BIMSA-Tsinghua Quantum Symmetry Seminar

TBD

Reference: https://doi.org/10.1090/jams/1023

Lean is a theorem prover that enables mathematical formalization with computer-verified correctness. In this talk, I will provide an overview of monoidal category theory as implemented in Mathlib, which is Lean’s user-maintained mathematics library. I will also explain how Lean’s metaprogramming framework handles the coherence theorem in monoidal categories, demonstrating the interaction between functional programming and formalized mathematics.

Speaker Intro:

Yuma Mizuno obtained his PhD from Tokyo Institute of Technology with research on cluster algebras. He is currently a postdoc at University College Dublin. He is also interested in Lean theorem prover and is working on formalization of 2-category theory.

Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated as Landau’s effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.

Speaker Intro:

Yu Zhao holds a B.Sc. in Physics from Fudan University and is currently a Ph.D. candidate in theoretical physics at Fudan under the supervision of Prof. Yidun Wan. His research focuses on exactly solvable models and the gauge structures of topological orders. He has authored five peer-reviewed papers in journals such as JHEP. He is now seeking a postdoctoral position.

We study bimodule quantum Markov semigroups, which describe the dynamics of quantum systems that preserve a large class of symmetries.

Mathematically, the symmetry is encoded by a finite index inclusion of von Neumann algebras N⊆M, and can be extracted from the standard invariant of the inclusion.

Using the standard invariant, we show how the generator of a bimodule quantum Markov semigroup can be expressed in terms of the Fourier multiplier.

This allows us to extend the notion of equilibrium states to bimodule equilibirum.

A bimodule quantum channel can have no stationary states but still admit a bimodule equilibrium.

Moreover, a bimodule quantum channel admits a bimodule equilibrium, we prove that the fixed points of the channel form a von Neumann subalgebra of M.

Finally, we discuss how well-known functional inequalities, such as the Poincaré inequality, the logarithmic Sobolev inequality and the Talagrand inequality, can be generalized to the bimodule equilibrium setting.

Reference: https://arxiv.org/abs/2504.09576

(by Jinsong Wu and Zishuo Zhao)

I will introduce a class of topological algebras that are appropriate analogues of C*-algebras over the p-adic integers. Thereafter, I will define a version of topological K-theory for such algebras, computing it in some simple cases.

Reference: https://arxiv.org/abs/2403.04046 (to appear at Trans. Amer. Math. Soc)

(by Alcides Buss, Luiz Felipe Garcia, Devarshi Mukherjee)

Speaker Intro:

He is a postdoc in the topology group at the University of Münster, working at the intersection of arithmetic and noncommutative geometry. He did his PhD in Göttingen and Copenhagen under the supervision of Ralf Meyer and Ryszard Nest.

The Berkovich projective line is an analytic space over a non-Archimedean field. Over the field of complex p-adic numbers Cp, it can be realized as an infinite R-tree with

a dense number of branching points and countable branches at each of these points. The automorphism group of the R-tree is in this case identified with PGL(2,Cp).

In this talk, we will review the representation theory of groups acting on R-trees. The theory is well understood in the case of locally compact trees. Interesting results can also be derived in

the case of R-trees with countable branches such as the Berkovich line.

Using this machinery, we exhibit a cross-product C*-algebra on the Berkovich line and construct an unbounded spectral triple on this space. This allows us to develop a noncommutative

harmonic analysis on the Berkovich projective line over Cp. We show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of this C*-algebra.

This is a joint work with Masoud Khalkhali

Reference: https://arxiv.org/abs/2411.02593

Speaker Intro:

Damien Tageddine (pronounced Ta:jédeen) has completed his PhD thesis at McGill University in 2024. He has been collaborating with Masoud Khalkhali (Western Ontario) since 2023. His main research interest is on noncommutative geometry; specially his work focuses on spaces closely tied to discrete structures (such as discrete groups, trees, R-trees, graphs etc…).

In this talk, we will present a precise computation of the (optimal) rate at which classical information can be transmitted through some quantum channels from a sender to a receiver when they share an unlimited amount of entanglement. The channels under investigation belong to a new class of unital quantum channels acting on matrix algebras, which act as multipliers when the matrix algebra is identified with a finite-dimensional fermion algebra. Our approach relies on an ergodic action of the discrete hypercube \(\{−1, 1\}^n\) on the fermion algebra.

If time permits, we will also discuss how this method can be greatly generalized to encompass any ergodic action of a compact abelian group, thereby introducing a broad class of quantum channels for which the aforementioned transmission rate can be explicitly computed for each ergodic action.

References:

https://arxiv.org/abs/2008.12019

and

https://doi.org/10.1016/j.jfa.2024.110790 (Journal of Functional Analysis)

Speaker Intro:

Cédric teaches to second-year students while also conducting research in Functional Analysis. He has already published over 20 papers, many of which appear in prestigious journals such as Advances in Mathematics, Journal of Functional Analysis, Transactions of the AMS.

TBD

Reference: https://arxiv.org/abs/2503.11149 (by Michael Brannan, Daniel Gromada, Junichiro Matsuda, Adam Skalski, Mateusz Wasilewski)

Paper’s abstract: We establish a quantum version of Frucht’s Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley graphs of the quantum group in question, and then providing a method to systematically combine them into a single quantum graph with the right symmetry properties. We also show that the dual Γˆ of any non-abelian finite group Γ is ``quantum rigid’’. That is, Γˆ always admits a quantum Cayley graph whose quantum automorphism group is exactly Γˆ.

Speaker Intro:

Mateusz is an assistant professor at the Institute of Mathematics of the Polish Academy of Sciences. From 2018 to 2021, he was a postdoc at KU Leuven (his supervisor was Stefaan Vaes). In June 2018, he obtained a PhD from the Institute of Mathematics of the Polish Academy of Sciences (under the supervision of Adam Skalski).

In this talk, I will discuss the modular data arising from the Drinfeld centers of near-group categories. I will begin by discussing a near-group category of type \(Z\)/4\(Z\)×\(Z\)/4\(Z\)+16 and explain how the modular data of its Drinfeld center can be computed by solving Izumi’s polynomial equations. I will then show that modular data of rank 10 can be obtained via condensation of its Drinfeld center and present an alternative realization of this data through the Drinfeld center of a fusion category of rank 4. Finally, I will discuss the modular data of the Drinfeld center of a near-group category of type \(Z\)/8\(Z\)+8 and demonstrate that the non-pointed factor of its condensation coincides with the modular data of the quantum group category \({C}\)(\({g}\)_2, 4). This talk is based on joint work with Zhiqiang Yu.

Reference: https://arxiv.org/abs/2412.20763

Speaker Intro:

Qing Zhang received her Ph.D. from Texas A&M University and is currently a Visiting Assistant Professor at the University of California, Santa Barbara. Her primary research focuses on tensor categories, with an emphasis on the classification and construction of modular and super-modular categories.

The Collatz map (or the 3n+1-map) f is defined on positive integers by setting f(n) equal to 3n+1 when n is odd and n/2 when n is even. The Collatz conjecture states that starting from any positive integer n, some iterate of f takes value 1. In this study, we discuss formulations of the Collatz conjecture by C∗-algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the C∗-algebra generated by each of these, we consider the condition that it has no non-trivial reducing subspaces. For (1), we prove that the condition implies the Collatz conjecture. In the cases (2) and (3), we prove that the condition is equivalent to the Collatz conjecture. For similar maps, we introduce equivalence relations by them and generalize connections between the Collatz conjecture and irreducibility of associated C∗-algebras.

References: arXiv:2411.08084

Speaker Intro:

Takehiko Mori was born and grew up in Japan. After receiving bachelor’s and master’s degrees from Chiba University, he enrolled in the doctoral program at the same university and continued. The area of expertise is operator algebras and their applications. He has been studying connections between operator algebras and dynamical systems.

The collection of non-degenerate braided fusion categories possesses a certain equivalence relation, the so-called Witt equivalence (invertible morphisms in the higher Morita category). The set of equivalence classes form an abelian group (the Witt group). Although this group has been calculated as an abstract group (as a direct sum of cyclic groups) not much is known about its generators.

Speaker Intro:

Professor Alexei Davydov got his PhD from Moscow University 1992, and worked at Moscow University as assistant professor from 1990 to 1998. After working at National University of Singapore, Macquarie University and Max Planck Institute for Mathematics, he became a visiting professor at University of New Hampshire. Professor Davydov have been working at Ohio University since 2012, and became a professor there since 2019. Professor Davydov is interested in algebra, representation theory and their connections to mathematical physics, specializing in Hopf algebras, quantum groups, tensor categories and their applications in conformal field theory.

Given a pair of inequivalent complex Hadamard matrices of order 4, we construct a potentially new class of infinite depth subfactors. We also examine `two subfactor theory’ when the complex Hadamard matrices of order n are equivalent. We apply our results to explicitly construct the value of the corresponding relative entropy and angle operators between the spin model subfactors. Joint work with Satyajit Guin and Guruprasad.

References:

https://doi.org/10.1142/S0129167X2450071X (International Journal of Mathematics)

and

https://arxiv.org/abs/2401.01664

Speaker Intro:

Keshab is an Assistant Professor in the Department of Mathematics and Statistics at the Indian Institute of Technology, Kanpur. He earned his Ph.D. from the Institute of Mathematical Sciences (HBNI), under the supervision of Prof. V. S. Sunder. His research focuses on topics in C*-algebras and von Neumann algebras, with a particular emphasis on Jones’ subfactor theory.

This talk is a continuation of the previous lecture. We will go over the following topics: Operadic formulation of the definition (introduction to operads, cat-operad models of e_2 and e_3), examples (pointed categories, Eilenberg-MacLane cohomology), basic constructions (monoidal centre, commutative algebras, local modules) and completely anisotropic categories (symmetric, braided)

Speaker Intro:

Professor Alexei Davydov got his PhD from Moscow University 1992, and worked at Moscow University as assistant professor from 1990 to 1998. After working at National University of Singapore, Macquarie University and Max Planck Institute for Mathematics, he became a visiting professor at University of New Hampshire. Professor Davydov have been working at Ohio University since 2012, and became a professor there since 2019. Professor Davydov is interested in algebra, representation theory and their connections to mathematical physics, specializing in Hopf algebras, quantum groups, tensor categories and their applications in conformal field theory.

Quantum symmetry in simple graphs is a well-explored topic within noncommutative mathematics. It has also gained increasing significance in quantum information theory in recent years. In this talk, we will introduce various notions of quantum symmetry in finite multigraphs in the category of compact quantum groups, naturally extending the well-known constructions done by Banica and Bichon in the context of simple graphs. Towards the end of the talk, we will discuss some examples and applications of our constructions.

References:

arXiv:2302.08726, arXiv:2403.00481 (with Debashish Goswami)

Speaker Intro:

Asfak completed his Ph.D. under the supervision of Professor Debashish Goswami at the Indian Statistical Institute, Kolkata, on December 1, 2023. His research primarily focuses on quantum symmetries of finite spaces and their applications across various areas of mathematics and physics. He is also exploring other areas of noncommutative mathematics as well. Currently, he is a postdoctoral fellow at the Indian Institute of Science Education and Research (IISER), Bhopal. The talk he will be presenting at the BIMSA-Tsinghua Quantum Symmetry Seminar is based on his doctoral research.

This is a first lecture in a series introducing the basic notions and facts needed for the Witt group of non-degenerate braided fusion categories. This introduction will be motivated (and guided) by the language of E_n-operads.

Speaker Intro:

Professor Alexei Davydov got his PhD from Moscow University 1992, and worked at Moscow University as assistant professor from 1990 to 1998. After working at National University of Singapore, Macquarie University and Max Planck Institute for Mathematics, he became a visiting professor at University of New Hampshire. Professor Davydov have been working at Ohio University since 2012, and became a professor there since 2019. Professor Davydov is interested in algebra, representation theory and their connections to mathematical physics, specializing in Hopf algebras, quantum groups, tensor categories and their applications in conformal field theory.

Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan’s property (T).

In this talk, I will focus on two main rigidity results for von Neumann algebras of wreath-like product groups obtained in joint work with Ionut Chifan, Denis Osin and Bin Sun. First, we show that any ICC group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group II1 factor L(G) remembers entirely the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T), confirming Connes’ rigidity conjecture from the early 1980s for these groups.

Second, for a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups. As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an ICC property (T) group G. This gives the first calculations of outer automorphism groups of II1 factors arising from property (T) groups, and can be viewed as a converse of Connes’ 1980 result showing that any such outer automorphism group is countable.

References:

arXiv:2111.04708 (Ann. of Math. 2023)

arXiv:2304.07457

arXiv:2402.19461

Speaker Intro:

Adrian Ioana is a Professor at the University of California, San Diego, acclaimed for his groundbreaking work in functional analysis, operator algebras, and ergodic theory, with a particular focus on von Neumann algebras and group actions. He was an invited speaker at the International Congress of Mathematicians in 2018.

Fusion 2-categories are used to describe the fusion of 2-spacetime-dimensional surface operator, and describe the symmetries in (2+1)d quantum field theories. I will explain how to classify fusion 2-categories using braided fusion categories and group cohomological data. This is the crux of the classification, and what makes it homotopy coherent, is an equivalence between the 3-groupoid of (multi)fusion 2-categories up to monoidal equivalences and a certain 3-groupoid of commuting squares of BZ/2-equivariant space. I will present the data of these such commuting squares, and finally touch on a few applications of the classification.

Reference: arXiv:2411.05907

Speaker Intro:

Matthew Yu is a postdoc at the mathematical institute at the University of Oxford. Before that, he earned his PhD at the University of Waterloo, while working at the Perimeter Institute for theoretical physics. He focuses on studying mathematical aspects of quantum field and string theory, bringing in ideas of homotopy theory and higher (fusion) categories.