that combines these techniques, achieves SOTA on several math and science benchmarks. I will describe the model, its capabilities and limitations.

Quantum statistical mechanics of charged black holes and strange metals
2022-10-05 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Please note this colloquium will be held at a special time: 4:00-5:00 pm.
Speaker: Subir Sachdev (Harvard)
Title: Quantum statistical mechanics of charged black holes and strange metals

Abstract: The Sachdev-Ye-Kitaev model was introduced as a toy model of interacting fermions without any particle-like excitations. I will describe how this toy model yields the universal low energy quantum theory of generic charged black holes in asymptotically 3+1 dimensional Minkowski space. I will also discuss how extensions of the SYK model yield a realistic theory of the strange metal phase of correlated electron systems.

Slides: cmsa22

Duality in Einstein’s Gravity
2022-10-06 10:30 - 23:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Uri Kol, CMSA

Title: Duality in Einstein’s Gravity
Abstract: Electric-Magnetic duality has been a key feature behind our understanding of Quantum Field Theory for over a century. In this talk I will describe a similar property in Einstein’s gravity. The gravitational duality reveals, in turn, a wide range of new IR phenomena, including aspects of the double copy for scattering amplitudes, asymptotic symmetries and more.

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces
2022-10-07 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Sam Bardwell-Evans, Boston University

Title: Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Abstract: In this talk, we construct special Lagrangian fibrations for log Calabi-Yau surfaces and scattering diagrams from Lagrangian Floer theory of the fibers. These scattering diagrams recover the algebro-geometric scattering diagrams of Gross-Pandharipande-Siebert and Gross-Hacking-Keel. The argument relies on a holomorphic/tropical disc correspondence to control the behavior of holomorphic discs, allowing us to relate open Gromov-Witten invariants to log Gromov-Witten invariants. This talk is based on joint work with Man-Wai Mandy Cheung, Hansol Hong, and Yu-Shen Lin.

Principal flow, sub-manifold and boundary
2022-10-07 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Zhigang Yao
Title: Principal flow, sub-manifold and boundary
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. I will discuss the problem of finding principal components to the multivariate datasets, that lie on an embedded nonlinear Riemannian manifold within the higher-dimensional space. The aim is to extend the geometric interpretation of PCA, while being able to capture the non-geodesic form of variation in the data. I will introduce the concept of a principal sub-manifold, a manifold passing through the center of the data, and at any point on the manifold extending in the direction of highest variation in the space spanned by the eigenvectors of the local tangent space PCA. We show the principal sub-manifold yields the usual principal components in Euclidean space. We illustrate how to find, use and interpret the principal sub-manifold, by which a principal boundary can be further defined for data sets on manifolds.

The Penrose Inequality as a Constraint on Low Energy Quantum Gravity
2022-10-11 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Swampland Seminar

Speaker: Aasmund Folkestad (MIT)

Title: The Penrose Inequality as a Constraint on Low Energy Quantum Gravity

Abstract: In this talk, I argue that the Penrose inequality (PI) can be used to constrain low energy theories compatible AdS/CFT, and possibly also quantum gravity in flat space. Focusing on AdS/CFT, it is shown that the PI can be violated for minimally coupled scalar fields, and I produce exclusion plots on couplings that respect the PI. I also present numerical evidence that top-down scalar theories and supersymmetric theories respect the PI. Finally, similar to the Breitenlohner-Freedman bound, I give a necessary condition for the stability AdS that constrains coupling constants (beyond the scalar mass).

Engineering topological phases with a superlattice potential
2022-10-12 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Jennifer Cano (Stony Brook and Flatiron Institute)
Title: Engineering topological phases with a superlattice potential

Abstract: We propose an externally imposed superlattice potential as a platform for manipulating topological phases, which has both advantages and disadvantages compared to a moire superlattice. In the first example, we apply the superlattice potential to the 2D surface of a 3D topological insulator. The superlattice potential creates tunable van Hove singularities, which, when combined with strong spin-orbit coupling and Coulomb repulsion give rise to a topological meron lattice spin texture. Thus, the superlattice potential provides a new route to the long sought-after goal of realizing spontaneous magnetic order on the surface of a 3D TI. In the second example, we show that a superlattice potential applied to Bernal-stacked bilayer graphene can generate flat Chern bands, similar to in twisted bilayer graphene, whose bandwidth can be as small as a few meV. The superlattice potential offers flexibility in both lattice size and geometry, making it a promising alternative to achieve designer flat bands without a moire heterostructure.

Complete disorder is impossible: Some topics in Ramsey theory
2022-10-12 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: James Cummings,Carnegie Mellon University
Title: Complete disorder is impossible: Some topics in Ramsey theory
Abstract: The classical infinite Ramsey theorem states that if we colour pairs of natural numbers using two colours, there is an infinite set all of whose pairs get the same colour. This is the beginning of a rich theory, which touches on many areas of mathematics including graph theory, set theory and dynamics. I will give an overview of Ramsey theory, emphasizing the diverse ideas which are at play in this area.

Strong Cosmic Censorship
2022-10-13 10:30 - 23:30 | Virtual
General Relativity Seminar

Speaker: Professor Oscar Dias (University of Southampton)

Title: Strong Cosmic Censorship
Abstract: Generically, strong cosmic censorship (SCC) is the statement that physics within general relativity should be predicted from initial data prescribed on a Cauchy hypersurface. In this talk I will review how fine-tuned versions of SCC have been formulated and evolved along the last decades up to the point where we believe that Christodoulou’s version is true in asymptotically flat spacetimes. However, I will also describe that in recent years it was found that this is no longer necessarily true for some other backgrounds, namely in some de Sitter (with a positive cosmological constant) spacetimes or even in rotating BTZ black holes in 3-dimensional Anti-de Sitter spacetime. Finally, I will discuss some possibilities (quantum effects, non-smooth initial data,…) that might restore SCC in those backgrounds where the standard formulation of the conjecture is violated.

Singularities of the quantum connection on a Fano variety
2022-10-14 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Speaker: Daniel Pomerleano, UMass Boston
Title: Singularities of the quantum connection on a Fano variety
Abstract: The small quantum connection on a Fano variety is one of the simplest objects in enumerative geometry. Nevertheless, it is the subject of far-reaching conjectures known as the Dubrovin/Gamma conjectures. Traditionally, these conjectures are made for manifolds with semi-simple quantum cohomology or more generally for Fano manifolds whose quantum connection is of unramified exponential type at q=infty.
I will explain a program, joint with Paul Seidel, to show that this unramified exponential type property holds for all Fano manifolds M carrying a smooth anticanonical divisor D. The basic idea of our argument is to view these structures through the lens of a noncommutative Landau-Ginzburg model intrinsically attached to (M, D).

Quantum magnet chains and Kashiwara crystals
2022-10-14 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Leonid Rybnikov, Harvard CMSA/National Research University Higher School of Economics
Title: Quantum magnet chains and Kashiwara crystals
Abstract: Solutions of the algebraic Bethe ansatz for quantum magnet chains are, generally, multivalued functions of the parameters of the integrable system. I will explain how to compute some monodromies of solutions of Bethe ansatz for the Gaudin magnet chain. Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the Deligne-Mumford moduli space of stable rational curves, which is unramified over the real locus of the Deligne-Mumford space. The monodromy action of the fundamental group of this space (called cactus group) on the eigenlines can be described very explicitly in purely combinatorial terms of Kashiwara crystals — i.e. combinatorial objects modeling the tensor category of finite-dimensional representations of a semisimple Lie algebra g. More specifically, this monodromy action is naturally equivalent to the action of the same group by commutors (i.e. combinatorial analog of a braiding) on a tensor product of Kashiwara crystals. This is joint work with Iva Halacheva, Joel Kamnitzer, and Alex Weekes.

Topological Wick Rotation and Holographic duality
2022-10-17 09:00 - 10:30 | Virtual
Quantum Matter Seminar
Speaker: Liang Kong (Sustech)
Title: Topological Wick Rotation and Holographic duality
Abstract: I will explain a new type of holographic dualities between
n+1D topological orders with a chosen boundary condition and nD
(potentially gapless) quantum liquids. It is based on the idea of
topological Wick rotation, a notion which was first used in
arXiv:1705.01087 and was named, emphasized and generalized later in
arXiv:1905.04924. Examples of these holographic dualities include the
duality between 2+1D toric code model and 1+1D Ising chain and its
finite-group generalizations (independently discovered by many
others); those between 2+1D topological orders and 1+1D rational
conformal field theories; and those between n+1D finite gauge theories
with a gapped boundary and nD gapped quantum liquids. I will also
briefly discuss some generalizations of this holographic duality and
its relation to AdS/CFT duality.

On the six-dimensional origin of non-invertible symmetries
2022-10-18 09:00 - 10:30 | Virtual
Quantum Matter Seminar
Speaker: Michele Del Zotto (Uppsala University)
Title: On the six-dimensional origin of non-invertible symmetries
Abstract: I will present a review about recent progress in charting non-invertible symmetries for four-dimensional quantum field theories that have a six-dimensional origin. These include in particular N=4 supersymmetric Yang-Mills theories, and also a large class of N=2 supersymmetric theories which are conformal and do not have a conventional Lagrangian description (the so-called theories of “class S”). Among the main results, I will explain criteria for identifying examples of systems with intrinsic and non-intrinsic non-invertible symmetries, as well as explore their higher dimensional origin. This seminar is based on joint works with Vladimir Bashmakov, Azeem Hasan, and Justin Kaidi.

The Mobility Edge of Lévy Matrices
2022-10-19 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Patrick Lopatto (Brown)
Title: The Mobility Edge of Lévy Matrices
Abstract: Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an alpha-stable law; such distributions have infinite variance when alpha is less than 2. Due to the ubiquity of heavy-tailed randomness, these models have been broadly applied in physics, finance, and statistics. When the entries have infinite mean, Lévy matrices are predicted to exhibit a phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the physical context for this conjecture, and describe a result establishing it for values of alpha close to zero and one. This is joint work with Amol Aggarwal and Charles Bordenave.

Towards Faithful Reasoning Using Language Models
2022-10-19 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Antonia Creswell, DeepMind
Title: Towards Faithful Reasoning Using Language Models
Abstract: Language models are showing impressive performance on many natural language tasks, including question-answering. However, language models – like most deep learning models – are black boxes. We cannot be sure how they obtain their answers. Do they reason over relevant knowledge to construct an answer or do they rely on prior knowledge – baked into their weights – which may be biased? An alternative approach is to develop models whose output is a human interpretable, faithful reasoning trace leading to an answer. In this talk we will characterise faithful reasoning in terms of logically valid reasoning and demonstrate where current reasoning models fall short. Following this, we will introduce Selection-Inference, a faithful reasoning model, whose causal structure mirrors the requirements for valid reasoning. We will show that our model not only produces more accurate reasoning traces but also improves final answer accuracy.

Symmetric Mass Generation
2022-10-19 16:00 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Yizhuang You, UC San Diego
Title: Symmetric Mass Generation

Abstract: Symmetric mass generation (SMG) is a novel mechanism for massless fermions to acquire a mass via a strong-coupling non-perturbative interaction effect. In contrast to the conventional Higgs mechanism for fermion mass generation, the SMG mechanism does not condense any fermion bilinear coupling and preserves the full symmetry. It is connected to a broad range of topics, including anomaly cancellation, topological phase classification, and chiral fermion regularization. In this talk, I will introduce SMG through toy models, and review the current understanding of the SMG transition. I will also mention recent numerical efforts to investigate the SMG phenomenon. I will conclude the talk with remarks on future directions.

Love Symmetry of Black Holes
2022-10-20 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Sergei Dubovsky (New York University)

Title: Love Symmetry of Black Holes

Abstract: Perturbations of massless fields in the Kerr-Newman black hole background enjoy a (“Love”) SL(2,ℝ) symmetry in the suitably defined near zone approximation. We show how the intricate behavior of black hole responses in four and higher dimensions can be understood from the SL(2,ℝ) representation theory. In particular, static perturbations of four-dimensional black holes belong to highest weight SL(2,ℝ) representations. It is this highest-weight property that forces the static Love numbers to vanish. We show that the Love symmetry is tightly connected to the enhanced isometries of extremal black holes. The Love symmetry also exhibits a peculiar UV/IR mixing.

Attempts at understanding human axial elongation and patterning
2022-10-20 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Sharad Ramanathan, Harvard

Title: Attempts at understanding human axial elongation and patterning

Abstract: Some of the most dramatic events during human development is the axial elongation of the embryo with concomitant changes in the geometry and composition of the underlying tissues. The posterior part of the embryo gives rise to the spinal cord, vertebral column, ribcage, back muscles, and dermis. In this talk, I will present our attempts at coaxing human embryonic stem cells to form these structures of the early human embryo that closely recapitulate the geometry, relative arrangements, composition, and dynamics of development of the early spinal cord flanked progenitors of the musculoskeletal system. Our goal was to do so, such that we could build hundreds of these organoids at a time. I will also present preliminary results for the use of this system to understand key events during early human development through imaging and genetic perturbations.

The index of M-theory
2022-10-21 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Nicolo Piazzalunga, Rutgers
Title: The index of M-theory
Abstract: I’ll introduce the higher-rank Donaldson-Thomas theory for toric Calabi-Yau threefolds, within the setting of equivariant K-theory. I’ll present a factorization conjecture motivated by Physics. As a byproduct, I’ll discuss some novel properties of equivariant volumes, as well as their generalizations to the genus-zero Gromov-Witten theory of non-compact toric varieties.

Explicit Ramsey Graphs and Two Source Extractors
2022-10-21 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: David Zuckerman, Harvard CMSA/University of Texas at Austin
Title: Explicit Ramsey Graphs and Two Source Extractors
Abstract: Ramsey showed that any graph on N nodes contains a clique or independent set of size (log N)/2. Erdos showed that there exist graphs on N nodes with no clique or independent set of size 2 log N, and asked for an explicit construction of such graphs. This turns out to relate to the question of extracting high-quality randomness from two independent low-quality sources. I’ll explain this connection and our recent exponential improvement in constructing two-source extractors.

Insulating BECs and other surprises in dipole-conserving systems
2022-10-24 09:00 - 10:30 | Virtual
Quantum Matter Seminar
Speaker: Ethan Lake (MIT)
Title: Insulating BECs and other surprises in dipole-conserving systems
Abstract: I will discuss recent work on bosonic models whose dynamics conserves both total charge and total dipole moment, a situation which can be engineered in strongly tilted optical lattices. Related models have received significant attention recently for their interesting out-of-equilibrium dynamics, but analytic and numeric studies reveal that they also possess rather unusual ground states. I will focus in particular on a dipole-conserving variant of the Bose-Hubbard model, which realizes an unusual phase of matter that possesses a Bose-Einstein condensate, but which is nevertheless insulating, and has zero superfluid weight. Time permitting, I will also describe the physics of a regime in which these models spontaneously fracture into an exotic type of glassy state.

Anomalies of Discrete Gauge Symmetries and their Cancellation in 6D F-theory
2022-10-24 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Swampland Seminar
Speaker: Paul-Konstantin Oehlmann(Northeastern)
Title: Anomalies of Discrete Gauge Symmetries and their Cancellation in 6D F-theory
Abstract: We consider 6D SUGRAs with a discrete gauge group G, engineered via F-theory compactifications on genus-one fibered threefolds. We argue that group G suffers from Dai-Freed anomalies that can be canceled via a discrete Green-Schwarz mechanism. We comment on the ambiguity to assign this GS term in the 7D Anomaly theory which leads to choices that are not all compatible with F-theory.
In F-theory we then deduce this Anomaly coefficient explicitly by computing the elliptic genera of the non-critical strings that couple to the 6D two-form fields: Their 2D worldsheet theories inherits a G Flavor symmetries whose t’Hooft anomaly cancels the 6D Dai-Freed anomaly in the bulk via inflow. This talk is based on work in preparation together with Markus Dierigl and Thorsten Schimmanek.

Unorientable Quantum Field Theories: From crosscaps to holography
2022-10-25 09:00 - 10:30 | Virtual
Quantum Matter Seminar
Speaker: João Caetano (CERN)
Title: Unorientable Quantum Field Theories: From crosscaps to holography
Abstract: In two dimensions, one can study quantum field theories on unorientable manifolds by introducing crosscaps. This defines a class of states called crosscap states which share a few similarities with the notion of boundary states. In this talk, I will show that integrable theories remain integrable in the presence of crosscaps, and this allows to exactly determine the crosscap state.

In four dimensions, the analog is to place the quantum field theory on the real projective space, the simplest unorientable 4-manifold. I will show how to do this in the example of N=4 Supersymmetric Yang-Mills, discuss its holographic description and present a new solvable setup of AdS/CFT.

Kähler bands—Chern insulators, holomorphicity and induced quantum geometry
2022-10-26 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar

Speaker: Bruno Mera, Tohoku University

Title: Kähler bands—Chern insulators, holomorphicity and induced quantum geometry

Abstract: The notion of topological phases has dramatically changed our understanding of insulators. There is much to learn about a band insulator beyond the assertion that it has a gap separating the valence bands from the conduction bands. In the particular case of two dimensions, the occupied bands may have a nontrivial topological twist determining what is called a Chern insulator. This topological twist is not just a mathematical observation, it has observable consequences—the transverse Hall conductivity is quantized and proportional to the 1st Chern number of the vector bundle of occupied states over the Brillouin zone. Finer properties of band insulators refer not just to the topology, but also to their geometry. Of particular interest is the momentum-space quantum metric and the Berry curvature. The latter is the curvature of a connection on the vector bundle of occupied states. The study of the geometry of band insulators can also be used to probe whether the material may host stable fractional topological phases. In particular, for a Chern band to have an algebra of projected density operators which is isomorphic to the W∞ algebra found by Girvin, MacDonald and Platzman—the GMP algebra—in the context of the fractional quantum Hall effect, certain geometric constraints, associated with the holomorphic character of the Bloch wave functions, are naturally found and they enforce a compatibility relation between the quantum metric and the Berry curvature of the band. The Brillouin zone is then endowed with a Kähler structure which, in this case, is also translation-invariant (flat). Motivated by the above, we will provide an overview of the geometry of Chern insulators from the perspective of Kähler geometry, introducing the notion of a Kähler band which shares properties with the well-known ideal case of the lowest Landau level. Furthermore, we will provide a prescription, borrowing ideas from geometric quantization, to generate a flat Kähler band in some appropriate asymptotic limit. Such flat Kähler bands are potential candidates to host and realize fractional Chern insulating phases. Using geometric quantization arguments, we then provide a natural generalization of the theory to all even dimensions.

References:
[1] Tomoki Ozawa and Bruno Mera. Relations between topology and the quantum metric for Chern insulators. Phys. Rev. B, 104:045103, Jul 2021.
[2] Bruno Mera and Tomoki Ozawa. Kähler geometry and Chern insulators: Relations between topology and the quantum metric. Phys. Rev. B, 104:045104, Jul 2021.
[3] Bruno Mera and Tomoki Ozawa. Engineering geometrically flat Chern bands with Fubini-Study Kähler structure. Phys. Rev. B, 104:115160, Sep 2021.

Clique listing algorithms
2022-10-26 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Virginia Vassilevska Williams (MIT)
Title: Clique listing algorithms
Abstract: A k-clique in a graph G is a subgraph of G on k vertices in which every pair of vertices is linked by an edge. Cliques are a natural notion of social network cohesiveness with a long history.
A fundamental question, with many applications, is “How fast can one list all k-cliques in a given graph?”.
Even just detecting whether an n-vertex graph contains a k-Clique has long been known to be NP-complete when k can depend on n (and hence no efficient algorithm is likely to exist for it). If k is a small constant, such as 3 or 4 (independent of n), even the brute-force algorithm runs in polynomial time, O(n^k), and can list all k-cliques in the graph; though O(n^k) time is far from practical. As the number of k-cliques in an n-vertex graph can be Omega(n^k), the brute-force algorithm is in some sense optimal, but only if there are Omega(n^k) k-cliques. In this talk we will show how to list k-cliques faster when the input graph has few k-cliques, with running times depending on the number of vertices n, the number of edges m, the number of k-cliques T and more. We will focus on the case when k=3, but we will note some extensions.
(Based on joint work with Andreas Bjorklund, Rasmus Pagh, Uri Zwick, Mina Dalirrooyfard, Surya Mathialagan and Yinzhan Xu)

From Engine to Auto
2022-10-26 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speakers: João Araújo, Mathematics Department, Universidade Nova de Lisboa and Michael Kinyon, Department of Mathematics, University of Denver

Title: From Engine to Auto

Abstract: Bill McCune produced the program EQP that deals with first order logic formulas and in 1996 managed to solve Robbins’ Conjecture. This very powerful tool reduces to triviality any result that can be obtained by encoding the assumptions and the goals. The next step was to turn the program into a genuine assistant for the working mathematician: find ways to help the prover with proofs; reduce the lengths of the automatic proofs to better crack them; solve problems in higher order logic; devise tools that autonomously prove results of a given type, etc.

In this talk we are going to show some of the tools and strategies we have been producing. There will be real illustrations of theorems obtained for groups, loops, semigroups, logic algebras, lattices and generalizations, quandles, and many more.
