nLab Center for Quantum and Topological Systems

The Center for Quantum and Topological Systems (CQTS) is a Research Center, launching 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 topological data analysis, ultimately aimed at addressing open questions in topological quantum computation. A unifying theme is the use of new methods from (“persistent”) Cohomotopy and generalized cohomology-theory, developed in high energy physics.

Conferences & Workshops

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June 2022

Homotopical perspectives on Topological data analysis

Organizers: Sadok Kallel and Hisham Sati

zoom link: nyu.zoom.us/j/91860528688

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 $K_{x}$.

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CQTS Colloquium

May 2022

• Urs Schreiber on joint work with Hisham Sati:

  New Foundations for Topological Data Analysis – The Power of Cohomotopy

  
  

  Abstract. 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.

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GTP Seminar

Geometry, Topology and Physics Seminar

Feb 2022

• 02 Feb 2022

  Luigi Alfonsi (University of Hertfordshire)

  Higher quantum geometry and global string duality

  zoom recording

  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

  zoom recording

  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.

March 2022

• 08 March 2022

  David White (Denison University, USA):

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

  zoom recording

  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

  zoom recording

  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.

April 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

  Abstract: 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)

  Equvariant 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.

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category: reference