Mar 16, 2020
Monday

09:15 AM  09:30 AM


Welcome to MSRI

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09:30 AM  10:30 AM


What is a homotopy coherent SO(3) action on a 3groupoid?
Noah Snyder (Indiana University)

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One consequence of the cobordism hypothesis is that there's a homotopy coherent action of O(n) on the space of fully dualizable objects in a symmetric monoidal ncategory. But what does this mean concretely? For example, by joint work with Douglas and SchommerPries there's an O(3) action on the 3groupoid of fusion categories, but how do you turn this abstract action into a concrete collection of statements about fusion categories? In this talk, I will explain joint work in progress with Douglas and SchommerPries answering this question by saying such an action is given by four explicit pieces of data. If time permits I will also discuss homotopy fixed points for SO(3) actions, and explain the relationship between homotopy fixed points and spherical structures on fusion categories.
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Slides
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10:30 AM  11:00 AM


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11:00 AM  12:00 PM


Cluster quantization from factorization homology
David Jordan (University of Edinburgh)

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Character varieties are moduli spaces of representations of fundamental groups of manifolds. In dimensions 2 and 3 these exhibit symplectic and Lagrangian structures respectively, and it is an important question in 3d and 4d TQFT to quantize these structures functorially. In the 90's and 00's, three quantization schemes were proposed: via socalled AlekseevGrosseSchomerus algebras, via skein theory, and via FockGoncharov cluster quantizations. The AGS and FG constructions were intrinsically algebraic, but lacked the desired topological functoriality, while the skein construction is intrinsically topological but algebraically inaccessible. It is by now understood how to obtain AGS algebras and skein theory in the context of factorization homology, in particular how to unify the two quantizations. In this talk I will explain joint work with Ian Le, Gus Schrader and Sasha Shapiro which constructs FockGoncharov cluster quantizations from factorization homology. An interesting new ingredient is that of parabolic induction domain walls labeling line defects on the surface.
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12:00 PM  02:00 PM


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02:00 PM  03:00 PM


Semisimple 4dimensional topological field theories cannot detect exotic smooth structure
David Reutter (MaxPlanckInstitut für Mathematik)

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A major open problem in quantum topology is the construction of an oriented 4dimensional topological field theory in the sense of AtiyahSegal which is sensitive to exotic smooth structure. In this talk, I will sketch a proof that no semisimple field theory can achieve this goal and that such field theories are only sensitive to the homotopy types of simply connected 4manifolds. This applies to all currently known examples of oriented 4dimensional TFTs valued in the category of vector spaces, including unitary field theories and onceextended field theories which assign algebras or linear categories to 2manifolds. If time permits, I will give a concrete expression for the value of a semisimple TFT on a simply connected 4manifold, explain how the presence of `emergent fermions’ in a field theory is related to its potential sensitivity to more than the homotopy type of a nonsimply connected 4manifold, and comment on implications for the theory of ribbon fusion categories. This is based on arXiv:2001.02288.
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03:00 PM  03:30 PM


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03:30 PM  04:30 PM


Invariants of 4manifolds from Hopf algebras
Xingshan Cui (Purdue University)

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The Kuperberg invariant is a quantum invariant of 3manifolds based on any finite dimensional Hopf algebras, which is also related to the TuraevViroBarrettWestbury invariant. We give a construction of an invariant of 4manifolds that can be viewed as a 4dimensional analog of the Kuperberg invariant. The algebraic data for the construction is what we call a Hopf triple which consists of three Hopf algebras and a bilinear form on each pair of the Hopf algebras satisfying certain compatibility conditions. Quasitriangular Hopf algebras are special examples of Hopf triples. Trisection diagrams of 4manifolds proposed by Gay and Kirby will be used in the construction of the invariant. Some interesting properties of the invariant will also be discussed.
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Mar 17, 2020
Tuesday

09:30 AM  10:30 AM


Gapped condensation in higher categories
Theo JohnsonFreyd (Perimeter Institute of Theoretical Physics)

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Idempotent (aka Karoubi) completion is used throughout mathematics: for instance, it is a common step when building a Fukaya category. I will explain the ncategory generalization of idempotent completion. We call it "condensation completion" because it answers the question of classifying the gapped phases of matter that can be reached from a given one by condensing some of the chemicals in the matter system. From the TFT side, condensation preserves full dualizability. In fact, if one starts with the ncategory consisting purely of \bC in degree n, its condensation completion is equivalent both to the ncategory of ndualizable \bClinear (n1)categories and to an ncategory of lattice condensed matter systems with commuting projector Hamiltonians. This establishes an equivalence between large families of TFTs and of gapped topological phases. Based on joint work with D. Gaiotto.
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10:30 AM  11:00 AM


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11:00 AM  12:00 PM


Tensor 2categories of Hall modules
Mark Penney (University of Waterloo)

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Tensor 2categories are expected to play a fundamental role in 4D quantum topology. The study of tensor 2categories is still in its early stages, especially when compared to the vast literature developed for tensor categories over the last few decades. In particular, the supply of examples is limited.
In this talk I will present an approach to constructing a wide variety of tensor 2categories via categorified Hall algebras. More specifically, 2Segal spaces provide a simplicial framework for unifying the various flavours of Hall algebra constructions. I will describe how to use the theory of 2Segal spaces to construct categorical Hopf algebras, whose modules form tensor 2categories.
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Notes
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Mar 18, 2020
Wednesday

09:30 AM  10:30 AM


Topological Field Theory
Michael Freedman (University of California, San Diego)

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10:30 AM  11:00 AM


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11:00 AM  12:00 PM


Fracton order: relation to and features beyond TQFT
Xie Chen (California Institute of Technology)

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In the recent study of quantum error correction codes, a class of new 3+1D models were discovered which host both features that are similar to the usual topological models and ones that are drastically different. On the one hand, similar to topological models, these socalled ‘fracton’ models have ground state degeneracy that is stable against any local perturbation. There are also fractional point excitations in the bulk, which cannot be created or destroyed on their own. On the other hand, the ground state degeneracy is not a topology dependent constant. Instead it grows with system size. At the same time, the fractional excitations have restricted motion, i.e. they cannot move freely in the full three dimensional space. It is obvious that the fracton models are beyond the framework of TQFT, but are also closely related to it in many important ways. In this talk, I will review some of the most interesting examples of fracton models and explain how we propose to interpret their nontrivial features as depending on not only the topology of the underlying manifold, but also its ‘foliation’ structure, where each foliation leaf carries a 2+1D TQFT of its own. While our understanding of the fracton models are very limited at this point, they are pointing to a way to generalize TQFT in higher dimensions.
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12:00 PM  02:00 PM


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02:00 PM  03:00 PM


COVID19: The Exponential Power of Now
Nicholas Jewell (London School of Hygiene and Tropical Medicine)

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Where are we with COVID19, and how are mathematical models and statistics helping us develop strategies to overcome the burden of infections. Nicholas P. Jewell is Chair of Biostatistics and Epidemiology at the London School of Medicine and Tropical Medicine and Professor of the Graduate School (Biostatistics and Statistics) at the University of California, Berkeley.
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Mar 19, 2020
Thursday

09:30 AM  10:30 AM


Bulk fields in conformal field theory
Christoph Schweigert (Universität Hamburg)

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In this talk, we present new results on bulk fields in conformal field theories and on correlation functions of bulk fields. We show that string net models provide an efficient approach, and we explain a new compact description of bulk fields which is also valid for logarithmic conformal field theories.
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10:30 AM  11:00 AM


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11:00 AM  12:00 PM


Fusion rings for quantum groups and DAHAs
Catharina Stroppel (Rheinische FriedrichWilhelmsUniversität Bonn)

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In this talk I will recall the fusion rings arising from quantum groups at roots of unity and explain their structure via actions of braid groups and Double affine Hecke algebras.
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12:00 PM  02:00 PM


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02:00 PM  03:00 PM


On minimal nondegenerate extensions of braided tensor categories
Dmitri Nikshych (University of New Hampshire)

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This is a report on the joint work of Alexei Davydov and the speaker. Let B be a braided tensor category. A nondegenerate braided category M containing B is called a minimal extension if the centralizer of B in M coincides with the symmetric center of B. We will discuss the existence problem for minimal extensions. When the symmetric center is pointed, this problem can be approached using the braided Picard group of B. We compute the (higher categorical) LanKongWen group of minimal extensions of a symmetric fusion category in this case.
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03:00 PM  03:30 PM


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03:30 PM  04:30 PM


Modules over factorization spaces, and moduli spaces of parabolic Gbundles
Emily Cliff (University of Sydney)

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Factorization spaces (introduced by Beilinson and Drinfeld as "factorization monoids") are nonlinear analogues of factorization algebras. They can be constructed using algebrogeometric methods, and can be linearised to produce examples of factorization algebras, whose properties can be studied using the geometry of the underlying spaces. In this talk, we will recall the definition of a factorization space, and introduce the notion of a module over a factorization space, which is a nonlinear analogue of a module over a factorization algebra. As an example and an application, we will introduce a moduli space of principal Gbundles with parabolic structures, and discuss how it can be linearised to recover modules of the factorization algebra associated to the affine Lie algebra corresponding to a reductive algebraic group G.
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Mar 20, 2020
Friday

09:30 AM  10:30 AM


A survey of factorization algebras in TFTs
Owen Gwilliam (University of Massachusetts Amherst)

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10:30 AM  11:00 AM


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11:00 AM  12:00 PM


The 1dimensional tangle hypothesis
David Ayala (Montana State University)

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Using factorization homology, I'll outline a conceptual construction of link invariants, generalizing the Jones and ReshetkhinTuraev polynomial invariants, using factorization homology. I'll explain how to supply a conceptual proof of the 1dimensional tangle hypothesis, using essentially the same construction. This is joint work with John Francis.
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Notes
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12:00 PM  02:00 PM


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02:00 PM  03:00 PM


Lowdimensional Gbordism and Gmodular TQFTs
Kevin Walker (Microsoft Research Station Q)

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Let G denote a class of manifolds (such as SO (oriented), O (unoriented), Spin, Pin+, Pin, manifolds with spin defects). We define a 2+1dimensional Gmodular TQFT to be one which lives on the boundary of a bordisminvariant 3+1dimensional GTQFT. Correspondingly, we define a Gmodular braided category to be a Gpremodular category which leads to a bordisminvariant 3+1dimensional TQFT. When G = SO, this reproduces the familiar WittenReshetikhinTuraev TQFTs and corresponding modular tensor categories. For other examples of G, nonzero Gbordism groups in dimensions 4 or lower lead to interesting complications (anomalies, mapping class group extensions, obstructions to defining the Gmodular theory on all Gmanifolds).
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