Quantum K-theory at roots of unity
2025-04-14 16:30 - 17:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Andrey Smirnov, University of North Carolina at Chapel Hill
Title: Quantum K-theory at roots of unity
Abstract: In this talk, I will discuss a version of quantum K-theory introduced by A.Okounkov, which can be defined through quasimap counts. In this framework, the quantum K-theory ring is obtained as a specialization of the equivariant quasimap count at $q=1$, where $q$ is the equivariant parameter associated with the torus action on the source of the quasimaps. A related, but less explored, structure emerges when $q$ is specialized at the roots of unity. I will outline the key ideas behind this construction and its implications. As an application, I’ll also describe the spectrum of $p$-curvature for the quantum connection, which offers a new proof of a recent result by P.Etingof and A.Varchenko. This talk is based on joint work with P. Koroteev.

Positive mass theorem for ALE(AE) and ALF(AF) Toric 4-Manifolds
2025-04-15 11:00 - 12:00 | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
General Relativity Seminar
Speaker: Aghil Alaee, Clark University
Title: Positive mass theorem for ALE(AE) and ALF(AF) Toric 4-Manifolds
Abstract: One of the fundamental conjectures in mathematical relativity is the positivity of total mass (if it is defined!) for complete non-compact Riemannian manifolds assuming appropriate lower bounds on scalar curvature. This conjecture has been proved for AE manifolds using several techniques, starting with the celebrated results of Schoen-Yau and Witten. There are counter-examples to this conjecture in the AF, ALF, and ALE cases. In this talk, we will refine this conjecture and prove it for toric 4-manifolds. The proof is robust and can be extended to higher dimensions if additional assumptions are added. This is a joint work with Marcus Khuri and Hari Kunduri.

Instantons on Taub-NUT space and Bow Construction
2025-04-15 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Anadil Saeed Rao, Northeastern University
Title: Instantons on Taub-NUT space and Bow Construction
Abstract: In this talk I will discuss Yang-Mills Instantons in a very special geometric setting: the Taub-NUT space, which itself is a Gravitational Instanton. In the first part of the talk I will review the essentials of gravitational instanton theory from the differential geometry/physics perspective and discuss the geometry of Taub-NUT space in detail. Then I will review the essentials of the ADHM-Nahm construction which describe Instantons and Monopoles in Euclidean R^4 and R^3 respectively. In the second part of my talk I will extend the ADHM-Nahm description to Instantons on Taub-NUT space and show how the ADHM-Nahm data may be presented in terms of combinatorial diagrams called Bows and their representations.

CMSA Q&A Seminar: Noah Golowich
2025-04-16 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Noah Golowich (MIT)
Topic: What is length generalization in large language models?

Fifth Annual Yip Lecture | Scott Aaronson (UT Austin): How Much Math Is Knowable?
2025-04-17 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Speaker: Scott Aaronson, Department of Computer Science, University of Texas, Austin
Scott Aaronson is the founding director at the Quantum Information Center at the University of Texas at Austin.
Date: April 17, 2025
Time: 4:00-5:00 pm ET (Reception following in the Math Common Room)
Location: Harvard Science Center Hall A

Title: How Much Math Is Knowable?
Abstract: Theoretical computer science has over the years sought more and more refined answers to the question of which mathematical truths are knowable by finite beings like ourselves, bounded in time and space and subject to physical laws. I’ll tell a story that starts with Gödel’s Incompleteness Theorem and Turing’s discovery of uncomputability. I’ll then introduce the spectacular Busy Beaver function, which grows faster than any computable function. Work by me and Yedidia, along with recent improvements by O’Rear and Riebel, has shown that the value of BB(745) is independent of the axioms of set theory; on the other end, an international collaboration proved last year that BB(5) = 47,176,870. I’ll speculate on whether BB(6) will ever be known, by us or our AI successors. I’ll next discuss the P!=NP conjecture and what it does and doesn’t mean for the limits of machine intelligence. As my own specialty is quantum computing, I’ll summarize what we know about how scalable quantum computers, assuming we get them, will expand the boundary of what’s mathematically knowable. I’ll end by talking about hypothetical models even beyond quantum computers, which might expand the boundary of knowability still further, if one is able (for example) to jump into a black hole, create a closed timelike curve, or project oneself onto the holographic boundary of the universe.

The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

The previous Yip Lecture featured Josh Tenenbaum (MIT) who spoke on How to grow a mind from a brain: From guessing and betting to thinking and talking

Member Seminar
2025-04-18 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Han Shao

Modeling the emergence of complex cortical structure from simple precursors in the brain: maps, hierarchies, and modules
2025-04-21 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Ila Fiete, MIT
Title: Modeling the emergence of complex cortical structure from simple precursors in the brain: maps, hierarchies, and modules
Abstract: Modular and hierarchical structures are ubiquitous in the brain. Two distinct hypotheses for such morphogenesis involve genetic specification (the positional information hypothesis) or spontaneous structure emergence from symmetry breaking (the pattern formation hypothesis). Indeed, there is rich evidence supporting both hypotheses in different systems, and more recently evidence that both systems might interact, for instance with genetic specification providing an initial but relatively low-information scaffold of positional guidance and pattern formation constructing sharper structures by bootstrapping from this guidance. In this talk, I will consider the emergence of two systems in the brain: the visual processing hierarchy with topographic structure, and a modular cognitive circuit consisting of functionally independent grid cell networks that compute spatial location from velocity cues as animals move and navigate the world. I will describe how simple activity-driven growth and competition rules can lead to the emergence of topographically ordered sensory processing hierarchies, and how genetically specified smooth gradients with purely local recurrent interactions on two scales can lead to global module emergence. In sum, simple growth rules, local interactions and smooth gradients can interact to produce rich emergent order on multiple scales in the form of maps, modules, and hierarchies, with predictions that bridge scales from genes to connectivity to function.

Hyperbolic equations in a double null gauge
2025-04-22 11:00 - 12:00 | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
General Relativity Seminar
Speaker: Christopher Stith, University of Michigan
Title: Hyperbolic equations in a double null gauge
Abstract: The hyperbolic nature of the Einstein equations is well-known and has been used in many different contexts. More recently, the double null gauge has proven to be a powerful tool for quantitative analysis of the Einstein equations. It has the advantage of reducing the equations for many dynamical quantities to ODEs along null curves, and the Bianchi equations to a first-order hyperbolic system. The double null gauge has been used extensively and to great effect in analyzing the structure of spacetime for many purposes, including (for instance) stability problems and trapped surface formation. However, the local existence problem for the Einstein equations in a double null gauge has never been treated in full in its own right. In this talk, we discuss how to formulate a general procedure for solving the linearized problem, namely, the local existence theory for systems of first-order hyperbolic equations in a double null gauge.

Classifying Space for Phases of Matrix Product States
2025-04-22 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speakers: Bryan Wang Peng Jun (Harvard) and Bowen Yang (CMSA)
Title: Classifying Space for Phases of Matrix Product States
Abstract: In this talk we will introduce recent work of Beaudry-Hermele-Pflaum-Qi-Spiegel-Stephen (arXiv:2501.14241), on constructing a classifying space for phases of matrix product states (MPS).

CMSA Q&A Seminar: Alexei Borodin
2025-04-23 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Alexei Borodin (MIT)
Topic: Connections between physics and probability

Machine learning for analytic calculations in theoretical physics
2025-04-23 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Matthias Wilhelm (University of Southern Denmark)
Title: Machine learning for analytic calculations in theoretical physics
Abstract: In this talk, we will present recent progress on applying machine-learning techniques to improve calculations in theoretical physics, in which we desire exact and analytic results. One example are so-called integration-by-parts reductions of Feynman integrals, which pose a frequent bottleneck in state-of-the-art calculations in theoretical particle and gravitational-wave physics. These reductions rely on heuristic approaches for selecting a finite set of linear equations to solve, and the quality of the heuristics heavily influences the performance. In this talk, we investigate the use of machine-learning techniques to find improved heuristics. We use funsearch, a genetic programming variant based on code generation by a Large Language Model, in order to explore possible approaches, then use strongly typed genetic programming to zero in on useful solutions. Both approaches manage to re-discover the state-of-the-art heuristics recently incorporated into integration-by-parts solvers, and in one example find a small advance on this state of the art.

Mass gap in AdS space
2025-04-24 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Ziming Ji, Northeastern University
Title: Mass gap in AdS space
Abstract: AdS space can be used as an IR regulator of QFT. The asymptotic conformal boundary in AdS space provides rich, unique observables. We study asymptotic free theories in two-dimensional AdS space. By changing the AdS curvature scale Lambda L, we observe boundary signals of quantum phase transitions where mass gaps are dynamically generated in the bulk. We also utilize supersymmetry to study gauge theories in AdS4. We argue a connection between the AdS partition function and the prepotential and use the F-maximization of the Nekrasov partition function to study supersymmetric boundary conditions and its connection to the Seiberg-Witten theory.

Adversarial KA
2025-04-25 03:00 - 16:00 | Virtual
Freedman CMSA Seminar
Speaker: Slava Dzhenzher, MIPT
Title: Adversarial KA
Abstract: Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or «expressing» functions, we test its robustness by analyzing its ability to withstand adversarial attacks. We find KA to be robust to countable collections of continuous adversaries, but unearth a question about the equi-continuity of the outer functions that, so far, obstructs taking limits and defeating continuous groups of adversaries. This question on the regularity of the outer functions is relevant to the debate over the applicability of KA to the general theory of NNs. Based on https://arxiv.org/abs/2504.05255

Bass-Note Spectra of locally uniform geometries
2025-04-28 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Peter Sarnak, IAS & Princeton University
Title: Bass-Note Spectra of locally uniform geometries
Abstract: We formulate and report on the problem of the Bass-Note Spectrum of an invariant operator as one varies over locally uniform geometries. In the Euclidean setting this recasts classical problems of Mahler from the geometry of numbers in a new light. For certain operators homogeneous dynamics can be used decisively. In the non-Euclidean setting of hyperbolic manifolds we review some recent developments using the conformal bootstrap method and of random covers to study the Bass-Note spectra. We highlight the theme and impact of rigidity.

Algebraic approach to the bow construction
2025-04-29 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speakers: Anadil Saeed Rao (Northeastern) and Vasily Krylov (CMSA/Harvard)
Title: Algebraic approach to the bow construction
Abstract: Anadil will finish his part of the talk. Vasily will then explain how to understand moduli spaces of instantons on R^4, monopoles on R^3, and instantons on Taub-NUT spaces algebraically as so-called bow varieties. We will see how affine type A Nakajima quiver varieties and Slodowy slices arise as special cases of bow varieties. Time permitting, we will also discuss applications to 3D mirror symmetry and Coulomb branches.

From superspace to twisted supergravity
2025-05-01 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Fabian Hahner, University of Washington
Title: From superspace to twisted supergravity
Abstract: In this talk, I will present a geometric perspective on the pure spinor superfield formalism, which proves fruitful for studying twisted supergravity. For eleven-dimensional supergravity, we use this technique to construct the full interacting theory together with all its twists in a uniform and geometric way as homotopy Poisson–Chern–Simons theories. In addition to simplifying the computation of twists immensely, this also provides fresh insights into the supergeometric origin of supergravity. Building on these ideas, we further construct local dg Lie algebras that recover conformal supergravity multiplets and their twists in terms of a geometric moduli problem on superspace.

Incentives for data sharing in federated learning
2025-05-02 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Han Shao, Harvard CMSA
Title: Incentives for data sharing in federated learning
Abstract: Federated learning has recently emerged as a powerful approach for enabling collaboration across large populations of learning agents. However, agents may have incentives to defect from the collaboration—that is, to withdraw or contribute less data than expected—due to the costs of data curation and privacy concerns. This raises several key questions: What happens when agents defect, and how can we prevent such defections?

Quantum Field Theory and Physical Mathematics
2025-05-05 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar

Thinking Outside the Ballot Box
2025-05-05 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Ariel Procaccia, Harvard University
Title: Thinking Outside the Ballot Box
Abstract: How should one design unprecedented democratic processes capable of handling enormous sets of alternatives like all possible policies, bills, or statements? I argue that this challenge can be addressed through a framework called generative social choice, which fuses the rigor of social choice theory with the flexibility and power of large language models. I then explore an application of generative social choice to the problem of identifying a proportionally representative slate of opinion statements. This includes a discussion of desired properties, an algorithm that provably achieves them, an implementation using GPT, and insights from an end-to-end pilot. By providing guarantees, generative social choice could alleviate concerns about AI-driven democratic innovation and help unlock its potential.

Geometry and Quantum Theory Seminar
2025-05-06 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138

Residues and homotopy Lie algebras
2025-05-08 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Zhenping Gui, Shanghai Institute for Mathematics and Interdisciplinary Sciences
Title: Residues and homotopy Lie algebras
Abstract: I will introduce the notion of a chiral operad for any compact Riemann surface. This operad consists of compositions of residue operations, which give rise to the Chevalley-Cousin complex and lead to the definition of chiral homology (derived conformal blocks). I will explain how to use this machinery to rigorously define certain Feynman integrals in 2D chiral CFTs. Subsequently, I will present a polysimplicial construction of a series of chain models for the configuration space of points in an affine space and study residue operations. These residue operations can be described by a homotopy Lie algebra structure, and the latter defines a higher-dimensional analog of the Chevalley-Cousin complex. This is based on joint work in progress with Charles Young and Laura Felder.

Asset pricing with heterogeneous agents
2025-05-09 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Sergiy Verstyuk, Harvard CMSA
Title: Asset pricing with heterogeneous agents
Abstract: This talk will introduce the basics of continuous-time finance, discuss important existing theories and models, as well as present some new asset pricing results in a setting with many heterogeneous investors. (Joint work with Puskar Mondal.)

An index for 2d invertible phases of quantum many-body systems
2025-05-12 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Nikita Sopenko, IAS
Title: An index for 2d invertible phases of quantum many-body systems

Factorizations for data analysis
2025-05-12 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Anna Seigal, Harvard University
Title: Factorizations for data analysis
Abstract: We can find structure in data by factoring it into building blocks, which should be interpretable for the context at hand. A classical example is principal component analysis (PCA), which uses the eigendecomposition of the covariance matrix to find axes of variation in a dataset. Starting from PCA, I will discuss matrix and tensor factorizations for data analysis, and the linear and multilinear algebra that underpins their theoretical properties. We will see examples from causal inference, independent component analysis, and dimensionality reduction.

Report on the Perimeter Institute Theory+AI Workshop
2025-05-14 12:00 - 13:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Conference Reports
Speaker: Hugo Cui, Harvard CMSA
Title: Report on the Perimeter Institute Theory+AI Workshop
Abstract: I will give a survey and brief summary of some of the talks given at the Theory+AI Workshop: Theoretical Physics for AI event organized by Perimeter Institute in April, on approaches to machine learning theory inspired from physics.
Link : https://events.perimeterinstitute.ca/event/993/

Resurgence, number theory, and quantum mirror curves
2025-05-15 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Claudia Rella (IHES)
Title: Resurgence, number theory, and quantum mirror curves
Abstract: Resurgence provides a powerful toolbox to access the non-perturbative sectors hidden within the divergent asymptotic series of quantum theories. Under some special assumptions, the non-perturbative data extracted via resurgent methods possess intrinsic number-theoretic properties that are deeply rooted in the symmetries and arithmetic of the geometry underlying the quantum theory. The framework of modular resurgence aims to formalise this observation. In this talk, after introducing the basics of modular resurgence, I will consider the TS/ST correspondence for toric Calabi-Yau threefolds and focus on the fermionic spectral traces of quantum mirror curves. Here, a complete realisation of the modular resurgence paradigm is found in the spectral theory of local P^2—where the bridge between non-perturbative physics and the arithmetic properties of the geometry takes the form of an exact strong-weak symmetry—and is now being generalised to all local weighted projective spaces. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work in progress.

Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining
2025-05-16 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Samy Jelassi, CMSA
Title: Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining
Abstract: Reinforcement Learning has become a crucial step in training state-of-the-art language models such as DeepSeek-R1 for solving mathematical problems. In this talk, I will first review the mechanisms of Reinforcement Learning fine-tuning. Then, I will present a systematic end-to-end study of RL fine-tuning for mathematical reasoning, training models entirely from scratch on different mixtures of fully open datasets and fine-tuning them with RL. Doing so allows us to investigate the effects of the pretraining data mixture on the behavior of RL, and its interaction with the model size and choices of the algorithm hyperparameters. Our study reveals that RL algorithms consistently converge towards a dominant output distribution, amplifying patterns in the pretraining data. We also find that models of different scales trained on the same data mixture will converge to distinct output distributions, suggesting that there are scale-dependent biases in model generalization.
The second part of the talk is based on a joint work with Rosie Zhao, Alex Meterez, Cengiz Pehlevan, Sham Kakade and Eran Malach: https://arxiv.org/abs/2504.07912

Higher Gauge Theory and Integrability
2025-05-22 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Joaquin Liniado, Instituto de Física La Plata
Title: Higher Gauge Theory and Integrability
Abstract: Integrable field theories are remarkable for possessing an infinite number of conserved quantities, which often allow for their exact solvability. In two dimensions, this structure is elegantly captured by the existence of a Lax connection, whose path ordered exponential allows for the systematic construction of an infinite number of conserved quantities. In 2019, Costello, Witten and Yamazaki introduced a four-dimensional holomorphic extension of Chern-Simons theory that provides the first attempt at explaining the appearance of the Lax connection, whose origin had remained somewhat mysterious until then.
In this talk, we present a generalization of these ideas to three-dimensional field theories, guided by the so-called “categorical ladder = dimensional ladder” principle. The central idea is that conserved quantities arise from surface-ordered exponentials of higher-rank tensors, defining a higher categorical notion of the Lax connection. We show that such a structure naturally emerges from a five-dimensional holomorphic extension of higher Chern-Simons theory. This work, carried out in collaboration with Hank Chen, provides a framework that enables the systematic construction of integrable field theories in three dimensions.

Summer School in Total Positivity and Quantum Field Theory
2025-06-02 09:00 - 2025-06-04 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Summer School in Total Positivity and Quantum Field Theory
Dates: June 2–4, 2025
Location: CMSA, 20 Garden Street, Cambridge MA

In the past decade, there has been a great deal of interest and progress in the study of algebro-combinatorial and geometric structures appearing across diverse areas of physics, from particle physics to cosmology. As these research programs expand, there is an ever-growing need for mathematicians and physicists to collaborate effectively and build a shared language. Join us at Harvard University’s Center of Mathematical Sciences and Applications for a week-long summer school dedicated to addressing these interdisciplinary connections. The school welcomes graduate students, postdocs, and early-career researchers drawn to the intersection of mathematics and physics. Whether you are an algebraic combinatorialist looking for a better grasp on the physics, a high energy theorist trying to figure out the math, or a newcomer to both fields, this summer school offers an ideal opportunity for you to learn.

Courses taught by both mathematicians and physicists will connect ideas from total positivity, matroid theory, discrete geometry, and real algebraic geometry with fundamental questions in quantum field theory. Topics will include amplituhedra, cluster algebras, and positive geometry as they relate to scattering amplitudes and cosmological correlators in high-energy physics. Our courses are designed to be accessible to a varied audience; speakers will be mindful of the diverse backgrounds of the participants from both fields.
Amid this exciting period of collaboration between mathematicians and physicists, we look forward to exploring these rich, cutting-edge topics with you.

Courses:

Positive Grassmannian and Cluster Algebras, Lara Bossinger (Instituto de Matemáticas Universidad Nacional Autónoma de México)
slides | exercises

Positive Geometry and Canonical Forms, Simon Telen (MPI Leipzig)
slides

Scattering Amplitudes and Amplituhedra, Marcus Spradlin (Brown)
exercises

Cosmology and Cosmological Polytopes, Nima Arkani-Hamed (IAS)

Organizers: Jonathan Boretsky (McGill University) | Matteo Parisi (Harvard CMSA and IAS Princeton) | Lauren Williams (Harvard University)

Youtube Playlist
Schedule
Monday, June 2, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras I

10:00–10:30 am
Coffee Break

10:30–11:10 am
Exercises

11:10 am–12:10 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes I

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes II

3:00–3:30 pm
Coffee Break

3:30–4:10 pm
Exercises

4:10–5:10 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes III

Tuesday, June 3, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras II

10:00–10:30 am
Coffee Break

10:30–11:30 am
Marcus Spradlin: Scattering Amplitudes and Amplituhedra I

11:30 am–12:10 pm
Exercises

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Simon Telen: Definitions and first examples of positive geometries

3:00–3:30 pm
Coffee Break

3:30–4:30 pm
Lightning Talks

4:30–5:30 pm
Simon Telen: Positive geometry of polytopes

Wednesday, June 4, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras III

10:00–10:30 am
Coffee Break

10:30–11:10 am
Exercises

11:10 am–12:10 pm
Marcus Spradlin: Scattering Amplitudes and Amplituhedra II

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Marcus Spradlin: Scattering Amplitudes and Amplituhedra III

3:00–3:30 pm
Coffee Break

3:30–4:30 pm
Simon Telen: Positive geometry of polypols

4:30–5:10 pm
Exercises

Image credit: Annabel Ma (Harvard College)

Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras
2025-06-30 09:00 - 2025-07-11 17:00 | Hybrid
Workshop on Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras
Dates: June 30 – July 11, 2025
Location: CMSA, 20 Garden Street, Cambridge MA and Max Planck Institute for Mathematics, Bonn, Germany
This event is a twinned workshop at the CMSA (Harvard) and the Max Planck Institute for Mathematics (Bonn). Lectures will alternate between the two sites, watched simultaneously on both sides, and there will be opportunities for dialogue between the locations. The first week will contain four pedagogical lecture series; lecturers and locations are
Michael Hopkins, Harvard (CMSA)Alexei Kitaev, Caltech (CMSA)Pieter Naaijkens, Cardiff (MPIM)Bruno Nachtergaele, UC Davis (MPIM)
The second week will consist of research talks.
Participants are strongly encouraged to attend at the location that minimizes travel and hence the ecological impact of the conference.
The application deadline was March 16, 2025.
Directions to CMSA
MPIM-Bonn location: https://www.mpim-bonn.mpg.de/qft25

Register for Zoom Webinar

Quantum Field Theory (QFT) and Quantum Statistical Mechanics are central to high energy physics and condensed matter physics; they also raise deep questions in mathematics. The application of operator algebras to these areas of physics is well-known. Recent developments indicate that to understand some aspects QFT properly a further ingredient is needed: homotopy theory and infinity-categories. One such development is the recognition that symmetry in a QFT is better described by a homotopy type rather than a group (so-called generalized symmetries). Another one is the work of Lurie and others on extended Topological Field Theory (TFT) and the Baez-Dolan cobordism hypothesis. Finally, there is a conjecture of Kitaev that invertible phases of matter are classified by homotopy groups of an Omega-spectrum. This workshop will bring together researchers and students approaching this physics using different mathematical techniques: operator algebras, homotopy theory, higher category theory, etc. The goal is to catalyze new interactions between different communities. At the workshop recent developments will be reviewed and hopefully progress can be made on two outstanding problems: the Kitaev conjecture as well as the long-standing goal of finding a proper mathematical formulation for QFT.
Organizers:

Dan Freed, Harvard University CMSA & Math
Dennis Gaitsgory, MPIM Bonn
Owen Gwilliam, UMass Amherst
Anton Kapustin, Caltech
Catherine Meusburger, University of Erlangen-Nürnberg

Talks are recorded and available on the CMSA Youtube Playlist.

BACKGROUND READING
Participants are encouraged to have some basic familiarity with the definition of a C*-algebra and quantum spin system. Some knowledge of quantum channels (completely positive trace-preserving maps) and quantum circuits will be useful. Some knowledge of Clifford algebras will also be helpful.
Possible references include:
1) arXiv:1311.2717 (Sections 2.1, 2.2, 2.4, and 2.5 up to Theorem 2.5.3)
2) Lectures by Daniel Spiegel on “C*-Algebraic Foundations of Quantum Spin Systems”, at the Summer School on C*-Algebraic Quantum Mechanics and Topological Phases of Matter, University of Colorado Boulder, July 29 to August 2, 2024. (lecture notes and video recordings: https://sites.google.com/colorado.edu/caqm).
3) https://nextcloud.tfk.ph.tum.de/etn/wp-content/uploads/2022/09/JvN_lecture_notes_S2016_abcde-1.pdf
4) https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras
5) Karoubi, K-theory, section III.3
6.) Alexei Kitaev: A norm bound for 1D local matrices (pdf)

Schedule Times are Eastern Time
download schedule pdf
Workshop on Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras
June 30 – July 11, 2025

Monday, June 30

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quantum Lattice Systems: observables, dynamics, ground states, GNS representation, ground state gap, examples

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories I
Abstract: This series will cover the relationship between gapped Hamiltonian lattice models and topological quantum field theories, with an emphasis on a conjecture of Kitaev.

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Motivation and introduction of basic setting
Abstract: (week 1 lectures) In this series of lectures, I will give an introduction to the operator-algebraic (Doplicher-Haag-Roberts) approach to study the superselection sectors of a (2D) gapped quantum spin system. The sectors have a rich mathematical structure of a braided monoidal category. This category describes all the algebraic properties of the ‘anyons’ or ‘charges’ such quantum spin systems can have. The aim of these lectures is to build up this theory from first principles, using simple examples of topologically ordered models to illustrate the main ideas. If time permits, I will elaborate on how this fits into the larger programme of the classification of gapped phases of matter, and long-range entangled states in particular. No prior knowledge of operator algebras or tensor categories is assumed.
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states I

Tuesday, July 1

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quasilocality: almost local observables and interactions, Lieb-Robinson bounds, quasi-adiabatic evolution, stability I

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories II

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Building the braided (fusion) category of superselection sectors I
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states II

Wednesday, July 2

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quantum Entanglement in many-body systems: short-range entangled states, topological entanglement, stability II

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories III

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Building the braided (fusion) category of superselection sectors II
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states III

Thursday, July 3

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quantum Phase Diagrams: order parameters, topological invariants, examples

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories IV

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Classification of phases and long-range entanglement
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states IV

No talks Friday July 4

Monday July 7

8:00–9:00 am

MPIM

Jackson van Dyke, TU Munich
Title: Moduli spaces of projective 3d TQFTs
Abstract: A gapped quantum system is well-approximated at low energy by a projective topological field theory. Therefore questions concerning the classification, symmetries, and anomalies of gapped quantum systems can be reinterpreted via the homotopy theory of the moduli space of such theories. I will describe a moduli space of 3-dimensional TQFTs, and the sense in which its homotopy theory informs us about the low energy behavior of gapped systems in 2+1 dimensions. This moduli space depends on the fixed target category: Explicitly, it is built from the classifying spaces of higher groups of automorphisms of ribbon categories. The emphasis will be on target categories which have convenient algebraic features, yet are analytically robust enough to allow for boundary/relative theories defined in terms of unitary representations on topological vector spaces.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Constantin Teleman, UC Berkeley
Title: Quantizing homotopy types
Abstract: Kontsevich (90’s) proposed a topological quantization of (sigma-models into) finite homotopy types to top dimensions (d, d+1). Its enhancement to a `fully extended’ TQFT was described later (Freed, Hopkins, Lurie and the speaker) in the target category of iterated algebras. Independently, Chas and Sullivan constructed a (partially defined) 2-dimensional TQFT (d=1) with target compact oriented manifolds. I will briefly review the features of the finite homotopy theory and its boundary conditions, with particular interest in Dirichlet conditions; their analogue in Chas-Sullivan theory (older work by Blumberg, Cohen and the speaker). Finally, I propose a generalization combining these to a higher-dimensional Chas-Sullivan theory.
Slides (link)

10:30–10:45 am

break

10:45–11:45 am

MPIM

Matthias Ludewig, University of Greifswald
Title: Generalized Kitaev Pairings and Higher Berry curvature in coarse geometry
Abstract: In Appendix C of his “Anyons” paper, Kitaev introduced the notion of a “generalized Chern number” for a 2-dimensional system by diving the system in three ordered parts and measuring a signed rotational flux. This construction has since been used by several authors to measure topological non-triviality of a physical system. In recent work with Guo Chuan Thiang, we observe that the recipe provided by Kitaev can be interpreted in coarse geometry as the pairing of a K-theory class with a coarse cohomology class. A corresponding index theorem then provides a proof that the set of values of this “Kitaev pairing” is always quantized, as already argued by Kitaev. In our work, we generalize Kitaev’s definition and the corresponding quantization result to arbitrary dimensions. By replacing a single Hamiltonian with a whole family of Hamiltonians (parametrized by a space X), we recover and extend the construction of “Higher Berry curvatures” by Kapustin and Spodyneiko. Given a coarse cohomology class, we obtain a characteristic class on the parameter space X, which is integral whenever integrated against a cycle in X that lies in the image of the homological Chern character (so, in particular, spheres in X).

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Theo Johnson-Freyd, Perimeter Institute
Title: Some thoughts about the Kapustin–Kitaev cobordism conjecture
Abstract: In 2013, Kitaev explained that, under some reasonable locality hypotheses, gapped invertible phases of bosonic lattice models in different dimensions are naturally organized into an Omega-spectrum. The following year, Kapustin conjectured that this spectrum is dual to a Thom spectrum, specifically (smooth) oriented bordism MSO, and that for fermionic lattice models one sees instead the dual to spin bordism. In 2016, Freed and Hopkins proved Kapustin’s conjecture for invertible phases of continuous unitary QFTs valued in an at-the-time conjectural universal target category. Freed and Hopkins put bordism categories into the statement of the problem, by working from the beginning with continuous QFTs. Kapustin’s conjecture for lattice models remains open.David Reutter and I, in ongoing work in progress, have investigating Kapustin’s conjecture from the perspective of deeper category theory. We have built the universal target category for phases satisfying a finite semisimplicity hypothesis, and we are working on relaxing finite semisimplicity. We can show that any spectrum of invertible finite-semisimple phases will indeed be dual to a Thom spectrum for some topological group G acting on the spectrum of spheres. For example, if one looks just at those bosonic phases which can be topologically condensed from the vacuum, G is almost the (oriented) piecewise linear group, whose Thom spectrum is the bordism spectrum MSPL is the (oriented) *piecewise* smooth manifolds; the difference between MSPL and MSO is only visible in dimensions 7 and above. I say almost because in fact our G is what you would get if you tried to build MSPL, but could only make finitary measurements, which surely is explained by our restriction to condensable semisimple TQFTs. We conjecture that MSPL, rather than MSO, classifies invertible gapped phases of bosonic lattice models.The general relation between MSPL and topological phases is explained by a certain “surgery exact sequence” for topological phases that mirrors the surgery sequence for MSPL. By studying this sequence, we can also answer the question of which invertible phases admit a gapped boundary condition. In particular that only (the trivial phase and) the Arf–Kervaire invariants admit finite-semisimple gapped boundary conditions.
SLIDES (pdf)

Tuesday, July 8

8:00–9:00 am

MPIM

David Reutter, University of Hamburg
Title: On the categorical spectrum of topological quantum field theories
Abstract: As originally suggested by Kitaev, invertible topological quantum field theories of varying dimensions should assemble into a spectrum/generalized homology theory. A candidate for such a spectrum of invertible TQFTs was proposed by Freed and Hopkins, with the defining property that (isomorphism classes of) n-dimensional invertible TQFTs are completely determined by their partition functions on closed n-manifolds. More generally, not-necessarily-invertible TQFTs should assemble into a ‘categorical spectrum’, an analogue of a spectrum with non-invertible cells at each level. In this talk, I will explain that there exists a unique such categorical spectrum satisfying a list of reasonable assumptions on the collection of (compact/very finite & discrete) TQFTs; one of these assumptions being that its invertibles agree with Freed and Hopkins’ suggestion. I will explain these assumptions, sketch how this categorical spectrum looks like in low-dimensions, outline its construction, and how it may be used to learn about gapped boundaries of anomaly theories in high dimensions. This is based on work in progress with Theo Johnson-Freyd.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Agnes Beaudry, UC Boulder
Title: An algebraic theory of planon-only fracton orders
Abstract: In this talk, I will describe an algebraic theory for planon-only abelian fracton orders. These are three-dimensional gapped phases with the property that fractional excitations are abelian particles restricted to move in parallel planes. The fusion and statistics data can be identified with a finitely generated module over a Laurent polynomial ring together with a U(1)-valued quadratic form. These systems thus lend themselves to an elegant algebraic theory which we expect will lead to easily computable phase invariants and a classification. As a starting point, we establish a necessary condition for physical realizability, the excitation-detector principle, which I will explain. We conjecture that this criterion is also sufficient for realizability. I will also discuss preliminary classification results.This talk is based on joint with Michael Hermele, Wilbur Shirley and Evan Wickenden.

10:30–10:45 am

break

10:45–11:45 am

MPIM

João Faria Martins, University of Leeds
Title: A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory
Abstract: Quinn’s Finite Total Homotopy TQFT is a topological quantum field theory defined for any dimension n of space, depending on the choice of a homotopy finite space B. For instance, B can be the classifying space of a finite group or a finite 2-group.In this talk, I will report on recent joint work with Tim Porter on once-extended versions of Quinn’s Finite Total Homotopy TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, yielding (0,1,2)-, (1,2,3)-, and (2,3,4)-extended TQFTs, given a homotopy finite space B. I will show how to compute these once-extended TQFTs when B is the classifying space of a homotopy 2-type, represented by a crossed module of groups.Reference: Faria Martins J, Porter T: “A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids.” arXiv:2301.02491 [math.CT]

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Emil Prodan, Yeshiva University
Title: Mapping the landscape of frustration-free models
Abstract: Frustration-free models are of great interest because they are amenable to specialized techniques and their understanding is more complete among the general quantum spin models. In this talk, I will establish an almost bijective relation between frustration-free families of projections and a subclass of hereditary subalgebras defined by an intrinsic property. This relation sets further synergies between frustration-free models and open projections in double duals, and subsets of pure states spaces. These connections enable a better understanding of the class of frustration-free models. For example, the open projections in the double dual derived from frustration-free models is dense in the norm-topology in the space of generic open projections, thus assuring us that, for many purposes, we can choose to work with frustration-free models without losing generality. Furthermore, the Cuntz semigroup, originally designed to classify the positive elements of C*-algebra, has been proven to also classify the open projections. Given the mentioned connections, we now have a new device to investigate the ground states of quantum spin models.
SLIDES (pdf)

Wednesday, July 9

8:00–9:00 am

MPIM

Alexander Schenkel, University of Nottingham
Title: C*-categorical prefactorization algebras for superselection sectors and topological order
Abstract: I will present a geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice Z^n. I will show that, under certain assumptions which are implied by Haag duality, the monoidal C*-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of Z^n. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder R^1 x S^{n-1}. While the sphere S^{n-1} arises geometrically as the angular coordinates of cones, the origin of the line R^1 is analytic and rooted in Haag duality. The usual braided (for n=2) or symmetric (for n>2) monoidal C*-categories of superselection sectors are recovered by removing a point of the sphere and using the equivalence between E_n-algebras and locally constant prefactorization algebras defined on open disks in R^n. The non-trivial homotopy groups of spheres induce additional algebraic structures on these E_n-monoidal C*-categories, which in the simplest case of Z^2 is given by a braided monoidal self-equivalence arising geometrically as a kind of ‘holonomy’ around the circle S^1.This talk is based on joint work with Marco Benini, Victor Carmona and Pieter Naaijkens.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Lukasz Fidkowski, University of Washington
Title: Non-invertible bosonic chiral symmetry on the lattice
Abstract: We construct a Hamiltonian lattice realization of the non-invertible chiral symmetry that mimics an axial rotation at a rational angle in a U(1) gauge theory with bosonic charged matter. We provide a heuristic argument that this setup allows a symmetric Hamiltonian which flows, at low energies, to a known field theory with this symmetry.

10:30–10:45 am

break

10:45–11:45 am

MPIM

Nils Carqueville, University of Vienna
Title: Gauging categorical symmetries
Abstract: Orbifold data are categorical symmetries that can be gauged in oriented defect topological quantum field theories. We review the general construction and apply it to 2-group symmetries of 3-dimensional TQFTs; upon further specialisation this leads to equivariantisation of G-crossed braided fusion categories. We also describe a proposal, via higher dagger categories, to gauging categorical symmetries in the context of other tangential structures. This is based on separate projects with Benjamin Haake and Tim Lüders.

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Nikita Sopenko, IAS
Title: Reflection positivity and invertible phases of 2d quantum many-body systems
Abstract: Reflection positivity is a property that is usually taken as an assumption in the classification of topological phases of matter via continuous quantum field theories. For general quantum many-body systems, this property does not hold. This raises the question of whether it somehow emerges in the effective theory from the microscopic description, thereby justifying the field-theoretic approach.In this talk, I will discuss reflection positivity in the context of invertible phases of two-dimensional lattice systems. I will explain why every such phase admits a reflection-positive representative, and why inverse phases are represented by complex conjugate states. I will also introduce an index that distinguishes these phases and is conjecturally related to the chiral central charge.

Thursday, July 10

8:00–9:00 am

MPIM

Ilka Brunner, Ludwig-Maximilians University of Munich
Title: Defects as functors between phases of Abelian gauged linear sigma models
Abstract: Defects act naturally on boundary conditions, providing functors between D-brane categories. In the context of gauged linear sigma models, one can use defects to transport branes from one phase to another. In this talk, I will show how to construct such defects explicitly.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

David Penneys, Ohio State
Title: Holography for bulk-boundary local topological order
Abstract: In previous joint work [arXiv:2307.12552] with C. Jones, Naaijkins and Wallick, we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order.In joint work with C. Jones and Naaijkens, we extend the LTO axioms to quantum spin systems equipped with a topological boundary, again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems.Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC), which arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. Their superselection sectors recover the underlying unitary category of our UBFC, and we conjecture the superselection category also captures the fusion and braiding.

10:30–10:45 am

break

10:45–11:45 am

MPIM

Christoph Schweigert, University of Hamburg
Title: Tensor network states: a topological field theory perspective.
Abstract: Projected entangled pair states (PEPS) and matrix product operators (MPO) are standard tools in quantum information theory and quantum many-body physics. We explain how to understand them in terms of Turaev-Viro models on manifolds with boundary. We then sketch how a recently developed categorical Morita theory for spherical module categories can be used to find generalizations of the standard PEPS tensors.

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Greg Moore, Rutgers
Title: p-form puzzles
Abstract: It is commonly stated that level k BF theory for a p-form (and a form of complementary dimension) is equivalent to a homotopy sigma model with target space K(A,p) where A is a cyclic group of order k. Some aspects of this standard statement are puzzling me. I’ll explain what they are. (Perhaps someone in the audience can resolve my puzzles.) Then I’ll revisit the (again standard) electromagnetic duality of p-form electrodynamics. The conclusion will be that a slightly modified version of Ray-Singer torsion is the partition function of an invertible topological field theory.

Friday, July 11Note: On Friday, there will be separate schedules for Bonn and CMSA.
To view the Bonn schedule, please visit the program page at: https://www.mpim-bonn.mpg.de/qft25

8:00–9:00 am

CMSA

Markus Pflaum, UC Boulder
Title: A tour d’horizon through homotopical aspects of C*-algebraic quantum spin systems
Abstract: In the talk I report on joint work with Beaudry, Hermele, Moreno, Qi and Spiegel, where a homotopy theoretic framework for studying state spaces of quantum lattice spin systems has been introduced using the language of C*-algebraic quantum mechanics. First some old and new results about the state space of the quasi-local algebra of a quantum lattice spin system when endowed with either the natural metric topology or the weak* topology will be presented. Switching to the algebraic topological side, the homotopy groups of the unitary group of a UHF algebra will then be determined and it will be indicated that the pure state space of any UHF algebra in the weak* topology is weakly contractible. In addition, I will show at the example of non-commutative tori that also in the case of a not commutative C*-algebra, the homotopy type of the state space endowed with the weak* topology can be non-trivial and is neither deformation nor Morita invariant. Finally, I indicate how such tools together with methods from higher homotopy theory such as E_infinity spaces may lead to a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter.

9:00–9:30 am

Breakfast break

9:30–11:00 am

CMSA

Speed Talks
Ben Gripaios, University of CambridgeTitle: Locality and smoothness of QFTs
Carolyn Zhang, Harvard UniversityTitle: SymTFT approach for (non-)invertible symmetries of mixed states
Roman Geiko, UCLATitle: Omega-spectrum of stabilizer invertible phases

11:00–11:15 am

break

11:15–12:45 pm

CMSA

Speed Talks continued
Eric Roon, Michigan State UniversityTitle: Finitely Correlated States Driven by Topological Dynamics
Dmitri Pavlov, Texas Tech UniversityTitle: The classification of two-dimensional extended conformal field theories
Bowen Shi, University of Illinois Urbana-ChampaignTitle: Mathematical Puzzles from the Entanglement Bootstrap: On Immersions and regular homotopySLIDES (pdf)
