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  • Workshop
    09:00 -15:30
    2022-12-01-2022-12-01
    CMSA, 20 Garden Street, Cambridge, MA 02138 USA

    On November 28 – Dec 1, 2022, the CMSA will host a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry.

    Organizers: An Huang (Brandeis University) | Siu-Cheong Lau (Boston University) | Tsung-Ju Lee (CMSA, Harvard) | Andrew Linshaw (University of Denver)

    Scientific Advisor: Shing-Tung Yau (Harvard, Tsinghua)

    Location: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

    Directions and Recommended Lodging

    The conference will be hybrid:  it will be both in-person and online.

    Registration is required.

    In-person registration

    Zoom webinar registration form: Zoom Webinar.

    The workshop is partially supported by Simons and NSF Grant DMS-2227199. There are funds available for participant support, which will be allocated in keeping with guidelines – students, recent PhDs, underrepresented groups, and people with no other federal support get priority.  

    Requests for support should be sent to the email address: rtcyms2022@gmail.com. The subject of your email should be “Request for support.” Please write a short description about your research and also attach your CV to the email.

    Speakers: 

    • Tomoyuki Arakawa (Kyoto)
    • Thomas Creutzig (Edmonton)
    • Jonathan Mboyo Esole (Northeastern)
    • Fei Han (National University of Singapore)
    • Shinobu Hosono (Gakushuin University)
    • Flor Orosz Hunziker (Colorado)
    • Cuipo Jiang (Shanghai)
    • Shashank Kanade (Denver)
    • Matt Kerr (Washington University in St. Louis)
    • Carl Lian (Humboldt-Universität zu Berlin)
    • Nai-Chung Conan Leung (CUHK)
    • Ivan Loseu (Yale)
    • Robert McRae (Tsinghua University)
    • Anne Moreau (Université Paris-Saclay, Orsay)
    • Tony Pantev (University of Pennsylvania)
    • Mauricio Romo (Tsinghua University)
    • Bailin Song (USTC)
    • Cumrun Vafa (Harvard University)
    • Chin-Lung Wang (National Taiwan University)
    • Weiqiang Wang (Virginia)
    • Yaping Yang (University of Melbourne)
    • Shing-Tung Yau (Tsinghua University)
    • Chenglong Yu (Tsinghua University)
    • Gufang Zhao (University of Melbourne)

     

    Schedule (Eastern Time)

    Schedule (pdf)

    11/28 (Monday)

    08:30am – 08:55amRefreshments
    08:55am – 09:00amOpening remarks by Horng-Tzer Yau
    09:00am – 09:45amShing-Tung Yau*Title: The Hull-Strominger system through conifold transitions

    Abstract: In this talk I discuss the geometry of C-Y manifolds outside of the Kähler regime and especially describe the Hull-Strominger system through the conifold transitions.

    10:00am – 10:45amChenglong Yu*Title: Commensurabilities among Lattices in PU(1,n)

    Abstract: In joint work with Zhiwei Zheng, we study commensurabilities among certain subgroups in PU(1,n). Those groups arise from the monodromy of hypergeometric functions. Their discreteness and arithmeticity are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of the lattices among them up to conjugation and finite index (commensurability) is not completed. When n=1, it is the commensurabilities of hyperbolic triangles. The cases of n=2 are almost resolved by Deligne-Mostow and Sauter’s commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some higher dimensional Calabi-Yau type varieties instead of complex reflection groups. We obtain some relations and commensurability indices for higher n and also give new proofs for existing pairs in n=2.

    11:00am – 11:45amThomas Creutzig*Title: Shifted equivariant W-algebras

    Abstract: The CDO of a compact Lie group is a family of VOAs whose top level is the space of functions on the Lie group. Similar structures appear at the intersections of boundary conditions in 4-dimensional gauge theories, I will call these new families of VOAs shifted equivariant W-algebras. I will introduce these algebras, construct them and explain how they can be used to quickly prove the GKO-coset realization of principal W-algebras.

    11:45am – 1:30 pmLunch
    01:30pm – 02:15pmCumrun VafaTitle: Reflections on Mirror Symmetry

    Abstract: In this talk I review some of the motivations leading to the search and discovery of mirror symmetry as well as some of the applications it has had.

    02:30pm – 03:15pmJonathan Mboyo EsoleTitle: Algebraic topology and matter representations in F-theory

    Abstract: Recently, it was observed that representations appearing in geometric engineering in F-theory all satisfy a unique property: they correspond to characteristic representations of embedding of Dynkin index one between Lie algebras. However, the reason why that is the case is still being understood. In this talk, I will present new insights, giving a geometric explanation for this fact using K-theory and the topology of Lie groups and their classifying spaces. In physics, this will be interpreted as conditions on the charge of instantons and the classifications of Wess-Zumino-Witten terms.

    03:15pm – 03:45 pmBreak
    03:45pm – 04:30pmWeiqiang WangTitle: A Drinfeld presentation of affine i-quantum groups

    Abstract: A quantum symmetric pair of affine type (U, U^i) consists of a Drinfeld-Jimbo affine quantum group (a quantum deformation of a loop algebra) U and its coideal subalgebra U^i (called i-quantum group). A loop presentation for U was formulated by Drinfeld and proved by Beck. In this talk, we explain how i-quantum groups can be viewed as a generalization of quantum groups, and then we give a Drinfeld type presentation for the affine quasi-split i-quantum group U^i. This is based on joint work with Ming Lu (Sichuan) and Weinan Zhang (Virginia).

    04:45pm – 05:30pmTony PantevTitle: Decomposition, anomalies, and quantum symmetries

    Abstract: Decomposition is a phenomenon in quantum physics which converts quantum field theories with non-effectively acting gauge symmetries into equivalent more tractable theories in which the fields live on a disconnected space. I will explain the mathematical content of decomposition which turns out to be a higher categorical version of Pontryagin duality. I will examine how this duality interacts with quantum anomalies and secondary quantum symmetries and will show how the anomalies can be canceled by homotopy coherent actions of diagrams of groups. I will discuss in detail the case of 2-groupoids which plays a central role in anomaly cancellation, and will describe a new duality operation that yields decomposition in the presence of anomalies. The talk is based on joint works with Robbins, Sharpe, and Vandermeulen.

     

    11/29 (Tuesday)

     

    Refreshments
    09:00am – 09:45amRobert MacRae*Title: Rationality for a large class of affine W-algebras

    Abstract: One of the most important results in vertex operator algebras is Huang’s theorem that the representation category of a “strongly rational” vertex operator algebra is a semisimple modular tensor category. Conversely, it has been conjectured that every (unitary) modular tensor category is the representation category of a strongly rational (unitary) vertex operator algebra. In this talk, I will describe my results on strong rationality for a large class of affine W-algebras at admissible levels. This yields a large family of modular tensor categories which generalize those associated to affine Lie algebras at positive integer levels, as well as those associated to the Virasoro algebra.

    10:00am – 10:45amBailin Song*Title: The global sections of chiral de Rham complexes on compact Calabi-Yau manifolds

    Abstract: Chiral de Rham complex is a sheaf of vertex algebras on a complex manifold. We will describe the space of global sections of the chiral de Rham complexes on compact Calabi-Yau manifolds.

    11:00am – 11:45amCarl Lian*Title: Curve-counting with fixed domain

    Abstract: The fixed-domain curve-counting problem asks for the number of pointed curves of fixed (general) complex structure in a target variety X subject to incidence conditions at the marked points. The question comes in two flavors: one can ask for a virtual count coming from Gromov-Witten theory, in which case the answer can be computed (in principle) from the quantum cohomology of X, or one can ask for the “honest” geometric count, which tends to be more subtle. The answers are conjectured to agree in the presence of sufficient positivity, but do not always. I will give an overview of some recent results and open directions. Some of this work is joint with Alessio Cela, Gavril Farkas, and Rahul Pandharipande.

    11:45am – 01:30pmLunch
    01:30pm – 02:15pmChin-Lung WangTitle: A blowup formula in quantum cohomology

    Abstract: We study analytic continuations of quantum cohomology $QH(Y)$ under a blowup $\phi: Y \to X$ of complex projective manifolds along the extremal ray variable $q^{\ell}$. Under $H(Y) = \phi^* H(X) plus K$ where $K = \ker \phi_*$, we show that (i) the restriction of quantum product along the $\phi^*H(X)$ direction, denoted by $QH(Y)_X$, is meromorphic in $x := 1/q^\ell$, (ii) $K$ deforms uniquely to a quantum ideal $\widetilde K$ in $QH(Y)_X$, (iii) the quotient ring $QH(Y)_X/\widetilde K$ is regular over $x$, and its restriction to $x = 0$ is isomorphic to $QH(X)$. This is a joint work (in progress) with Y.-P. Lee and H.-W. Lin.

    02:30pm – 03:15pmIvan LoseuTitle: Quantizations of nilpotent orbits and their Lagrangian subvarieties

    Abstract: I’ll report on some recent progress on classifying quantizations of the algebras of regular functions of nilpotent orbits (and their covers) in semisimple Lie algebras, as well as the classification of quantizations of certain Lagrangian subvarieties. An ultimate goal here is to understand the classification of unitary representations of real semisimple Lie groups.

    03:15pm – 03:45pmBreak
    03:45pm – 04:30pmMatt Kerr*Title: $K_2$ and quantum curves

    Abstract: The basic objects for this talk are motives consisting of a curve together with a $K_2$ class, and their mixed Hodge-theoretic invariants.

    My main objective will be to explain a connection (recently proved in joint work with C. Doran and S. Sinha Babu) between (i) Hodge-theoretically distinguished points in the moduli of such motives and (ii) eigenvalues of operators on L^2(R) obtained by quantizing the equations of the curves.

    By local mirror symmetry, this gives evidence for a conjecture in topological string theory (due to M. Marino, A. Grassi, and others) relating enumerative invariants of toric CY 3-folds to spectra of quantum curves.

    04:45pm – 05:30pmFlor Orosz HunzikerTitle: Tensor structures associated to the N=1 super Virasoro algebra

    Abstract:  We have recently shown that there is a natural category of representations associated to the N=1 super Virasoro vertex operator algebras that have braided tensor structure. We will describe this category and discuss the problem of establishing its rigidity at particular central charges. This talk is based on joint work in progress with Thomas Creutzig, Robert McRae and Jinwei Yang.

     

     

     

    11/30 (Wednesday)

    08:30am – 09:00amRefreshments
    09:00am – 09:45amTomoyuki ArakawaTitle: 4D/2D duality and representation theory

    Abstract: This talk is about the 4D/2D duality discovered by Beem et al. rather recently in physics. It associates a vertex operator algebra (VOA) to any 4-dimensional superconformal field theory, which is expected to be a complete invariant of thl theory. The VOAs appearing in this manner may be regarded as chiralization of various symplectic singularities and their representations are expected to be closely related with the Coulomb branch of the 4D theory. I will talk about this remarkable 4D/2D duality from a representation theoretic perspective.

    10:00am – 10:45amShashank KanadeTitle: Combinatorics of principal W-algebras of type A

    Abstract: The combinatorics of principal W_r(p,p’) algebras of type A is controlled by cylindric partitions. However, very little seems to be known in general about fermionic expressions for the corresponding characters. Welsh’s work explains the case of Virasoro minimal models W_2(p,p’). Andrews, Schilling and Warnaar invented and used an A_2 version of the usual (A_1) Bailey machinery to give fermionic characters (up to a factor of (q)_\infty) of some, but not all, W_3(3,p’) modules. In a recent joint work with Russell, we have given a complete set of conjectures encompassing all of the remaining modules for W_3(3,p’), and proved our conjectures for small values of p’. In another direction, characters of W_r(p,p’) algebras also arise as appropriate limits of certain sl_r coloured Jones invariants of torus knots T(p,p’), and we expect this to provide further insights on the underlying combinatorics.

    11:00am – 11:45amGufang ZhaoTitle: Quasimaps to quivers with potentials

    Abstract: This talk concerns non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential. The construction borrows ideas from the theory of gauged linear sigma models as well as recent development in shifted symplectic geometry and Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials are discussed. This is based on work in progress, in collaboration with Yalong Cao.

    11:45am – 01:30pmGroup Photo, Lunch
    01:30pm – 02:15pmYaping YangTitle: Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds

    Abstract: Let X be a smooth local toric Calabi-Yau 3-fold. On the cohomology of the moduli spaces of certain sheaves on X, there is an action of the cohomological Hall algebra (COHA) of Kontsevich and Soibelman via “raising operators”. I will discuss the “double” of the COHA that acts on the cohomology of the moduli space by adding the “lowering operators”. We associate a root system to X. The double COHA is expected to be the shifted Yangian of this root system. We also give a prediction for the shift in terms of an intersection pairing. We provide evidence of the aforementioned expectation in various examples. This is based on my joint work with M. Rapcak, Y. Soibelman, and G. Zhao

    02:30pm – 03:15pmFei HanTitle: Graded T-duality with H-flux for 2d sigma models

    Abstract: T-duality in string theory can be realised as a transformation acting on the worldsheet fields in the two-dimensional nonlinear sigma model. Bouwknegt-Evslin-Mathai established the T-duality in a background flux for the first time upon compactifying spacetime in one direction to a principal circle by constructing the T-dual maps transforming the twisted cohomology of the dual spacetimes. In this talk, we will describe our recent work on how to promote the T-duality maps of Bouwknegt-Evslin-Mathai in two aspects. More precisely, we will introduce (1) graded T-duality, concerning the graded T-duality maps of all levels of twistings; (2) the 2-dimensional sigma model picture, concerning the double loop space of spacetimes. This represents our joint work with Mathai.

    03:15pm – 3:45pmBreak
    03:45pm – 04:30pmMauricio RomoTitle: Networks and BPS Counting: A-branes view point

    Abstract: I will review the countings of BPS invariants via exponential/spectral networks and present an interpretation of this counting as a count of certain points in the moduli space of A-branes corresponding to degenerate Lagrangians.

    04:45pm – 05:30pmShinobu HosonoTitle: Mirror symmetry of abelian fibered Calabi-Yau manifolds with ρ = 2

    Abstract: I will describe mirror symmetry of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces, which have Picard number two. Finding a mirror family over a toric variety explicitly, I  observe that mirror symmetry of all related Calabi-Yau manifods arises from the corresponding boundary points, which are not necessarily toric boundary points.  Calculating Gromov-Witten invariants up to genus 2, I find that the generating functions are expressed elliptic (quasi-)modular forms, which reminds us the modular anomaly equation found for elliptic surfaces. This talk is based on a published work with Hiromichi Takaki (arXiv:2103.08150).

    06:00pmBanquet @ Royal East Restaurant, 782 Main St, Cambridge, MA 02139

     

    12/1 (Thursday)

    08:30am – 09:00amRefreshments
    09:00am – 09:45amConan Nai Chung Leung*Title: Quantization of Kahler manifolds

    Abstract: I will explain my recent work on relationships among geometric quantization, deformation quantization, Berezin-Toeplitz quantization and brane quantization.

    10:00am – 10:45amCuipo Jiang*Title: Cohomological varieties associated to vertex operator algebras

    Abstract: We define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point.  We study its basic properties. As examples, we construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational $C_2$-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. This talk is based on joint work with Antoine Caradot and Zongzhu Lin.

    11:00am – 11:45amAnne Moreau*Title: Action of the automorphism group on the Jacobian of Klein’s quartic curve

    Abstract: In a joint work with Dimitri Markouchevitch, we prove that the quotient variety of the 3-dimensional Jacobian of the plane Klein quartic curve by its full automorphism group of order 336 is isomorphic to the 3-dimensional weighted projective space with weights 1,2,4,7.

    The latter isomorphism is a particular case of the general conjecture of Bernstein and Schwarzman suggesting that a quotient of the n-dimensional complex space by the action of an irreducible complex crystallographic group generated by reflections is a weighted projective space.

    In this talk, I will explain this conjecture and the proof of our result. An important ingredient is the computation of the Hilbert function of the algebra of invariant theta-functions on the Jacobian.

    11:45am – 11:50amClosing remarks
    11:50amFree discussions and departure

    * = Online speaker

    CMSA COVID-19 Policies

     

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  • Member Seminar
    11:00 -12:00
    2022-12-02
    CMSA, 20 Garden Street, Cambridge, MA 02138 USA

    Member Seminar

    Speaker: Alejandro Poveda

    Title: Compactness and Anticompactness Principles in Set Theory

    Abstract: Several fundamental properties in Topology, Algebra or Logic are expressed in terms of Compactness Principles.For instance, a natural algebraic question is the following: Suppose that G is an Abelian group whose all small subgroups are free – Is the group G free? If the answer is affirmative one says that compactness holds; otherwise, we say that compactness fails. Loosely speaking, a compactness principle is anything that fits the following slogan: Suppose that M is a mathematical structure (a group, a topological space, etc) such that all of its small substructures N have certain property $\varphi$; then the ambient structure M has property $\varphi$, as well. Oftentimes when these questions are posed for infinite sets the problem becomes purely set-theoretical and axiom-sensitive. In this talk I will survey the most paradigmatic instances of compactness and present some related results of mine. If time permits, I will hint the proof of a recent result (joint with Rinot and Sinapova) showing that stationary reflection and the failure of the Singular Cardinal Hypothesis can co-exist. These are instances of two antagonist set-theoretic principles: the first is a compactness principle while the second is an anti-compactness one. This result solves a question by M. Magidor from 1982.

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  • Swampland Seminar
    11:00 -12:00
    2022-12-05
    17 Oxford St, Cambridge, MA 02138

    Swampland Seminar

    Location: Jefferson Physical Laboratory, Seminar room J356

    Speaker: Markus Dierigl (LMU Munich)

    Title: IIB Explored – Dualities, Bordisms, and the Swampland

    Abstract: In this talk I will discuss the application of the Cobordism Conjecture to type IIB supergravity with non-trivial duality bundle. Calculating the relevant bordism groups we find that they are highly non-trivial and would predict the presence of various global symmetries in the underlying theory. Since quantum gravity theories do not allow for global symmetries, we discuss which defects need to be included to break them completely. Interestingly, we find many backgrounds that are well-known in the F-theory literature, such as [p,q]-7-branes, non-Higgsable clusters, as well as S-folds and their generalizations to higher codimensions. Further including worldsheet reflections, predicts the existence of a new non-supersymmetric 7-brane with interesting properties and applications, which I will discuss in more detail.

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  • Topological Quantum Matter Seminar
    10:00 -11:00
    2022-12-07
    CMSA, 20 Garden Street, Cambridge, MA 02138 USA

    Topological Quantum Matter Seminar

    Speaker: Vasil Rokaj (Harvard)

    Title: Controlling Quantum Matter with Quantum Cavity Fields

    Abstract: Cavity modification of material properties and phenomena is a novel research field motivated by the advances in strong light-matter interactions [1]. For condensed matter systems it has been demonstrated experimentally that the transport properties of 2D materials can be modified via coupling to vacuum fields [2,3]. While in polaritonic chemistry it has been shown that ground state chemical properties can be controlled with cavity fields [4]. In the first part of my talk, I will present how the quantized cavity field can alter the conduction properties of a condensed matter system by focusing on the paradigmatic Sommerfeld model of the free electron gas [5]. The exact analytic solution of the Sommerfeld model in the cavity will be presented as well as its fundamental properties. Then, in the second part of the talk, I will focus on a many-particle system of cold ions in a harmonic trap coupled to the cavity field. I will show how this system couples collectively to the cavity and that hybrid states between light and matter, known as polaritons, emerge. The formation of polaritons leads to the modification of the properties of the cold ions and enhances the localization of the many-body wave function [6]. Connections to experiments will be discussed as well.

    [1] F. Garcia-Vidal, C. Ciuti, T. W. Ebbesen, Science, 373, 178 (2021)

    [2] G. L. Paravicini-Bagliani et al., Nat. Phys. 15, 186-190 (2019)

    [3] F. Appugliese et al., Science 375 (6584), 1030-1034 (2022)

    [4] T. W. Ebbesen, Acc. Chem. Res. 49, 11, 2403–2412 (2016)

    [5] V. Rokaj, M. Ruggenthaler, F. G. Eich, A. Rubio, Phys. Rev. Research 4, 013012 (2022)

    [6] V. Rokaj, S.I. Mistakidis, H.R. Sadeghpour, arXiv:2207.03436 (2022)

  • New Technologies in Mathematics Seminar
    14:00 -15:00
    2022-12-07
    CMSA, 20 Garden Street, Cambridge, MA 02138 USA

    New Technologies in Mathematics Seminar

    Speaker: Cyril Zhang, Microsoft Research

    Title: How do Transformers reason? First principles via automata, semigroups, and circuits

    Abstract: The current “Transformer era” of deep learning is marked by the emergence of combinatorial and algorithmic reasoning capabilities in large sequence models, leading to dramatic advances in natural language understanding, program synthesis, and theorem proving. What is the nature of these models’ internal representations (i.e. how do they represent the states and computational steps of the algorithms they execute)? How can we understand and mitigate their weaknesses, given that they resist interpretation? In this work, we present some insights (and many further mysteries) through the lens of automata and their algebraic structure.

    Specifically, we investigate the apparent mismatch between recurrent models of computation (automata & Turing machines) and Transformers (which are typically shallow and non-recurrent). Using tools from circuit complexity and semigroup theory, we characterize shortcut solutions, whereby a shallow Transformer with only o(T) layers can exactly replicate T computational steps of an automaton. We show that Transformers can efficiently represent these shortcuts in theory; furthermore, in synthetic experiments, standard training successfully finds these shortcuts. We demonstrate that shortcuts can lead to statistical brittleness, and discuss mitigations.

    Joint work with Bingbin Liu, Jordan Ash, Surbhi Goel, and Akshay Krishnamurthy.

  • Probability Seminar

    Probability Seminar
    Title TBA

    15:30 -16:30
    2022-12-07
    1 Oxford Street, Cambridge MA 02138

    Probability Seminar

    Note location change: Science Center ROom 300H

    Speaker: Lior Alon (MIT)

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