nLab Center for Quantum and Topological Systems

The Center for Quantum and Topological Systems (nyuad.nyu.edu/cqts) is a Research Center, launched in 2022, within the Research Institute of New York University in Abu Dhabi.

CQTS hosts cross-disciplinary research on topological quantum systems, such as topological phases of matter understood via holography and using tools from algebraic topology, ultimately aimed at addressing open questions in topological quantum computation. A unifying theme is the use of new methods from (persistent) Cohomotopy (aka framed Cobordism, aka absolute $\mathbb{F}_1$-algebraic K-theory) and generalized Twisted Equivariant Differential (TED) cohomology, developed in string theory.

Conferences & Workshops

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Jun 2022

Homotopical perspectives on Topological data analysis

Organizers: Sadok Kallel and Hisham Sati

Schedule for 02 June 2022:

• 15:00 - 16:00 GST/UTC+4

  Ling Zhou (The Ohio State University, USA)

  Persistent homotopy groups of metric spaces

  By capturing both geometric and topological features of datasets, persistent homology has shown its promise in applications. Motivated by the fact that homotopy in general contains more information than homology, we study notions of persistent homotopy groups of compact metric spaces, together with their stability properties in the Gromov-Hausdorff sense. Under fairly mild assumptions on the spaces, we proved that the classical fundamental group has an underlying tree-like structure (i.e. a dendrogram) and an associated ultrametric. We then exhibit pairs of filtrations that are confounded by persistent homology but are distinguished by their persistent homotopy groups. We finally describe the notion of persistent rational homotopy groups, which is easier to handle but still contains extra information compared to persistent homology.

  
  

• 16:00 - 17:00 GST/UTC+4

  Wojciech Chacholski (KTH, Sweden)

  Realisations of Posets

  My presentation is based on an article with the same title coauthored with A. Jin and F. Tombari (arXiv:2112.12209).

  Encoding information in form of functors indexed by the poset of $r$-tuples of real numbers (persistence modules) is attractive for three reasons:

  a) metric properties of the poset are essential to study distances on persistence modules

  b) the poset of $r$-tuples of real numbers has well behaved discrete approximations which are used to provide finite approximations of persistence modules

  c) the mentioned discretizations and approximations have well studied algebraic and homological properties as they can be identified with multi graded modules over polynomial rings.

  In my talk I will describe a construction called realisation, that transforms arbitrary posets into posets which satisfy all three requirements above and hence are particularly suitable for persistence methods.Intuitively the realisation associates a continuous structure to a locally discrete poset by filling in empty spaces. For example the realisation of the poset of natural numbers is the poset of non-negative reals. I will focus on illustrating how homological techniques, such as Koszul complexes, can be used to study persistence modules indexed by realisations.

  
  

• 17:30 - 18:30 GST/UTC+4

  Grégory Ginot (Université Paris 13, France)

  Homotopical and sheaf theoretic point of view on multi-parameter persistence.

  In this talk we will highlight the study of level set persistence through the prism of sheaf theory and a special type of 2-parameter persistence: Mayer-Vietoris systems and a pseudo-symetry between those. This is based on joint work with Berkouk and Oudot.

  
  

• 18:30 - 19:30 GST/UTC+4

  Rick Jardine (University of Western Ontario, Canada)

  Thoughts on big data sets

  This talk describes work in progress. The idea is to develop methods for analyzing a very large data sets $X \subset \mathbb{R}^{N}$ in high dimensional spaces. There are well-known pitfalls to avoid, including the inability to computationally analyze TDA constructions for $X$ on account of its size, the “curse of high dimensionality”, and the failure of excision for standard TDA constructions. We discuss the curse of high dimensionality and define a hypercube metric on $\mathbb{R}^{N}$ that may lessen its effects. The excision problem for the Vietoris-Rips construction can be addressed by expanding the TDA discussion to filtered subobjects $K$ of Vietoris-Rips constructions. Unions of such subobjects satisfy excision in path components (clusters) and homology groups, by classical results. The near-term goal is to construct, for each data point $x$, a “computable” filtered subcomplex $K_{x} \subset V(X)$ with $x \in K_{x}$, which would capture spatial local behaviour of the data set $X$ near $x$. A large (but highly parallelizable) algorithm finds a nearest neighbour, or a set of $k$-nearest neighbours for a fixed data point $x \in X$. Some variant of this algorithm may lead to a good construction of the local subcomplex $K_{x}$.

Jan 2023

• 12-15 Jan 2023

  M-Theory and Mathematics 2023 – Classical and Quantum Aspects

  on M-theory (non-perturbative string theory and related quantum field theories)

$\phantom{-----}$ [logo adapted from JMP 62 (2021) 042301]

Feb 2023

24 Feb 2023

CQTS and TII Workshop 2023

joint workshop with the Quantum Research Center (QRC) at the Technology Innovation Institute (TII) in Abu Dhabi

on quantum materials, quantum computation and quantum programming

• 9:00 - 9:05

  Hisham Sati:

  Welcome

• 9:05 - 9:15

  Luigi Amico:

  Introduction to Quantum Physics @ TII

• 9:20 - 9:45

  Juan Polo Gomez:

  Atomtronics

• 9:50 - 10:10

  Enrico Domanti:

  Coherence of confined matter in lattice gauge theories at the mesoscopic scale

• 10:15 - 10:40

  Gianluigi Catelani:

  Superconducting qubits

Break: 10:40 - 11:10

• 11:10 - 11:15

  Leandro Aolita:

  Intro to Quantum Algorithms @TII

• 11:20 - 11:50

  Ingo Roth:

  Quantum device characterization

• 11:55 - 12:25

  Egor Tiunov:

  Quantum-inspired algorithms

• 12:30 - 13:00

  Thais de Lima Silva:

  Quantum algorithms: Quantum Signal Processing

Lunch: 13:00 – 2:15

• 14:15 - 14:25

  Hisham Sati:

  Introducing research and researchers @CQTS

• 14:30 - 14:50

  Amaria Javed:

  Quantum information processing via NLS

• 15:00 - 15:20

  Marwa Mannaï:

  Tuning topological quantum materials

• 15:30 - 15:50

  Mitchell Riley:

  Verified quantum programming with linear HoTT

Break: 3:50-4:20 pm

• 4:20 - 4:40

  Vivek Singh:

  Topological Quantum field theory for TQC

• 16:50 - 17:10

  Sachin Valera:

  Topological Qubits from Anyons

• 17:20 - 17:50

  Urs Schreiber:

  Towards verified hardware-aware topological quantum programming

Mar 2023

15-18 Mar 2023 (ongoing)

Geometric/Topological Quantum Field Theories and Cobordisms

on functorial quantum field theory, knot homology and cobordism theory/cobordism categories/cobordism hypothesis

$\phantom{-----}$ [logo adapted from arXiv:2103.01877]

$\;\;$

Mee Seong Im, Mikhail Khovanov, Vivek Singh, Sergei Gukov, Anna Beliakova, Khaled Qazaqzeh

Domenico Fiorenza, Carlo Collari, Sadok Kallel Nafaa Chbili, Christian Blanchet, David Jaz Myers

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ Nitu Kitchloo, Daniel Berwick-Evans, Adrian Clough

Sachin Valera, Alonso Perez Lona, Urs Schreiber $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ Daniel Grady, Christoph Schweigert

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ Hisham Sati, Konrad Waldorf, Dmitri Pavlov

• Mikhail Khovanov (Columbia University):

  Universal construction, foams and link homology

  cf.: arXiv:1808.09662, arXiv:2011.11077

  In this series of three talks we will explain the foam approach to link homology. Bigraded link homology theories categorify the Jones polynomial and other Reshetikhin-Turaev link invariants, such as the HOMFLY-PT polynomial. Foams, which are polyhedral 2D complexes embedded in 3-space allow to construct state spaces for planar graphs which are then used to define link homology groups. The most explicit and efficient way to define graph state spaces is via evaluation of the closed foams (Robert-Wagner formula).

  A) This formula will be first explained in the less technical unoriented $SL(3)$ case. Resulting graph state spaces are then related to the Four-Color Theorem and Kronheimer-Mrowka homology for 3-orbifolds.

  B) A step in that construction requires building a topological theory (a lax TQFT) from an evaluation of closed objects, such as closed $n$-manifolds. We will explain the setup for topological theories, including in two dimensions, recovering the Deligne categories and their generalizations. In one dimension and adding defects, these topological theories are related to noncommutative power series, pseudocharacters, and, over the Boolean semiring, to regular languages and automata.

  C) Robert-Wagner $GL(N)$ foam evaluation and its application to constructing link homology theories will be explained as well.

• Nitu Kitchloo (John Hopkins University):

  Symmetry breaking and homotopy types for link homologies

  cf.: arXiv:1910.07443, arXiv:1910.07444, arXiv:1910.07516

  I will describe how the spaces that record symmetry breaking data in a $U(n)$-gauge theory (for arbitrary $n$) can be used to construct homotopy types that are invariants for links in $\mathbb{R}^3$. In particular, I will show how one may recover Khovanov-Rozansky link homology and $\mathfrak{sl}(n)$ link homology by evaluating this homotopy type under suitable cohomology theories.

• Sergei Gukov (DIAS, Dublin and Caltech)

  Machine learning and hard problems in topology

  cf.: arXiv:2010.16263

• Nils Carqueville (University of Vienna):

  Extended defect TQFTs

  slides: pdf, pdf

  cf.: arXiv:2201.03284

  According to the cobordism hypothesis with singularities, fully extended topological quantum field theories with defects are equivalently described in terms of coherent full duality data for objects and (higher) morphisms as well as appropriate homotopy fixed point structures. We discuss the 2-dimensional oriented case in some detail and apply it to truncated affine Rozansky-Witten models, which are under very explicit computational control. This is joint work with Ilka Brunner, Pantelis Fragkos, and Daniel Roggenkamp.

• Arun Debray (Purdue University):

  Twisted string bordism and a potential anomaly in $E_8 \times E_8$ heterotic string theory

  cf.: arXiv:2210.04911

  Quantum field theories can have an inconsistency called an anomaly, formulated as an invertible field theory in one dimension higher. Theorems of Freed-Hopkins-Teleman and Freed-Hopkins classify invertible field theories in terms of bordism groups. In this talk, I’ll apply this to the low-energy approximation of $E_8 \times E_8$ heterotic string theory; Witten proved anomaly cancellation in a restricted setting, but we perform a twisted string bordism computation to show that the relevant group of 11-dimensional invertible field theories does not vanish, and therefore there could be an anomaly in $E_8 \times E_8$ heterotic string theory. Standard computational techniques for twisted string bordism do not work for this problem; I will also discuss work joint with Matthew Yu using Baker-Lazarev’s R-module Adams spectral sequence to simplify a large class of twisted spin and string bordism computations.

• Matthew Young (Utah State University):

  Non-semisimple TFT and $U(1 \vert 1)$ Chern-Simons theory

  cf.: arXiv:2210.04286 and super Chern-Simons theory

  Chern-Simons theory, as introduced by Witten, is a three dimensional quantum gauge theory associated to a compact simple Lie group and a level. The mathematical model of this theory as a topological field theory was introduced by Reshetikhin and Turaev and is at the core of modern quantum topology. The goal of this talk is to explain a non-semisimple modification of the construction of Reshetikhin and Turaev which realizes Chern-Simons theory with gauge supergroup $U(1\vert 1)$, as studied in the physics literature by Rozansky-Saleur and Mikhaylov. The key new algebraic structure is a relative modular structure on the category of representations of the quantum group of $\mathfrak{gl}(1\vert 1)$. Based on joint work with Nathan Geer.

• Mee Seong Im (United States Naval Academy):

  Correspondence between automata and one-dimensional Boolean topological theories and TQFTs

  cf.: arXiv:2301.00700

  Automata are important objects in theoretical computer science. I will describe how automata emerge from topological theories and TQFTs in dimension one and carrying defects. Conversely, given an automaton, there’s a canonical Boolean TQFT associated with it. In those topological theories, one encounters pairs of a regular language and a circular regular language that describe the theory.

• Alexander Schenkel (University of Nottingham):

  Quantum field theories on Lorentzian manifolds

  cf.: arXiv:2208.04344 and homotopical AQFT

  This talk provides an introduction and survey of recent developments in algebraic QFT on Lorentzian manifolds. I will outline an axiomatization of such QFTs in terms of operad theory and illustrate this formalism through classification results in low dimensions. One of the central axioms is a certain local constancy condition, called the time-slice axiom, that encodes a concept of time evolution. Using model categorical localization techniques, I will show that this i.g. homotopy-coherent time evolution admits a strictification in many relevant cases. I will conclude this talk by explaining similarities and differences between algebraic QFT and other approaches such as factorization algebras and functorial field theories.

• Konrad Waldorf (University of Greifswald):

  The stringor bundle

  cf.: arXiv:2206.09797

  The stringor bundle plays the role of the spinor bundle, but in string theory instead of quantum mechanics. It has been anticipated in pioneering work of Stolz and Teichner as a vector bundle on loop space. I will talk about joint work with Matthias Ludewig and Peter Kristel that provides a fully rigorous and neat presentation of the stringor bundle as an associated 2-vector bundle, via a representation of the string 2-group on a von Neumann algebra.

• Domenico Fiorenza (Sapienza University of Rome):

  String bordism invariants in dimension 3 from $U(1)$-valued TQFTs

  cf.: arXiv:2209.12933

  The third string bordism group is known to be $\mathbb{Z}/$24. Using Waldorf’s notion of a geometric string structure on a manifold, Bunke-Naumann and Redden have exhibited integral formulas involving the Chern-Weil form representative of the first Pontryagin class and the canonical 3-form of a geometric string structure that realize the isomorphism between the third string bordism group and $\mathbb{Z}/24$ (these formulas have been recently rediscovered by Gaiotto, Johnson-Freyd & Witten). In the talk I will show how these formulas naturally emerge when one considers the $U(1)$-valued 3d TQFTs associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure. Joint work with Eugenio Landi (arXiv:2209.12933).

• Christian Blanchet (Université Paris Cité):

  Heisenberg homologies of surface configurations

  cf.: arXiv:2206.11475

  The Heisenberg group of a surface is the central extension of its one-dimensional homology associated with the intersection cocycle. We show that a representation of the Heisenberg group defines local coefficients on the unordered configuration space of points in the surface. We study the corresponding homologies, the Mapping Class Group action and the connection with quantum constructions. This is based on joint work with Awais Shaukat and Martin Palmer.

• Nafaa Chbili (United Arab Emirates University):

  Quasi-alternating links, Examples and obstructions

  cf.: arXiv:2009.08624

  Quasi-alternating links represent an important class of links that has been introduced by Ozsváth and Szabó while studying the Heegaard Floer homology of the branched double-covers of alternating links. This new class of links, which share many homological properties with alternating links, is defined in a recursive way which is not easy to use in order to determine whether a given link is quasi-alternating. In this talk, we shall review the main obstruction criteria for quasi-alternating links. We also discuss how new examples of quasi-alternating links can constructed.

• Dmitri Pavlov (Texas Tech University):

  The geometric cobordism hypothesis

  slides: pdf pdf

  cf.: arXiv:2011.01208, arXiv:2111.01095

  I will explain my recent joint work with Daniel Grady on locality of functorial field theories (arXiv:2011.01208) and the geometric cobordism hypothesis (arXiv:2111.01095). The latter generalizes the Baez–Dolan cobordism hypothesis to non-topological field theories, in which bordisms can be equipped with geometric structures, such as smooth maps to a fixed target manifold, Riemannian metrics, conformal structures, principal bundles with connection, or geometric string structures. Applications include a generalization of the Galatius–Madsen–Tillmann–Weiss theorem, a solution to a conjecture of Stolz and Teichner on representability of concordance classes of functorial field theories, and a construction of power operations on the level of field theories (extending the recent work of Barthel-Berwick-Evans-Stapleton).

  I will illustrate the general theory by constructing the prequantum Chern-Simons theory as a fully extended nontopological functorial field theory.

  If time permits, I will discuss the ongoing work on defining quantization of field theories in the setting of the geometric cobordism hypothesis.

• Daniel Grady (Wichita State University):

  Deformation classes of invertible field theories and the Freed-Hopkins conjecture

  In their seminal paper, Freed and Hopkins proved a classification theorem that identifies deformation classes of certain invertible topological field theories with the torsion subgroup of some generalized cohomology of a Thom spectrum. They conjectured that the identification continues to hold for non-topological field theories, if one passes from the torsion subgroup to the full generalized cohomology group of the Thom spectrum. In this talk, I will discuss a result which provides an affirmative answer to this conjecture. The method of proof uses recent joint work with Dmitri Pavlov on the geometric cobordism hypothesis.

• Daniel Berwick-Evans (University of Illinois Urbana-Champaign)

  How do field theories detect the torsion in topological modular forms?

  cf.: arXiv:2303.09138

  Since the mid 1980s, there have been hints of a deep connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner’s conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) in which cocycles are 2-dimensional supersymmetric field theories. Basic properties of these field theories lead to expected integrality and modularity properties, but the abundant torsion in TMF has always been mysterious. In this talk, I will describe deformation invariants of 2-dimensional field theories that realize certain torsion classes in TMF. This leads to a description of the generator of $\pi_3(TMF) =\mathbb{Z}/24$ in terms of the supersymmetric sigma model with target $S^3$.

• Christoph Schweigert (Hamburg University)

  String-net methods for CFT correlators

  cf.: arXiv:2302.01468

  Based on a graphical calculus for pivotal bicategories, we develop a string-net construction of a modular functor. We show that a rigid separable Frobenius functor between strictly pivotal bicategories induces a linear map between the corresponding bicategorical string-net spaces that is compatible with the mapping class group actions and with sewing. This result implies that correlators of two-dimensional conformal field theories factorize over string-net spaces constructed from defect data.

• Anna Beliakova (University of Zurich):

  On algebraisation of low-dimensional Topology

  cf.: arXiv:2205.11385

  Categories of $n$-cobordisms (for $n=2$, $3$ and $4$) are among the most studied objects in low dimensional topology. For $n=2$ we know that $2Cob$ is a monoidal category freely generated by its commutsative Frobenius algebra object: the circle. This result also classifies all TQFT functors on $2Cob$. In this talk I will present similar classification results for special categories of 3- and 4-cobordisms. They were obtained in collaboration with Marco De Renzi and are based on the work of Bobtcheva and Piergallini. Frobenius algebra in these cases will be replaced by a braided Hopf algebra.

  I plan to finish by relating our results with the famous problem in combinatorial group theory — the Andrews–Curtis conjecture.

• Carlo Collari (University of Pisa):

  Weight systems which are quantum states

  slides: pdf

  cf. arXiv:2210.05399

  Roughly speaking, a weight system is a function from a space of chord diagrams to the complex numbers. Weight systems can be used to recover invariants for the relevant kind of knotted object (eg. knots, links, braids etc.) from the Kontsevich integral. The work of Sati and Schreiber highlighted the connection between (horizontal) chord diagrams and higher observables in quantum brane physics. This motivates the question: “which weight systems are quantum states?” Corfield, Sati and Schreiber showed that all $\mathfrak{gl}(n)$ weight systems associated to the defining representation are indeed quantum states. In this talk I will present an extension of their result to more general weight systems.

  The plan of the talk is the following; first, I will introduce the mathematical problem. Then, I will review the proof given by Corfield, Sati and Schreiber that $\mathfrak{gl}(n)$ weight systems associated to the defining representation are quantum states. Finally, I will show how this result can be extended to weight systems associated to exterior and symmetric powers of the defining representation.

• Ralph Blumenhagen (Max-Planck-Institute for Physics, Munich):

  Nullifying Cobordism in Quantum Gravity

  cf.: arXiv:2208.01656, arXiv:2303.03423

  In the swampland program one tries to delineate effective theories consistent with quantum gravity from those which are not by so-called swampland conjecture. As a consequence of the absence of global symmetries in QG, one such conjecture is saying that the cobordism group has to vanish. In mathematics very often these groups do not vanish right away. Physics tells us that this can be ameliorated by either gauging or breaking of the corresponding global symmetries.

  First, we show how the gauging fits into some known constraints in string theory, the so-called tadpole cancellation conditions. Mathematically, this is reflecting a well known connection between certain K-theory and cobordism groups. Second, we report on new results related to the breaking of a global symmetry via codimension one defects. In fact, going beyond topology a very similar mechanism arises for (for a long time puzzling) rolling solutions in string theory, giving rise to the notion of a dynamical cobordism conjecture.

Apr 2023

• 27 Apr - 1 Mar 2023 (upcoming)

  NYUAD Hackaton on Quantum Computing 2023

  on quantum computing/quantum programming

May 2023

• 22 - 26 May 2023 (upcoming)

  Quantum Information, Quantum Matter and Quantum Gravity

  on quantum information, quantum matter and quantum gravity

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Quantum Colloquium

Weekly colloquium, broadly on quantum systems, with focus on quantum computation and specifically on topological quantum computation and dependently typed quantum programming languages.

May 2022

• Urs Schreiber on joint work with Hisham Sati:

  New Foundations for Topological Data Analysis – The Power of Cohomotopy

  
  

  The aim of topological data analysis (TDA) is to provide qualitative analysis of large data/parameter sets in a way which is robust against uncertainties and noise. This is accomplished using tools and theorems from the mathematical field of algebraic topology. While a tool called persistent homology has become the signature method of TDA, it tends to produce answers that are either hard to interpret (persistent cycles) or impossible to compute (well groups). Both problems are solved by a variant method $[$FK17$]$ which we may call persistent cohomotopy: A first result shows $[$FKW18$]$ that this new method provides computable answers to the concrete question of detecting whether there exist data+parameters that meet a prescribed target indicator precisely, even in the presence of uncertainty and noise. More generally, efficient data analysis will require further refining persistent cohomotopy to equivariant cohomotopy and/or twisted cohomotopy $[$SS20$]$. Curiously, these flavors of cohomotopy theory have profound relations to formal high energy physics and quantum materials, connecting to which might help to further enhance the power of topological data analysis.

Sep 2022

• 13 Sep 2022

  Mitchell Riley (NYU Abu Dhabi, CQTS):

  Dependent Type Theories à la Carte

  slides: pdf

  on realizing linear homotopy type theory

• 14 Sep 2022

  Sachin Valera (NYU Abu Dhabi, CQTS):

  A Quick Introduction to the Algebraic Theory of Anyons

  slides: pdf

  on anyon braiding described by braided fusion categories

• 14 Sep 2022

  Urs Schreiber (NYU Abu Dhabi, CQTS):

  Initial Researchers’ Meeting – Motivation, Strategy & Technology

  slides: pdf

  outline of a research program on Topological Quantum Programming in TED-K

Oct 2022

• 11 Oct 2022

  Tim Byrnes (NYU Shanghai, CQTS):

  Topological quantum states for quantum computing and metrology

  video: rec

  • Part I – Quantum teleportation of Majorana Zero Modes

    slides: pdf

    on Phys. Rev. Lett. 126, 090502 (2021)

  • Part II – Quantum Hall effect in Bose-Einstein condensates

    slides: pdf

    on Phys. Rev. A 92.023629 (2015) and Phys. Rev. B 99, 184427 (2019)

Nov 2022

• 07 Nov 2022

  Jiannis Pachos (Leeds University, UK):

  Non-abelian topological Berry-phases

  video: rec

  Combining physics, mathematics and computer science, topological quantum information [1] is a rapidly expanding field of research focused on the exploration of quantum evolutions that are resilient to errors. In this talk I will present a variety of different topics starting from introducing anyonic models, topological phases of matter, Majorana fermions, characterising knot invariants, their quantum simulation with anyons and finally the possible realisation of anyons in the laboratory.

  $\,$

  [1] Jiannis K. Pachos, Introduction to Topological Quantum Computation, Cambridge University Press (2012) $[$doi:10.1017/CBO9780511792908$]$

• 14 Nov 2022

  Sachin Valera (NYU Abi Dhabi, CQTS):

  A Quick Introduction to the Algebraic Theory of Anyons (Part II)

  slides: pdf

  on anyon braiding described by braided fusion categories

• 21 Nov 2022

  Andrew Kent (Center for Quantum Phenomena, NYU)

  A new spin on magnetism with applications in information processing

  slides: pdf

  video: rec, YT

  Recent advances in magnetism research are likely to have an important impact on electronics and information processing. These advances use the electron magnetic moment (spin) to transmit, write and store information. They enable new devices that operate at high speed with very low energy consumption. The information is stored in the orientation of electron magnetic moments in magnetic materials and can persist without power; energy is only needed to write and read the information. Important physics concepts include the interconversion of electrical (charge) currents into spin currents, the efficiency of the interconversion, controlling the currents, spin polarization direction, and the associated spin torques on magnetic order. Magnetic skyrmions are also of interest both because of their stability — they are topologically protected objects — and because their nucleation and motion can be controlled using spin currents. In this talk I will highlight the new physics concepts that have enabled these advances and discuss some of their applications in information processing.

cf.: J. Appl. Phys. 130 (2021) [doi:10.1063/5.0046950]

• 28 Nov 2022

  Asif Equbal (NYU Abu Dhabi, CQTS)

  Molecular spin qubits for future quantum technology

  slides: pdf

  video: rec, YT

  cf.: spin resonance qbits

  Spins are a purely quantum mechanical phenomenon and have been proposed as one of the several candidates for qubits in quantum information science. Quantum computers based on spin qubits were first proposed by DiVincenzo, who established five necessary criteria for building a quantum computer. The technology to control the quantum states of nuclear and electron spins and the theory of spin-spin and spin-magnetic field interactions are well developed, but a quantum computer based on spin qubits has not yet been realized. Why is this?

  In this talk, I will discuss the challenges in developing spin qubits that meet DiVincenzo’s criteria for quantum computers. First, I will explain in a pedagogical way how to manipulate spins in an external magnetic field that form the building block of quantum logic gates. I will then provide some insight into my own recent research on the development of optically polarized molecular spin qubits in solids.

Dec 2022

• 12 Dec 2022

  Leandro Aolita

  Quantum Algorithms, from noisy intermediate scale devices through the early fault-tolerant era

  video: rec, YT

  Reaching long-term maturity in quantum computation science and technology relies on the field delivering practically useful application in a short term. In this colloquium, I will discuss ideas for the noisy intermediate scale (NISQ) and early fault-tolerant eras. I will divide my talk into two parts. In the first part, I will make a brief non-technical introduction to the field, its relevance to the UAE, and the main lines of research of the Quantum Algorithms division at QRC-TII.

  In the second one, I will try to convey some level of technical detail about our work. In particular, I will first present a hybrid classical-quantum algorithm to simulate high-connectivity quantum circuits from low-connectivity ones. This provides a versatile toolbox for both error-mitigation and circuit boosts useful for NISQ computations. Then, I will move on to algorithms for the forthcoming quantum hardware of the early fault-tolerant era: I will present a new generation of high-precision algorithms for simulating quantum imaginary-time evolution (QITE) that are significantly simpler than current schemes based on quantum amplitude amplification (QAA). QITE is central not only to ground-state optimisations but also to partition-function estimation and Gibbs-state sampling, with a plethora of computational applications.

Jan 2023

• 30 Jan 2023

  Vivek Singh (CQTS @ NYU Abu Dhabi)

  Chern-Simons theory, Knot polynomials & Quivers

  slides: pdf

  video: rec, YT

  cf. arXiv:2103.10228

  First, I will give a brief introduction to knot theory and its connection to Chern-Simons quantum field theory. Then I discuss the method of obtaining polynomial invariants and limitations towards tackling classification of knots. In particular, we will highlight our new results on weaving knots and review the recent developments on Knot-Quiver correspondence.

Feb 2023

• 13 Feb 2023

  Kazuki Ikeda (Co-design Center for Quantum Advantage, Stony Brook University, USA)

  Demonstration of Quantum Energy Teleportation by Superconducting Quantum Processors and Implications for Communications and High Energy Physics

  Quantum energy teleportation is a theoretical concept in quantum physics that describes the transfer of energy from one location to another without the need for a physical medium to carry it. This is made possible by means of universal properties of quantum entanglement and measurement of quantum states. The role of QET in physics and information engineering is largely unexplored, as the theory has not received much attention for long time since it was proposed about 15 years ago. To validate it on a real quantum processor, my research has tested the energy teleportation protocol in its most visible form for the first time on IBM’s superconducting quantum computer. In my colloquium talk, I will explain the historical background of quantum energy teleportation, quantum circuits and quantum operations. Moreover I will present a concrete setup for a long-distance and large-scale quantum energy teleportation with real quantum networks.

  In addition, I will present the results of quantum simulations with relativistic field theory as a study based on the high-energy physics perspective and the symmetry-protected topological (SPT) phase of matter of quantum energy teleportation. The models will describe include the two dimensional QED (the massive Thirring model), the AKLT model, the Haldane model, and the Kitaev model. Those results show that the phase diagrams of the field theory and SPT phase are closely related to energy teleportation.

  In summary my talk will provide a novel suggestion that quantum energy teleportation paves a new pathway to a link between quantum communication on real quantum network, phase diagram of quantum many-body system, and quantum computation.

• 20 Feb 2023

  Zachary Murray

  Constructive Real Numbers in the Agda Proof Assistant

  cf. arXiv:2205.08354

  video: rec, YT

  Proof assistant software enables the development of proofs in a manner such that a computer can verify their validity. As proof assistants commonly take the form of a programming language, users face programming-related problems, such as the naturality of expressing ideas and algorithms in the language, usability, and performance. We will investigate these issues as they occur in developing Errett Bishop’s constructive real numbers in the Agda proof assistant and functional programming language, with an introduction to each.

• 27 Feb 2023

  Aeysha Khalique (National University of Science and Technology, Islamabad):

  Computational Tasks through Non-Universal Quantum Computation

  video: YT

  Quantum Mechanics offers phenomena which defy our everyday observation. These are not just theoretical principles but have wide range applications in quantum computation and quantum information, making some tasks possible which are impossible to be done classically. This talk will take you to the journey through quantum computation, starting from underlying principles to the applications, including my own own contribution to it.

Mar 2023

• 6 Mar 2023

  Altaf Nizamani and Qirat Iqbal (University of Sindh, Pakistan):

  Quantum Technology with Trapped Ions

  video: YT

  Quantum technology is a rapidly advancing field that is poised to revolutionize numerousindustries, including computing, communications, sensing, and cryptography. At its core, quantum technology relies on the principles of quantum mechanics, which allow for the creation of devices that operate on the quantum level. These devices based on quantum technology can perform tasks that are impossible or prohibitively difficult for classical devices. One of the most promising applications of quantum technology is in quantum computing, quantum communications, and quantum sensors.

  Trapped ions are one of the promising platform for quantum computing and sensing. In this approach, individual ions are trapped in a vacuum chamber using electromagnetic fields and manipulated using lasers to perform quantum operations. As a quantum system, trapped ions offer several advantages. First, they have long coherence times, meaning that the quantum state of the ion can be preserved for a longer period, allowing for more complex calculations. Second, trapped ions can be precisely controlled and manipulated, allowing for the implementation of high-fidelity quantum gates. Finally, trapped ions can be entangled with one another, allowing for the implementation of quantum algorithms that are impossible to simulate on classical computers. Trapped ions also have great potential as quantum sensors. By using the properties of the ions to measure changes in their environment, trapped ions can detect minute changes in temperature, magnetic fields, and electric fields, among other things. This makes them useful for applications in precision measurement, such as in atomic clocks, gravitational wave detection, and magnetometry.

  One of the major challenges facing trapped ion systems is scalability. While individual ions have been used to perform simple quantum algorithms, scaling the system up to include a large number of ions is a difficult task. However, recent advances in ion trap technology have made it possible to trap larger numbers of ions and transport them in 2D and 3D space to perform more complex operations for quantum computation and sensing experiments. Realization of such devices is not far away. As compared to present atomic clocks, a new generation of quantum-enhanced clocks is now emerging showing significantly improved accuracy. Sensitive physical measurements are an essential component of modern science and technology. Developments in quantum sensors will outdate their classical counterparts.

  We will present recent developments and opportunities in quantum technology applications based on trapped ions, including quantum computation and sensing.

• 13 Mar 2023

  Roger S. K. Mong (Pittsburgh Quantum Institute, USA)

  Detecting topological order from modular transformations of ground states on the torus

  cf.: arXiv:2203.04329

  Every two-dimensional topological phase is associated with some topological quantum field theory (TQFT), or more formally a modular tensor category. The ground states of a topological phase encode information about the TQFT, which makes them useful in determining the TQFT data, such as anyon mutual statistics and self statistics. For example, many numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), which are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We rederive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular $S$- and $T$-matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity means that there are some sets of TQFTs that cannot be distinguished from one another solely by the overlap of MESs.

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Geometry, Topology & Physics (GTP) Seminar

Weekly seminar, broadly on topics in geometry, (algebraic) topology and theoretical/mathematical physics, with some focus on applicability to high energy physics/string theory and quantum systems.

Feb 2022

• 02 Feb 2022

  Luigi Alfonsi (University of Hertfordshire)

  Higher quantum geometry and global string duality

  video: rec

  In this talk I will discuss the relation between higher geometric quantisation and the global geometry underlying string dualities. Higher geometric quantisation is a promising framework that makes quantisation of classical field theories achievable. This can be obtained by quantising either an ordinary prequantum bundle on the ∞-stack of solutions of the equations of motion or a categorified prequantum bundle on a generalised phase space. I will discuss how the higher quantum geometry of string theory underlies the global geometry of T-duality. In particular, I will illustrate how a globally well-defined moduli stack of tensor hierarchies can be constructed and why this is related to a higher gauge theory with the string 2-group. Finally, I will interpret the formalism of Extended Field Theory as an atlas description of the higher quantum geometry of string theory.

• 23 Feb 2022

  Dmitri Pavlov (Texas Tech University)

  The geometric cobordism hypothesis

  video: rec

  I will explain my recent work with Daniel Grady on locality of functorial field theories (arXiv:2011.01208) and the geometric cobordism hypothesis (arXiv:2111.01095). The latter generalizes the Baez–Dolan cobordism hypothesis to nontopological field theories, in which bordisms can be equipped with geometric structures, such as smooth maps to a fixed target manifold, Riemannian metrics, conformal structures, principal bundles with connection, or geometric string structures.

  Applications include a generalization of the Galatius–Madsen–Tillmann–Weiss theorem, a solution to a conjecture of Stolz and Teichner on representability of concordance classes of functorial field theories, a construction of power operations on the level of field theories (extending the recent work of Barthel–Berwick-Evans–Stapleton), and a recent solution by Grady of a conjecture by Freed and Hopkins on deformation classes of reflection positive invertible field theories. If time permits, I will talk about the planned future work on nonperturbative quantization of functorial field theories and generalized Atiyah–Singer-style index theorems.

Mar 2022

• 08 March 2022

  David White (Denison University, USA):

  The Kervaire Invariant, multiplicative norms, and N-infinity operads

  video: rec

  In a 2016 Annals paper, Hill, Hopkins, and Ravenel solved the Kervaire Invariant One Problem using tools from equivariant stable homotopy theory. This problem goes back over 60 years, to the days of Milnorand the discovery of exotic smooth structures on spheres. Of particular importance it its solution were equivariant commutative ring spectra and their multiplicative norms. A more thorough investigation of multiplicative norms, using the language of operads, was recently conducted by Blumberg and Hill, though the existence in general of their new “N-infinity” operads was left as a conjecture. In this talk, I will provide an overview of the Kervaire problem and its solution, I will explain where the operads enter the story, and I will prove the Blumberg-Hill conjecture using a new model structure on the categoryof equivariant operads.

• 16 March 2022

  Guo Chuan Thiang (Beijing University)

  How open space index theory appears in physics

  video: rec

  The incredible stability of quantum Hall systems and topological phases indicates protection by an underlying index theorem. In contrast to Atiyah-Singer theory for compactified problems, what is required is an index theory on noncompact Riemannian manifolds, with interplay between discrete and continuous spectra. Input data comes not from a topological category a la TQFT, but a metrically-coarsened one. This is the subject of coarse geometry and index theory, and I will explain their experimental manifestations.

• 30 March 2022

  Martin Palmer (Romanian Academy)

  Mapping class group representations via Heisenberg, Schrödinger and Stone-von Neumann

  zoom link

  One of the first interesting representations of the braid groups is the Burau representation. It is the first of the family of Lawrence representations, defined topologically by viewing the braid group as the mapping class group of a punctured disc. Famously, the Burau representation is almost never faithful, but the $k = 2$ Lawrence representation is always faithful: this is a celebrated theorem of Bigelow and Krammer and implies immediately that braid groups are linear (act faithfully on finite-dimensional vector spaces). Motivated by this, and by the open question of whether mapping class groups are linear, I will describe recent joint work with Christian Blanchet and Awais Shaukat in which we construct analogues of the awrence representations for mapping class groups of compact, orientable surfaces. Tools include twisted Borel-Moore homology of con guration spaces, Schrödinger representations of discrete Heisenberg groups and the Stone-von Neumann theorem.

Apr 2022

• 06 April 2022

  Kiyonori Gomi (Tokyo Institute of Technology)

  Differential KO-theory via gradations and mass terms

  zoom link

  Differential generalized cohomologies refine generalized cohomologies on manifolds so as to retain information on differential forms. The aim of my talk is to describe formulations of differential KO-theory based on gradations and mass terms. The formulation based on mass terms is motivated by a conjecture of Freed and Hopkins about a classification of invertible quantum field theories and by a model of the Anderson dual of cobordism theory given by Yamashita and Yonekura. I will start with an account of this background, and then describe the formulation of differential KO-theory. In the formulation a key role is played by a uperconnection associated to a mass term. This is a joint work with Mayuko Yamashita.

• 13 April 2022

  Mario Velásquez (Universidad Nacional de Colombia)

  The Baum-Connes conjecture for groups and groupoids

  In this talk we present some basics definitions around the Baum-Connes conjecture in the context of groups and groupoids, in particular we define the reduced $C^\ast$-algebra $C_r^*(G)$ of a groupoid G. When a group (or groupoid) satisfies this conjecture we present how we can compute the topological K-theory of $C_r^*(G)$ via a classifying space. We also present some explicit computations and an application about Fredholm boundary conditions in manifolds with corners.

• 27 April 2022

  Amnon Neeman (Australian National University)

  Bounded t-structures and stability conditions

  We will give a gentle introduction to the topic. We will review the definitions of derived and triangulated categories, of t-structures an of stability conditions. The only new result will come at the very end of the talk, a theorem saying that there are no stability condition on the derived category of bounded complexes of vector bundles on a singular scheme.

May 2022

• 11 May 2022

  Alex Fok (NYU Shanghai)

  Equivariant twisted KK-theory of noncompact Lie groups

  The Freed-Hopkins-Teleman theorem asserts a canonical link between the equivariant twisted K-theory of a compact Lie group equipped with the conjugation action by itself and the representation theory of its loop group. Motivated by this, we will present results on the equivariant twisted KK-theory of a noncompact semisimple Lie group $G$. We will give a geometric description of generators of the equivariant twisted KK-theory of G with equivariant correspondences, which are applied to formulate the geometric quantization of quasi-Hamiltonian manifolds with proper G-actions. We will also show that the Baum-Connes assembly map for the $C^\ast$-algebra of sections of the Dixmier-Douady bundle which realizes the twist is an isomorphism, and discuss a conjecture on representations of the loop group $L G$. This talk is based on joint work with Mathai Varghese.

Sep 2022

• 21 Sep 2022

  Grigorios Giotopoulos (NYU Abi Dhabi)

  Braided Homotopy Lie Algebras and Noncommutative Field Theories

  on: arXiv:2112.00541

• 28 Sep 2022

  David Jaz Myers (NYU Abu Dhabi, CQTS)

  Objective Cohomology – Towards topological quantum computation

  slides: pdf

  In this talk, we will see the homotopy type theory point of view on defining pptwisted cohomology]] classes by means of bundle gerbes. We’ll take an increasingly less leisurely tour up the tower of cohomology degrees, seeing characters, principal bundles, central extensions, and characteristic classes along the way. Finally, we will go through the construction of the cohomology of the braid groups valued in the complex numbers, twisted by a complex character of the braid group. Through the work of many people, and in particular Feigin, Schechtman, Varchenko, the actions of the braid group of $d$ “defects” on the twisted complex cohomology of the braid group of $n$ “particles” is the monodromy action of the Knizhnik-Zamolodchikov connection on a space of conformal blocks. At CQTS we use this as a way to go from abstract homotopy type theory to protocols for topological quantum computation.

Oct 2022

• 05 Oct 2022

  Adrian Clough (NYU Abu Dhabi, CQTS)

  The smooth Oka principle

  video: rec

  notes: pdf

  cf.: Clough 2021

The infinity topos of differentiable sheaves contains all smooth manifolds as a full subcategory and has excellent formal properties. In particular, it admits an intrinsic notion of underlying homotopy type of any differentiable sheaf, which coincides with classical constructions such as taking smooth total singular complexes. Moreover, there is a canonical sense in which the mapping sheaf between any two differentiable sheaves may have the correct homotopy type. This latter notion is reminiscent of the Oka principle in complex geometry. In this talk I will show how to exhibit the Oka principle in the smooth setting using model structures and other homotopical calculi on the infinity topos of differentiable sheaves.

• 12 Oct 2022

  Salah Mehdi (U Lorraine and NYU Abu Dhabi)

  Algebraic and geometric aspects of the Dirac equation

  on arXiv:2102.03562

• 19 Oct 2022

  Liang Kong (SIQSE and SUST)

  Topological Wick Rotation and Holographic Dualities

  video: rec

  slides: pdf

  on topological order, braided fusion categories and the holographic principle

• 26 Oct 2022

  Urs Schreiber (NYU Abu Dhabi, CQTS)

  Quantum Programming via Linear Homotopy Types

  slides: see those for external talk at QTML2022

  We first recall basic notions of quantum logic gates and quantum circuits, highlighting the conceptually more subtle issues of classical effects (measurements) and control (state preparation). Then we briefly review the formulation of computational effects and control via adjunctions and monads on data type type systems, in order to finally indicate basics of our observation that in any decent type system which has classically dependent linear data types, the relevant language structures for describing classical/quantum effects emerge naturally.

Nov 2022

• 09 Nov 2022

  Zhen Huan (HUST)

  Twisted Real quasi-elliptic cohomology

  video: rec

  Quasi-elliptic cohomology is closely related to Tate K-theory. It is constructed as an object both reflecting the geometric nature of elliptic curves and more practicable to study than most elliptic cohomology theories. It can be interpreted by orbifold loop spaces and expressed in terms of equivariant K-theories. We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of the Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal. In this talk we construct twisted Real quasi-elliptic cohomology as the twisted KR-theory of loop groupoids. The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory, we construct Real analogues of the string power operation of quasi-elliptic cohomology. We also explore the relation of the theory to the Tate curve. This is joint work with Matthew Young. [arXiv:2210.07511]

• 23 Nov 2022

  Valentino Foit (NYUAD)

  Brownian loops and conformally invariant systems

  slides: pdf

  video: rec, YT

  The Brownian loop soup (BLS) is a stochastic system that is constructed from random loops in the plane and is invariant under conformal transformations. Correlation functions of certain observables can be used to formulate the BLS as a Conformal Field Theory (CFT). I will give an overview of CFTs in two dimensions and point out their relation to certain stochastic systems. Then I will discuss the BLS including some recent progress, such as the operator content, the continuous spectrum, and hints of an extended symmetry algebra.

• 30 Nov 2022

  Allan Merino,

  Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup

  video: rec

  One of the main problems in representation theory is to determine the set of equivalence classes of irreducible unitary representations of a Lie group. Using the Weil representation, Roger Howe established a one-to-one correspondence (known as the local theta correspondence) between particular representations of two subgroups $G$ and $G'$ forming a dual pair in $Sp(W)$. This correspondence provides a nice way to construct unitary representations of small Gelfand-Kirillov dimension.

  In his wonderful paper “Remarks on classical invariant theory”, Roger Howe suggested that his classical duality should be extendable to superalgebras/supergroups. In a recent work with Hadi Salmasian, we obtained a classification of irreducible reductive dual pairs in a real or complex orthosymplectic Lie supergroup $SpO(V)$. Moreover, we proved a “double commutant theorem” for all dual pairs in a real or complex orthosymplectic Lie supergroup.

  In my talk, I will spend quite some time explaining how the Howe duality works in the symplectic case and then talk about the results we obtained in our paper with H. Salmasian. [arXiv:2208.09746]

Dec 2022

• 07 Dec 2022

  Emily Riehl (Johns Hopkins University)

  $\infty$-Category theory for undergraduates

  video: rec, YT

  cf.: arXiv:2302.07855

  At its current state of the art, $\infty$-category theory is challenging to explain even to specialists in closely related mathematical areas. Nevertheless, historical experience suggests that in, say, a century’s time, we will routinely teach this material to undergraduates. This talk describes one dream about how this might come about — under the assumption that 22nd century undergraduates have absorbed the background intuitions of homotopy type theory/univalent foundations.

• 14 Dec 2022

  Eric Finster (University of Birmingham)

  The $(\infty,1)$-category of Types

  slides: pdf

  video: rec, YT

  A major outstanding difficulty in Homotopy Type Theory is the description of coherent higher algebraic structures. As an example, we know that the algebraic structure possessed by the collection of types and functions between them is not a traditional 1-category, but rather an (∞,1)-category. In this talk, I will describe how the addition of a finite collection of additional definitional equalities designed to render the notion of “opetopic type” definable in fact allows one to construct the (∞,1)-category structure on the universe of types.

Jan 2023

• 25 Jan 2023

  Thomas Creutzig (University of Alberta):

  Representation Theory of affine vertex algebras

  Recently there has been increased interest in non-semisimple braided tensor categories. Vertex algebras are a rich source of such categories and so I will give an overview on the representation theory of affine vertex algebras with a focus on the simplest example of $\mathfrak{sl}(2)$. As we will see, already in this example quite rich and non-semisimple categories of modules appear.

• 01 Feb 2023

  David Jaz Myers (CQTS @ NYU Abu Dhabi):

  Simplicial, Differential, and Equivariant Homotopy Type Theory

  video: rec, YT

  cf.: arXiv:2301.13780

  on cohesive homotopy type theory with two commuting notions of cohesion

Feb 2023

• 8 Feb 2023

  Ruizhi Huang (Chinese Academy of Sciences)

  Fractional structures on bundle gerbe modules and fractional classifying spaces

  video: rec

  cf.: arXiv:2203.14439

  Both higher structures and bundle gerbe modules play important roles in modern geometry and mathematical physics. Bundle gerbe modules is a twisted version of vector bundles, and was introduced by Bouwknegt-Carey-Mathai-Murray-Stevenson in 2002. In particular, they introduced the twisted Chern character from the perspective of Chern-Weil theory. In a recent joint work with Han and Mathai, we study the homotopy theory aspects of the twisted Chern classes of torsion bundle gerbe modules. Using Sullivan’s rational homotopy theory, we realize the twisted Chern classes at the level of classifying spaces. The construction suggests a notion, which we call fractional U-structure serving as a universal framework to study the twisted Chern classes of torsion bundle gerbe modules from the perspective of classifying spaces. Based on this, we introduce and study higher fractional structures on torsion bundle gerbe modules parallel to the higher structures on ordinary vector bundles.

• 15 Feb 2023

  Eugene Rabinovich (University of Notre Dame, USA)

  Classical Bulk-Boundary Correspondences via Factorization Algebras

  cf. arXiv:2202.12332 (a form of Poisson holography)

  A factorization algebra is a cosheaf-like local-to-global object which is meant to model the structure present in the observables of classical and quantum field theories. In the Batalin-Vilkovisky (BV) formalism, one finds that a factorization algebra of classical observables possesses, in addition to its factorization-algebraic structure, a compatible Poisson bracket of cohomological degree +1. Given a “sufficiently nice” such factorization algebra on a manifold $N$, one may associate to it a factorization algebra on $N\times \mathbb{R}_{\geq 0}$.

  The aim of the talk is to explain the sense in which the latter factorization algebra “knows all the classical data” of the former.

  This is the bulk-boundary correspondence of the title. Time permitting, we will describe how such a correspondence appears in the deformation quantization of Poisson manifolds.

• 22 Feb 2023

  Dmitry Kozlov

  Applied and Computational Topology

  • video: YT

  We will give a brief introduction to the subject of Applied and Computational Topology. The survey of the subject’s main ideas and tools will be complemented with applications to discrete mathematics and to theoretical distributed computing. We will conclude with stating an open problem in combinatorial topology which is related to the complexity of the Weak Symmetry Breaking distributed task.

Mar 2023

• 1 Mar 2023

  Lukas Müller

  Deformation quantization and categorical factorization homology

  video: YT

  cf. arXiv:2107.12348

  Moduli spaces of flat principal bundles on surfaces are a prominent object in mathematical physics, algebraic geometry and geometric representation theory. In particular they are the phase space of 3-dimensional Chern-Simons theory on a surface times an interval and hence equipped with a symplectic structure going back to the work of Atiyah and Bott. Various deformation quantizations of the algebra of functions have been constructed. Ben-Zvi, Brochier & Jordan constructed “local to global” quantizations using factorization homology of representation categories of quantum groups. Local to global constructions in this setting only work if the higher geometric structure of the moduli space of flat bundles is taken into account, i.e. it is treated as a moduli stack. In this setting the algebra of functions does not contain all the information and should be replaced by the category of quasicoherent sheaves.

  In my talk we will explore categorifications of deformation quantization as deformations of symmetric monoidal categories (algebras over the $E_\infty$-operad) into $E_i$-categories and their interplay with factorization homology. The main result is that 2-dimensional factorization homology “commutes” with quantization in a way relating $E_0$-quantizations to braided ($E_2$) quantizations. We will illustrate our results with examples from Poisson geometry and quantum groups. As a specific application we show that deformation quantizations of the moduli space of flat bundles based on Kontsevich integrals constructed by Li, Bland & Ševera are equivalent to quantizations constructed by Alekseev, Grosse & Schomerus based on quantum groups. The talk is based on joint work in progress with Eilind Karlsson, Corina Keller, and Jan Pulmann.

• 8 Mar 2023

  Gereon Quick (Norwegian University of Science and Technology):

  Geometric Hodge filtered complex cobordism

  cf.: arXiv:2210.13259

  video: YT

  Differential cohomology theories on smooth manifolds play an important role in mathematical physics and other areas of mathematics. In their seminal work, Hopkins and Singer showed that every topological cohomology theory has a differential refinement. In this talk, I will first report on joint work with Mike Hopkins on a similar refinement of complex cobordism on complex manifolds which takes the Hodge filtration into account. I will then present joint work with Knut Haus in which we give a concrete geometric cycle model for this theory. This allows us to give a concrete description of an Abel-Jacobi type secondary invariant for topologically trivial cobordism cycles.

External presentations

External presentations reporting on work at CQTS.

Sep 2022

• 15 Sep 2022 at PlanQC 2022

  Urs Schreiber on joint work with Hisham Sati:

  Topological Quantum Programming in TED-K

  slides: pdf (view full screen)

  • video: YT

  • extended abstract: arXiv:2209.08331

• 16 Sep 2022 at Math Faculty Meeting, NYU Abu Dhabi

  Urs Schreiber on joint work with Hisham Sati:

  Practical Foundations for Topological Quantum Programming

  slides: pdf

Nov 2022

• 12 Nov 2022 at Workshop on Quantum Software, satellite of QTML 2022 (Naples, Italy)

  Urs Schreiber on joint work with D. J. Myers, M. Riley and H. Sati:

  Quantum Data Types via Linear Homotopy Type Theory

  slides: pdf

  The proper concept of data types in quantum programming languages, hence of their formal verification and categorical semantics, has remained somewhat elusive, as witnessed by the issue of “dynamic lifting” encountered in the Quipper language family. In this talk I explain our claim that a powerful quantum data type-system elegantly solving these problems is naturally provided by the linear homotopy type theory recently realized by M. Riley. Together with our previous claim that homotopy type theory natively knows about the fine detail of $\mathfrak{su}$(2)-anyon braid quantum gates, this shows that linear homotopy type theory is a natural substrate for typed quantum programming languages aware of topological quantum hardware.

Dec 2022

• 17 Dec 2022 at AQIS 2022

  Sachin Valera:

  Braidless Topological Quantum Teleportation

  poster: pdf

  on quantum teleportation with/of anyons

Jan 2023

• 15 Jan 2023

  Hisham Sati:

  M-theory and matter via Twisted Equivariant Differential (TED) K-theory

  talk at M-Theory and Mathematics 2023, NYU Abu Dhabi

  [links]

• 15 Jan 2023

  Urs Schreiber:

  Topological Quantum Gates from M-Theory

  talk at M-Theory and Mathematics 2023, NYU Abu Dhabi

  [links]

Feb 2023

• 20 Feb 2023

  Amaria Javed:

  Simulating an all-optical quantum controlled-NOT gate using soliton scattering by a reflectionless potential well

  talk at UAE U Nonlinear Physics Group Conference, Al Ain

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Members

Principal Investigator:

• Hisham Sati

co-PIS:

• Tim Byrnes

• Asif Equbal

• for more see here

Senior Researcher:

• Urs Schreiber

Researchers:

• Adrian Clough

• Tatiana Ezubova

• Grigorios Giotopoulos

• Amaria Javed

• Marwa Mannaï

• David Jaz Myers

• Mitchell Riley

• Vivek Kumar Singh

• Sachin Valera

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category: reference