## News

June 13, 2022

### Representation Theory, Calabi–Yau Manifolds, and Mirror Symmetry

On November 28 – Dec 1, 2022, the CMSA will host a Workshop on Representation...

June 10, 2022

### Math Science Lectures in Honor of Raoul Bott

(Dates TBA) In March 2023 Andrea Bertozzi (UCLA) will give two lectures hosted by the...

October 5, 2022

### Third Annual Yip Lecture

Andrew Strominger will give the Third Annual Yip Lecture on February 2, 2023. Time: 7:00-8:00...

## Upcoming Events

### December 2, 202211:00 am

Speaker : Alejandro Poveda , Harvard CMSA

### December 5, 202211:00 am

Speaker : Markus Dierigl , LMU Munich

### December 7, 202210:00 am

Speaker : Vasil Rokaj , Harvard

### December 7, 20222:00 pm

Speaker : Cyril Zhang , Microsoft Research

### December 7, 20223:30 pm

Speaker : Lior Alon , MIT

### Title: Title TBA

< 2021 >
November
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October
1
• Swampland Seminar

##### Bubble instability of mIIA on AdS_4 x S^6

13:00 -14:00
2021-11-01

Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.

2
• Joint Harvard-CUHK-YMSC Differential Geometry

##### Joint Harvard-CUHK-YMSC Differential GeometryCounting invariant curves on a Calabi-Yau threefold with an involution

09:30 -10:30
2021-11-02

Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface, the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.

• Algebraic Geometry in String Theory Seminar

##### Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero

13:00 -14:00
2021-11-02

Abstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)

3
• Colloquia

##### Hitchin map as spectrum of equivariant cohomology

09:30 -10:30
2021-11-03
• Quantum Matter

##### 11/3/2021 Quantum Matter in Mathematics and Physics

14:00 -15:30
2021-11-03

Title: Non-Invertible Duality Defects in 3+1 Dimensions

Abstract:  For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions and determine the fusion rule. From modular invariance and a direct analysis of one-form symmetry-protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. By further assuming time-reversal symmetry, we find that the presence of certain duality defects implies that the low-energy phase has to be gapless unless the one-form symmetry is spontaneously broken. We give an explicit realization of this duality defect in the free Maxwell theory where the duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides.

• New Technologies in Mathematics Seminar

##### When Computer Algebra Meets Satisfiability: A New Approach to Combinatorial Mathematics

17:12 -18:12
2021-11-03

Abstract: Solvers for the Boolean satisfiability (SAT) problem have been increasingly used to resolve problems in mathematics due to their excellent search algorithms.  This talk will describe a new method for mathematical search that couples SAT solvers with computer algebra systems (CAS), thereby combining the expressiveness of CASs with the search power of SAT solvers.  This paradigm has led to a number of results on long-standing mathematical questions such as the first computer-verifiable resolution of Lam’s problem and the discovery of a new infinite class of Williamson matrices.

4
• Quantum Matter

##### 11/4/2021 Quantum Matter in Mathematics and Physics

10:30 -12:00
2021-11-04

Title: Fusion Category Symmetries in Quantum Field Theory

Abstract: Topological defects provide a modern perspective on symmetries in quantum field theory. They generalize the familiar invertible symmetries described by groups to non-invertible symmetries described by fusion categories. Such generalized symmetries are ubiquitous in quantum field theory and provide new constraints on renormalization group flows and the IR phase diagram. In this talk I’ll review some recent progress in identifying and understanding fusion category symmetries in 1+1d conformal field theories. Time permitting, I’ll also comment on higher dimensional generalizations.

• General Relativity Seminar

##### The stability of charged black holes

13:00 -14:00
2021-11-04

Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

• Interdisciplinary Science Seminar

##### Interdisciplinary Science Seminar11/4/21 CMSA Interdisciplinary Science Seminar

18:46 -20:46
2021-11-04

Title: Exploring Invertibility in Image Processing and Restoration

Abstract: Today’s smartphones have enabled numerous stunning visual effects from denoising to beautification, and we can share high-quality JPEG images easily on the internet, but it is still valuable for photographers and researchers to keep the original raw camera data for further post-processing (e.g., retouching) and analysis. However, the huge size of raw data hinders its popularity in practice, so can we almost perfectly restore the raw data from a compressed RGB image and thus avoid storing any raw data? This question leads us to design an invertible image signal processing pipeline. Then we further explore invertibility in other image processing and restoration tasks, including image compression, reversible image conversion (e.g., image-to-video conversion), and embedding novel views in a single JPEG image. We demonstrate that customized invertible neural networks are highly effective in these inherently non-invertible tasks.

5
• Member Seminar

##### The Greene-Plesser Construction Revisited

09:30 -11:30
2021-11-05

Member Seminar

Speaker: Chuck Doran

Title: The Greene-Plesser Construction Revisited

Abstract: The first known construction of mirror pairs of Calabi-Yau manifolds was the Greene-Plesser “quotient and resolve” procedure which applies to pencils of hypersurfaces in projective space. We’ll review this approach, uncover the hints it gives for some more general mirror constructions, and describe a brand-new variant that applies to pencils of hypersurfaces in Grassmannians. This last is joint work with Tom Coates and Elana Kalashnikov (arXiv:2110.0727).

6
7
8
9
• Combinatorics Physics and Probability

##### Combinatorics Physics and ProbabilityGradient flows on totally nonnegative flag varieties

09:30 -10:30
2021-11-09

Abstract: One can view a partial flag variety in C^n as an adjoint orbit inside the Lie algebra of n x n skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. We classify gradient flows on adjoint orbits in various metrics which are compatible with total positivity. As applications, we show how the classical Toda flow fits into this framework, and prove that a new family of amplituhedra are homeomorphic to closed balls. This is joint work with Anthony Bloch.

• Algebraic Geometry in String Theory Seminar

##### Cosection localization for virtual fundamental classes of d-manifolds and Donaldson-Thomas invariants of Calabi-Yau fourfolds

10:30 -11:30
2021-11-09

Speaker: Michail Savvas, UT Austin

Title: Cosection localization for virtual fundamental classes of d-manifolds and Donaldson-Thomas invariants of Calabi-Yau fourfolds

Abstract: Localization by cosection, first introduced by Kiem-Li in 2010, is one of the fundamental techniques used to study invariants in complex enumerative geometry. Donaldson-Thomas (DT) invariants counting sheaves on Calabi-Yau fourfolds were first defined by Borisov-Joyce in 2015 by combining derived algebraic and differential geometry.
In this talk, we develop the theory of cosection localization for derived manifolds in the context of derived differential geometry of Joyce. As a consequence, we also obtain cosection localization results for (-2)-shifted symplectic derived schemes. This provides a cosection localization formalism for the Borisov-Joyce DT invariant. As an immediate application, the stable pair invariants of hyperkähler fourfolds, constructed by Maulik-Cao-Toda, vanish, as expected.

Organizer: Seminars
• Algebraic Geometry in String Theory Seminar

##### Cosection localization for virtual fundamental classes of d-manifolds and Donaldson-Thomas invariants of Calabi-Yau fourfolds

10:30 -11:30
2021-11-09

Abstract: Localization by cosection, first introduced by Kiem-Li in 2010, is one of the fundamental techniques used to study invariants in complex enumerative geometry. Donaldson-Thomas (DT) invariants counting sheaves on Calabi-Yau fourfolds were first defined by Borisov-Joyce in 2015 by combining derived algebraic and differential geometry.
In this talk, we develop the theory of cosection localization for derived manifolds in the context of derived differential geometry of Joyce. As a consequence, we also obtain cosection localization results for (-2)-shifted symplectic derived schemes. This provides a cosection localization formalism for the Borisov-Joyce DT invariant. As an immediate application, the stable pair invariants of hyperkähler fourfolds, constructed by Maulik-Cao-Toda, vanish, as expected.

10
• Colloquia

##### Hypergraph decompositions and their applications

09:30 -10:30
2021-11-10
• Joint Harvard-CUHK-YMSC Differential Geometry

##### Joint Harvard-CUHK-YMSC Differential GeometryHigher rank DT theory from rank 1

16:00 -17:00
2021-11-10

Abstract: Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X. Along the way we also show they are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

11
• Active Matter Seminar

##### Nonreciprocal matter: living chiral crystals

13:00 -14:30
2021-11-11

Abstract: Active crystals are highly ordered structures that emerge from the nonequilibrium self-organization of motile objects, and have been widely studied in synthetic and bacterial active matter. In this talk, I will describe how swimming sea star embryos spontaneously assemble into chiral crystals that span thousands of spinning organisms and persist for tens of hours. Combining experiment, hydrodynamic theory, and simulations, we demonstrate that the formation, dynamics, and dissolution of these living crystals are controlled by the natural development of the embryos. Remarkably, due to nonreciprocal force and torque exchange between the embryos, the living chiral crystals exhibit self-sustained oscillations with dynamic signatures recently predicted to emerge in materials with odd elasticity.

• General Relativity Seminar

##### Sharp decay for Teukolsky equation in Kerr spacetimes

13:00 -14:00
2021-11-11

Abstract: Teukolsky equation in Kerr spacetimes governs the dynamics of the spin $s$ components, $s=0, \pm 1, \pm 2$ corresponding to the scalar field, the Maxwell field, and the linearized gravity, respectively. I will discuss recent joint work with L. Zhang on proving the precise asymptotic profiles for these spin $s$ components in Schwarzschild and Kerr spacetimes.

• Interdisciplinary Science Seminar

##### Interdisciplinary Science Seminar11/11/21 Interdisciplinary Science Seminar

14:22 -15:22
2021-11-11

Title: The Kervaire conjecture and the minimal complexity of surfaces

Abstract: We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture.
The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surfaces in HNN extensions. This gives a conceptually simple proof of Klyachko’s theorem that confirms the Kervaire conjecture for any G torsion-free. I will also explain new results obtained using this approach.

12
• Member Seminar

##### Universal relations between entanglement, symmetries, and entropy

09:30 -10:30
2021-11-12

Member Seminar

Speaker: Gabriel Wong

Title: Universal relations between entanglement, symmetries, and entropy

Abstract: Entanglement is an essential property of quantum systems that distinguishes them from classical ones.   It is responsible for the nonlocal character of quantum information and provides a resource for quantum teleportation and quantum computation. In this talk I will provide an introduction to quantum entanglement and explain the essential role it plays in two seemingly unrelated subjects: implementation of measurement-based quantum computation and microstate counting of black holes in quantum gravity.   Time permitting, I will also discuss attempts to characterize entanglement in string theory. A unifying theme that illuminates the entanglement structure of these diverse systems is the role of surface symmetries and (entanglement) edge modes. We will explain how these universal aspects of entanglement are captured in the framework of extended topological quantum field theory.

• Quantum Matter

##### 11/12/2021 Quantum Matter in Mathematics and Physics

14:30 -16:00
2021-11-12

Title: A degeneracy bound for homogeneous topological order

13
14
15
• Swampland Seminar

##### 11/15/2021 – Swampland Seminar

13:00 -14:00
2021-11-15

This week’s seminar will be an open mic discussion which will be led by Nima Arkani-Hamed (IAS), and by Gary Shiu (UW-Madison), and the topic will be “Swampland constraints, Unitarity and Causality”. They will start with a brief introduction sharing their thoughts about the topic and moderate a discussion afterwards.

16
• Algebraic Geometry in String Theory Seminar

##### Gromov-Witten theory of complete intersections

09:30 -10:30
2021-11-16

Abstract: I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. This is joint work with Hülya Argüz, Rahul Pandharipande, and Dimitri Zvonkine (arxiv:2109.13323).

• Algebraic Geometry in String Theory Seminar

##### 11/16/21 Algebraic Geometry in String Theory

09:30 -10:30
2021-11-16
• Combinatorics Physics and Probability

##### Combinatorics Physics and ProbabilityA tale of two balloons

12:30 -13:30
2021-11-16

Abstract: From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees.
Joint work with Omer Angel and Gourab Ray.

• Quantum Matter

##### 11/16/2021 Quantum Matter in Mathematics and Physics

15:00 -16:30
2021-11-16

Title: Quantum Geometric Aspects of Chiral Twisted Graphene Models

Abstract: “Moire” materials produced by stacking monolayers with small relative twist angles are of intense current interest for the range of correlated electron phenomena they exhibit. The quench of the kinetic energy means that the interacting physics is controlled by the interplay between the interaction scale and intrinsic quantum geometries of the flat band states, in particular the Berry curvature and the Fubini-Study metric, which are in general spatially non-uniform. We show that the analytical solution of the twisted bilayer graphene wavefunction in the chiral limit has a special band geometry, endowing the Brillouin zone with a complex structure. This talk focus on the origin of the momentum space complex structure, concrete models that realize it, and its implications to electron-electron interactions. We first show the momentum space complex structure in Chern number C=1 flatbands implies the Bloch wavefunction to exhibit an exact correspondence to the lowest Landau level in the dual momentum space [2]. We present a generalization of the Haldane pseudopotential concept to deal with interacting problems in these bands and discuss experimental implications [2]. We also present an analytically solvable multi-layer generalized chiral graphene model, which exhibits arbitrarily high Chern number and ideal quantum geometries [3]. Numerical studies of interacting particles indicate model fractional Chern insulators without Landau level analogues, characterized by exact degeneracies and infinite particle entanglement spectra gaps [3]. References:

[1] Jie Wang, Yunqin Zheng, Andrew J. Millis, Jennifer Cano (Phys. Rev. Research 3, 023155)
[2] Jie Wang, Jennifer Cano, Andrew J. Millis, Zhao Liu, Bo Yang (arXiv: 2105.07491, to appear in PRL)
[3] Jie Wang, Zhao Liu (arXiv: 2109.10325)

• Quantum Matter

##### 11/16/21 Quantum Matter in Mathematics and Physics

15:00 -16:30
2021-11-16
• More events
• Algebraic Geometry in String Theory Seminar
##### Gromov-Witten theory of complete intersections
09:30 -10:30
2021-11-16

Abstract: I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. This is joint work with Hülya Argüz, Rahul Pandharipande, and Dimitri Zvonkine (arxiv:2109.13323).

• Algebraic Geometry in String Theory Seminar
##### 11/16/21 Algebraic Geometry in String Theory
09:30 -10:30
2021-11-16
• Combinatorics Physics and Probability
##### Combinatorics Physics and ProbabilityA tale of two balloons
12:30 -13:30
2021-11-16

Abstract: From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees.
Joint work with Omer Angel and Gourab Ray.

• Quantum Matter
##### 11/16/2021 Quantum Matter in Mathematics and Physics
15:00 -16:30
2021-11-16

Title: Quantum Geometric Aspects of Chiral Twisted Graphene Models

Abstract: “Moire” materials produced by stacking monolayers with small relative twist angles are of intense current interest for the range of correlated electron phenomena they exhibit. The quench of the kinetic energy means that the interacting physics is controlled by the interplay between the interaction scale and intrinsic quantum geometries of the flat band states, in particular the Berry curvature and the Fubini-Study metric, which are in general spatially non-uniform. We show that the analytical solution of the twisted bilayer graphene wavefunction in the chiral limit has a special band geometry, endowing the Brillouin zone with a complex structure. This talk focus on the origin of the momentum space complex structure, concrete models that realize it, and its implications to electron-electron interactions. We first show the momentum space complex structure in Chern number C=1 flatbands implies the Bloch wavefunction to exhibit an exact correspondence to the lowest Landau level in the dual momentum space [2]. We present a generalization of the Haldane pseudopotential concept to deal with interacting problems in these bands and discuss experimental implications [2]. We also present an analytically solvable multi-layer generalized chiral graphene model, which exhibits arbitrarily high Chern number and ideal quantum geometries [3]. Numerical studies of interacting particles indicate model fractional Chern insulators without Landau level analogues, characterized by exact degeneracies and infinite particle entanglement spectra gaps [3]. References:

[1] Jie Wang, Yunqin Zheng, Andrew J. Millis, Jennifer Cano (Phys. Rev. Research 3, 023155)
[2] Jie Wang, Jennifer Cano, Andrew J. Millis, Zhao Liu, Bo Yang (arXiv: 2105.07491, to appear in PRL)
[3] Jie Wang, Zhao Liu (arXiv: 2109.10325)

• Quantum Matter
##### 11/16/21 Quantum Matter in Mathematics and Physics
15:00 -16:30
2021-11-16
17
• Colloquia

##### Curve counting on surfaces and topological strings

09:30 -10:30
2021-11-17
18
• Interdisciplinary Science Seminar

##### Interdisciplinary Science Seminar11/18/2021 Interdisciplinary Science Seminar

14:25 -15:25
2021-11-18

Title: Amplituhedra, Scattering Amplitudes and Triangulations

Abstract: In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – recently introduced by physicists as new geometric constructions encoding interactions of elementary particles in certain Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is connected to computing scattering amplitudes of N=4 super Yang-Mills theory. Triangulations of polygons are encoded in the associahedron studied by Stasheff in the sixties; in the case of polytopes, triangulations are captured by secondary polytopes constructed by Gelfand et al. in the nineties. Whereas a “secondary” geometry describing triangulations of Amplituhedra is still not known, and we pave the way for such studies. We will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with a dual object we define – the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between triangulations of Amplituhedra of “m=2” type and the ones of a seemingly unrelated object – the Hypersimplex. The latter is a polytope which has been central in many contexts, such as matroid theory, torus orbits in the Grassmannian, and tropical geometry. Based on joint works with Lauren Williams, Melissa Sherman-Bennett, Tomasz Lukowski [arXiv:2104.08254, arXiv:2002.06164].

• Quantum Matter

##### 11/18/21 Quantum Matter in Mathematics and Physics

14:30 -16:00
2021-11-18
• Quantum Matter

##### 11/18/2021 Quantum Matter in Mathematics and Physics

14:30 -16:00
2021-11-18

Title: Exact Eigenstates in Non-Integrable Systems: A violation of the ETH

Abstract: We find that several non-integrable systems exhibit some exact eigenstates that span the energy spectrum from lowest to the highest state. In the AKLT Hamiltonian and in several others “special” non-integrable models, we are able to obtain the analytic expression of states exactly and to compute their entanglement spectrum and entropy to show that they violate the eigenstate thermalization hypothesis. This represented the first example of ETH violation in a non-integrable system; these types of states have gained notoriety since then as quantum Scars in the context of Rydberg atoms experiments. We furthermore show that the structure of these states, in most models where they are found is that of an almost spectrum generating algebra which we call Restricted Spectrum Generating Algebra. This includes the (extended) Hubbard model, as well as some thin-torus limits of Fractional Quantum Hall states. Yet in other examples, such as the recently found chiral non-linear Luttinger liquid, their structure is more complicated and not understood.

• Special Seminar

##### Special SeminarMacroscopic properties of buyer-seller networks in online marketplaces

17:52 -18:52
2021-11-18

Abstract:  Online marketplaces are the main engines of legal and illegal e-commerce, yet the aggregate properties of buyer-seller networks behind them are poorly understood. We analyse two datasets containing 245M transactions (16B USD)  between 2010 and 2021 involving online marketplaces: 28 dark web marketplaces (DWM), unregulated markets whose main currency is Bitcoin, and 144 product markets of one regulated e-commerce platform. We show how transactions in online marketplaces exhibit strikingly similar patterns of aggregate behavior despite significant differences in language, products, time, regulation, oversight, and technology. We find remarkable regularities in the distributions of (i) transaction amounts, (ii) number of transactions, (iii) inter-event times, (iv) time between first and last transactions. We then show how buyer behavior is affected by the memory of past interactions, and draw on these observations to propose a model of network formation able to reproduce the main stylised facts of the data. Our findings have important implications for understanding market power on online marketplaces as well as inter-marketplace competition.

• More events
• Interdisciplinary Science Seminar
##### Interdisciplinary Science Seminar11/18/2021 Interdisciplinary Science Seminar
14:25 -15:25
2021-11-18

Title: Amplituhedra, Scattering Amplitudes and Triangulations

Abstract: In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – recently introduced by physicists as new geometric constructions encoding interactions of elementary particles in certain Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is connected to computing scattering amplitudes of N=4 super Yang-Mills theory. Triangulations of polygons are encoded in the associahedron studied by Stasheff in the sixties; in the case of polytopes, triangulations are captured by secondary polytopes constructed by Gelfand et al. in the nineties. Whereas a “secondary” geometry describing triangulations of Amplituhedra is still not known, and we pave the way for such studies. We will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with a dual object we define – the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between triangulations of Amplituhedra of “m=2” type and the ones of a seemingly unrelated object – the Hypersimplex. The latter is a polytope which has been central in many contexts, such as matroid theory, torus orbits in the Grassmannian, and tropical geometry. Based on joint works with Lauren Williams, Melissa Sherman-Bennett, Tomasz Lukowski [arXiv:2104.08254, arXiv:2002.06164].

• Quantum Matter
##### 11/18/21 Quantum Matter in Mathematics and Physics
14:30 -16:00
2021-11-18
• Quantum Matter
##### 11/18/2021 Quantum Matter in Mathematics and Physics
14:30 -16:00
2021-11-18

Title: Exact Eigenstates in Non-Integrable Systems: A violation of the ETH

Abstract: We find that several non-integrable systems exhibit some exact eigenstates that span the energy spectrum from lowest to the highest state. In the AKLT Hamiltonian and in several others “special” non-integrable models, we are able to obtain the analytic expression of states exactly and to compute their entanglement spectrum and entropy to show that they violate the eigenstate thermalization hypothesis. This represented the first example of ETH violation in a non-integrable system; these types of states have gained notoriety since then as quantum Scars in the context of Rydberg atoms experiments. We furthermore show that the structure of these states, in most models where they are found is that of an almost spectrum generating algebra which we call Restricted Spectrum Generating Algebra. This includes the (extended) Hubbard model, as well as some thin-torus limits of Fractional Quantum Hall states. Yet in other examples, such as the recently found chiral non-linear Luttinger liquid, their structure is more complicated and not understood.

• Special Seminar
##### Special SeminarMacroscopic properties of buyer-seller networks in online marketplaces
17:52 -18:52
2021-11-18

Abstract:  Online marketplaces are the main engines of legal and illegal e-commerce, yet the aggregate properties of buyer-seller networks behind them are poorly understood. We analyse two datasets containing 245M transactions (16B USD)  between 2010 and 2021 involving online marketplaces: 28 dark web marketplaces (DWM), unregulated markets whose main currency is Bitcoin, and 144 product markets of one regulated e-commerce platform. We show how transactions in online marketplaces exhibit strikingly similar patterns of aggregate behavior despite significant differences in language, products, time, regulation, oversight, and technology. We find remarkable regularities in the distributions of (i) transaction amounts, (ii) number of transactions, (iii) inter-event times, (iv) time between first and last transactions. We then show how buyer behavior is affected by the memory of past interactions, and draw on these observations to propose a model of network formation able to reproduce the main stylised facts of the data. Our findings have important implications for understanding market power on online marketplaces as well as inter-marketplace competition.

19
• Member Seminar

##### China’s financial regulatory reform, financial opening-up, and Central Bank Digital Currency (CBDC)

09:30 -10:30
2021-11-19

Member Seminar

Speaker: Kan Lin

Title: China’s financial regulatory reform, financial opening-up, and Central Bank Digital Currency (CBDC)

Abstract: In this talk, I will explain the overall situation of China’s financial industry and review the development of China’s financial regulatory system reform from 1949 to 2021. Then, I will explain the policies of the 3 stages of financial opening-up, 2001–08, 2008–18, 2018≠present. In particular, the latest round of opening-up from 2018 has brought great opportunities for foreign institutions. China has the world’s largest banking industry with assets totaling \$53 trillion, and accounts for 1/3 of the growth in global insurance premiums over the next 10 years. I will also introduce the progress of research & development of China’s Central Bank Digital Currency (CBDC, or E-CNY). By October 2021, 140 million people had opened E-CNY wallets, and 1.6 million merchants could accept payments using eCNY wallets, including utilities, catering services, transportation, retail, and government services.

• General Relativity Seminar

##### On Curvature Propagation and ‘Breakdown’ of the Einstein Equations on U(1) Symmetric Spacetimes

13:00 -14:00
2021-11-19

Abstract: The analysis of global structure of the Einstein equations for general relativity, in the context of the initial value problem, is a difficult and intricate mathematical subject. Any additional structure in their formulation is welcome, in order to alleviate the problem.  It is expected that the initial value problem of the Einstein equations on spacetimes admitting a translational, fixed-point free, spatial U(1) isometry group are globally well-posed. In our previous works, we discussed the special structure provided by the dimensional reduction of 3+1 dimensional U(1) symmetric Einstein equations to 2+1 Einstein-wave map system and demonstrated global existence in the equivariant case for large data.  In this talk, after discussing some preliminaries and background, we shall discuss about yet another structure of the U(1) symmetric Einstein equations, namely the analogy with Yang-Mills theory via the Cartan formalism and reconcile with the dimensionally reduced field equations. We shall also discuss implications for ‘breakdown’ criteria of U(1) symmetric Einstein equations.

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• Swampland Seminar

##### Taming the Landscape

13:00 -14:00
2021-11-22

Abstract: In this talk I will introduce a generalized notion of finiteness that provides a structural principle for the set of effective theories that can be consistently coupled to quantum gravity. More concretely, I will propose a ‘tameness conjecture’ that states that all scalar field spaces and coupling functions that appear in such an effective theory must be definable in an o-minimal structure. The fascinating field of tame geometry has seen much recent progress and I will argue that the results can be used to support the above swampland conjecture. The strongest evidence arises from a new finiteness theorem for the flux landscape which is shown using the tameness of the period map.

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• Algebraic Geometry in String Theory Seminar

##### Wall crossing for moduli of stable log varieties

09:30 -10:30
2021-11-23
• Combinatorics Physics and Probability

##### Combinatorics Physics and ProbabilityPrague dimension of random graphs

09:30 -10:30
2021-11-23

Abstract: The Prague dimension of graphs was introduced by Nesetril, Pultr and Rodl in the 1970s: as a combinatorial measure of complexity, it is closely related to clique edges coverings and partitions. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order n/(log n) for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size O(log n).

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• Quantum Matter

##### 11/24/21 Quantum Matter in Mathematics and Physics

10:30 -12:00
2021-11-24
• Quantum Matter

##### 11/24/2021 Quantum Matter in Mathematics and Physics

10:30 -12:00
2021-11-24

Title: Multipartitioning topological phases and quantum entanglement

Abstract: We discuss multipartitions of the gapped ground states of (2+1)-dimensional topological liquids into three (or more) spatial regions that are adjacent to each other and meet at points. By considering the reduced density matrix obtained by tracing over a subset of the regions, we compute various correlation measures, such as entanglement negativity, reflected entropy, and associated spectra. We utilize the bulk-boundary correspondence to achieve such multipartitions and construct the reduced density matrix near the entangling boundaries. We find the fingerprints of topological liquid in these quantities, such as (universal pieces in) the scaling of the entanglement negativity, and a non-trivial distribution of the spectrum of the partially transposed density matrix.

• Joint Harvard-CUHK-YMSC Differential Geometry

##### Joint Harvard-CUHK-YMSC Differential GeometryQuantum cohomology as a deformation of symplectic cohomology

16:00 -17:00
2021-11-24

Abstract: Let X be a compact symplectic manifold, and D a normal crossings symplectic divisor in X. We give a criterion under which the quantum cohomology of X is the cohomology of a natural deformation of the symplectic cochain complex of X \ D. The criterion can be thought of in terms of the Kodaira dimension of X (which should be non-positive), and the log Kodaira dimension of X \ D (which should be non-negative). We will discuss applications to mirror symmetry. This is joint work with Strom Borman and Umut Varolgunes.

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• Swampland Seminar

13:00 -14:00
2021-11-29

Abstract: In this talk I will review massive type IIA flux compactifications that seem to give rise to infinite families of supersymmetric 4d AdS vacua. These vacua provide an interesting testing ground for the swampland program. After reviewing potential shortcomings of this setup, I will discuss recent progress on overcoming them and getting a better understanding of these solutions.

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• Combinatorics Physics and Probability

##### Combinatorics Physics and ProbabilityResistance curvature – a new discrete curvature on graphs

09:30 -10:30
2021-11-30

Abstract: The last few decades have seen a surge of interest in building towards a theory of discrete curvature that attempts to translate the key properties of curvature in differential geometry to the setting of discrete objects and spaces. In the case of graphs there have been several successful proposals, for instance by Lin-Lu-Yau, Forman and Ollivier, that replicate important curvature theorems and have inspired applications in a variety of practical settings.
In this talk, I will introduce a new notion of discrete curvature on graphs, which we call the resistance curvature, and discuss some of its basic properties. The resistance curvature is defined based on the concept of effective resistance which is a metric between the vertices of a graph and has many other properties such as a close relation to random spanning trees. The rich theory of these effective resistances allows to study the resistance curvature in great detail; I will for instance show that “Lin-Lu-Yau >= resistance >= Forman curvature” in a specific sense, show strong evidence that the resistance curvature converges to zero in expectation for Euclidean random graphs, and give a connectivity theorem for positively curved graphs. The resistance curvature also has a naturally associated discrete Ricci flow which is a gradient flow and has a closed-form solution in the case of vertex-transitive and path graphs.
Finally, if time permits I will draw a connection with the geometry of hyperacute simplices, following the work of Miroslav Fiedler.
This work was done in collaboration with Renaud Lambiotte.

• Algebraic Geometry in String Theory Seminar

09:30 -10:30
2021-11-30
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