# nLab Center for Quantum and Topological Systems

Contents

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.

# Contents

## Conferences & Workshops

### 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

$\phantom{-----}$ [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

Break: 10:40 - 11:10

• 11:10 - 11:15

Intro to Quantum Algorithms @TII

• 11:20 - 11:50

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

Introducing research and researchers @CQTS

• 14:30 - 14:50

Quantum information processing via NLS

• 15:00 - 15:20

Tuning topological quantum materials

• 15:30 - 15:50

Verified quantum programming with linear HoTT

Break: 3:50-4:20 pm

### Mar 2023

15-18 Mar 2023 (ongoing)

Geometric/Topological Quantum Field Theories and Cobordisms

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

$\;\;$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ 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

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.

• 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.

• 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.

• Nafaa Chbili (United Arab Emirates University):

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.

• Carlo Collari (University of Pisa):

Weight systems which are quantum states

slides: pdf

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

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)

### May 2023

• 22 - 26 May 2023 (upcoming)

Quantum Information, Quantum Matter and Quantum Gravity

## 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.

### 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

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

slides: pdf

### 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$]$

• 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]

### 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

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.

• 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.

## 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.

### 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

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

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

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

### Oct 2022

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

Algebraic and geometric aspects of the Dirac equation

• 19 Oct 2022

Liang Kong (SIQSE and SUST)

Topological Wick Rotation and Holographic Dualities

video: rec

slides: pdf

### Nov 2022

• 23 Nov 2022

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

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.

### Feb 2023

• 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.

## External presentations

External presentations reporting on work at CQTS.

### Sep 2022

• 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

## Members

Principal Investigator:

co-PIS:

Senior Researcher:

Researchers:

category: reference

Last revised on March 22, 2023 at 13:34:32. See the history of this page for a list of all contributions to it.