A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory
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 shown 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]
tba (OS Arithm. Geometry and Representation Theory)
Habiro Cohomology
The Cousin maps and relation with the Sen operator
Venue: Endenicher Allee 60, Room 1.007, Math. Center, University of Bonn
Global formality for embeddings of varieties
I will discuss a joint work in progress with Damien Calaque aimed at globalizing a local formality result by Calaque-Felder-Ferrario-Rossi. Concretely, for an embedding of smooth varieties $X \hookrightarrow Y$, I will explain the construction of a global $L_\infty$-quasi-isomorphism between the polyvector field-valued differential-graded Lie algebra controlling the Kodaira-Spencer deformations of $X$ inside of $Y$ and the Hochschild cochain complex of an $A_\infty$-category describing the $A_\infty$-bimodule deformations of the structure sheaf of $X$ associated with the relative Kodaira-Spencer class. I will report on how this formality map is used to derive certain general isomorphisms between Ext-algebras of embeddings of varieties.
Lectures on The interrelation between Hilbert’s Tenth Problem and arithmetic geometry
Lectures 3 & 4 by Carlo
Thursday 26 June, 10:30-12:00 & 13:00-14:30
These lectures focus on the use of elliptic curves in proving H10/R for any finitely generated ring R; using many existing reductions, one arrives at a problem of controlling Selmer groups of an analogue of the Manin elliptic curve, which is done by new tools using additive combinatorics.
Local to Global results in Floer theory from neck-stretching
Deformations of log-symplectic forms
By a log-symplectic form I mean a meromorphic closed non-degenerate 2-form omega on a complex manifold X that has only simple poles along a divisor D. I will discuss how the de Rham class of omega in H^2(X\D) determines the deformations of the triple (X,D,omega). Some open problems about topology of the divisor deforming D will be discussed.
Double ∞-categories
I will give a leisurely introduction to double ∞-categories, provide a range of examples, and explain why these structures are useful. In particular, I will sketch how they play a key role in an abstract framework that systematically develops different generalizations of ∞-category theories (like equivariant and enriched versions). This was part of my PhD thesis.
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
Examples of knot invariants using Skein relations
Free group actions on symmetric spaces-hyperbolic geometry and Atiyah-Singer index theory
We present two important actions of free groups on hyperbolic 3-space and Hermitian symmetric space via Kleinian group theory
and surface group representations into Hermitian Lie groups. The first one is related to the Thurston's conjecture of generalization of the double limit
theorem to 3-manifolds with compressible boundary, and the second one is related to the Atiyah-Patodi-Singer index theory on flat bundles over surfaces with boundary.
Arithmetic functions at integer polynomials
In joint work with Christopher Frei we use the circle method to estimate averages of arithmetic functions over values of random integer polynomials.
A tour d’horizon through homotopical aspects of C*-algebraic quantum spin systems
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.
Speed talks (only MPIM)
Speed talks at MPIM Bonn by
Lucas Hataishi (University of Oxford)
Enoch Leung (Max-Planck Institute for Mathematics in the Sciences, Leipzig)
Shuhei Oyama (University of Vienna)
William Stewart (University of Texas at Austin)
Matthew Yu (University of Oxford)
Thomas Wasserman (University of Oxford)
tba
Gauging categorical symmetries
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.
Generalized Kitaev Pairings and Higher Berry curvature in coarse geometry
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).
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