Abstracts

David Aretz
K-theory for super algebras
While there is a well-known connection between Clifford algebras and topological K-theory, I will describe a version of K-theory for all super algebras that behaves more like algebraic K-theory. Specifically, I will introduce a spectral refinement of Karoubi K-theory, and explain how this construction provides a direct link between algebraic and topological Bott periodicity. (Based on forthcoming work with Luuk Stehouwer.)

Hannes Berkenhagen
Equivariance of the loop spinor bundle under diffeomorphisms
It is an ongoing project to transgress 2-vector bundles to loop spaces. Of particular interest is the spinor bundle on the loop space, which is expected to be the transgression of the stringor bundle. In this talk I want to explain the construction of the spinor bundle on the loop space via Fock spaces and implementers. The main result is a new lift of the action of the group of orientation-preserving diffeomorphisms of the circle on the loop space to the loop spinor bundle.

Janina Bernardy
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Christian Blohmann
The smooth noncommutative geometry of a differentiable stack
The convolution C*-algebra of a Lie groupoid G is often thought of as the noncommutative geometry of the differentiable stack presented by G. However, C*-convolution is not functorial and forgets the smooth structure of G. We have proved that the bornological convolution algebras of Lie groupoids and convolution bimodules of groupoid bibundles define a symmetric monoidal functor from the 2-category of differentiable stacks to the Morita 2-category of complete bornological algebras. I will explain the main result, give a list of examples, and suggest applications to a noncommutative HKR-theorem, the Baum-Connes conjecture, the representation theory of stacky Lie groups (such as the string group), and the geometric quantization of Poisson manifolds. This is joint work with David Aretz.

Severin Bunk
Connections on ∞-bundles
Principal bundles with higher-categorical structure groups are becoming increasingly important in mathematics and its applications, including in the differential geometry of supergravity and higher symmetries in quantum field theory. I will survey a general theory of p-form connections on such bundles, based on derived differential geometry. Along the way, will use Nuiten's family version of the Lurie-Pridham Theorem to define the Atiyah L-infinity-algebroid of a higher principal bundle and introduce the p-th order infinitesimal path groupoid of a manifold. If time permits, I will sketch how our notion of p-form connection reproduces the spaces of p-form connections on higher U(1)-bundles. This is joint work with Lukas Müller, Joost Nuiten, and Richard Szabo.

Pedro Brunialti Lima de Andrade
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Domenico Fiorenza
Drawing ambidexterity: a colorful introduction to Freed-Hopkins-Lurie-Telmann quantization functor
In their seminal paper Topological quantum field theories from compact Lie group, Freed, Hopkins, Lurie, and Telmann describe a "quantization map" from spans of finite homotopy types over a symmetric monoidal category C to the category C itself. For instance, when C is the category Vect of finite dimensional vector spaces over some field k, the quantization map is a symmetric monoidal functor Sum_1: Spans(Vect) --> Vect. Crucial to this construction, and later explicitly emphasized by Hopkins and Lurie, is the ambidexterity of certain pullback functors. We will show how objects of Spans(Vect) are naturally realized as boundary conditions for a fully extended 2d TQFT with defects. From this point of view, the quantization map reduces to an instance of the usual "sandwich construction" in TQFT. Based on Fabio Trova's PhD thesis, on a tentative past project with Ilka Brunner and Nils Carqueville, and on ongoing work in progress with Daniele Migliorati, Alex Takeda, and Thomas Wasserman.

Grigorios Giotopoulos
Cyclified Cohomotopy Flux Quantization of IIA Supergravity via dimensional reduction of Hypothesis H
It is well-known, but remains underappreciated, that higher gauge theories require flux quantization laws for their global completion. A famous proposal for type IIA 10D supergravity is flux quantization in twisted K-theory, whose lift to 11D, however, is problematic. Here we proceed the other way around: After recalling the recently established S^4 flux quantizable superspace formulation of 11D supergravity, we explain how the (appropriately defined) dimensional reduction of its S^1-symmetric backgrounds yields a manifestly cyc(S^4) flux quantizable superspace formulation of IIA supergravity. We note that the latter choice of flux quantization law is close, but not quite the famously hypothesized twisted K-theory. Based on joint work with Hisham Sati and Urs Schreiber.

Kalin Krishna
Bundle up the Arrows: We’re Going Higher
Principal bundles have been an indispensable tool in modern geometry and physics for decades. Yet the emergence of higher structures and higher gauge theory warrants a higher analogue of these bundles. We introduce a notion of higher principal bundles and see how these bundles arise from, and also act as arrows between, higher Lie groupoids. We will show why our notion is a natural generalization to the higher setting. We will also compare some existing notions of higher bundles and morphisms with our framework. As an extension, our framework allows one to create an infinity category of higher Lie groupoids. We will also touch upon how double Lie groupoids and their morphisms, naturally gives rise to higher principal bundles of Lie 2 groupoids. The talk is based on two ongoing joint works: one with C. Blohmann and C. Zhu, and one with D. Alvarez and S. Ronchi.

Gerd Laures
Sheaves of vertex algebras and topological modular forms with level structures
We explore connections between topological modular forms with level N structure and modules over certain vertex algebras. Starting from an almost complex manifold X, we extend its chiral de Rham complex by an even lattice algebra and use this to build a module whose associated graded coincides with the character of the TMF_1(N)-Euler class of X. Along the way, we also discuss how the modular group SL_2(Z) acts in this setting.

Tim Lueders
Towards generalized Higher Dagger Structures
In this talk, I will outline recent progress regarding the theory of higher volutive/dagger structures, with a special emphasis on the role of duality. A particular aspect of interest is the question to which extent higher volutive/dagger structures carry over from rigid settings to frameworks which allow for more infinite-dimensional objects. We will present evidence that suggests the existence of rich generalizations in this direction. Some of the material discussed in the talk is based on arXiv:2504.17764 (joint with Nils Carqueville), arxiv:2602.15667, and work in progress.

Alessandro Nanto
Transgression of 2-vector bundles
In this talk, I'll briefly present the on-going project with Hannes Berkenhagen and Konrad Waldorf to construct a transgression of 2-vector bundles, the desired properties for this construction and it's relation to the established transgression of abelian bundle gerbes.

Ulrich Pennig
Equivariant higher twisted K-theory of SU(n)
Loop groups have a rich representation theory encoded in their Verlinde ring, which is determined by their positive energy representations at a fixed level. A surprising connection links these to topology: twisted equivariant K-theory, a refinement of equivariant topological K-theory that allows for local coefficient systems called twists, can be used to recover this ring by deep results of Freed, Hopkins and Teleman. In this talk I will discuss an operator-algebraic model for equivariant non-classical twists over the groups SU(n) induced by exponential functors on the category of vector spaces. These twists are represented by certain Fell bundles, which can be viewed as operator-algebraic generalisations of bundle gerbes. The C*-algebraic picture allows a full computation of the associated K-groups. I will draw some parallels of our results with the FHT theorem. This is joint work with D. Evans.

Christian Saemann
Adjusted Connections on Higher Principal Bundles
The definition of general connections on higher principal bundles and (higher) principal groupoid bundles is surprisingly subtle. Achieving the level of generality needed for applications in theoretical physics necessitates an additional algebraic datum, known as an adjustment. In the first part, I explain how this adjustment datum arises and how it enters the description of the differential cocycles for higher principal 2-bundles with connection. I then summarise current results concerning the existence and algebraic interpretation of adjustments. In a second part, I show that working in the category of dg-functors yields a natural and uniform description of differential cocycles for principal groupoid bundles with connection. Here, adjustments are Cartan connections on the structure groupoid. Throughout the talk, I illustrate the constructions with concrete examples motivated by applications in mathematical physics, particularly in string and M-theory.

Rhiannon Savage
C-infinity-bornological rings
In this talk, we outline the development of a new model for derived differential geometry using an extension of C-infinity-rings that we call C-infinity-bornological rings. This new theory embeds into the theory of derived bornological geometry recently proposed by Ben-Bassat, Kelly, and Kremnizer. We also discuss how we can use an Artin-Lurie style representability theorem to show that the derived moduli stack of solutions to non-linear elliptic PDEs is representable by a derived C-infinity-bornological affine scheme.

Urs Schreiber
Bulk-Edge Correspondence via Higher Gauge Theory
More profound than bulk topological order of quantum materials is only its unwinding via gapless excitations along boundaries of the sample. We recast this bulk-edge correspondence — for the experimentally relevant case of fractional quantum Hall systems — in terms of higher gauge theory for loop ∞ -groups of spheres. Finally we geometrically engineer this phenomenon by complete topological quantization of M1@M5-brane probes of 11D supergravity globally completed by proper flux quantization in twisted equivariant differential (TED) Cohomotopy. Talk notes are available at ncatlab.org/schreiber/show/BBC.

Carlos Shahbazi
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Pelle Steffens
Six operations in Differential Geometry
Recently, there has been a lot of interest in abstract six-functor formalisms: machines that take as input geometric spaces of some kind and output (higher) categories, usually some version of sheaves. This assignment should be appropriately local, and comes with an intricate system of functorialities and coherences. Sheaves of spectra on locally compact spaces are the go-to example, and general six-functor formalism allow one transfer notions familiar from that context, such as proper maps, manifold bundles and Poincaré duality, to a wide range of settings. Six-functor formalisms have been constructed in algebraic and analytic geometry, leading to powerful categorical invariants. I will discuss a new six-functor formalism in differential geometry, which roughly takes a manifold to the derived oo-category of bornological modules for its sheaf of smooth functions, and give some applications to the smooth representation theory of Lie groups and Lie groupoids, to TQFTs and to geometric PDEs.

Jonas Stelzig
Higher operations in complex geometry
ABC-Massey products are 3-1 operations in the cohomology of complex manifolds defined as a `secondary' variant of ordinary Massey products which depends on the complex structure. By a classic result of Deligne-Griffiths-Morgan-Sullivan, compact Kähler manifolds are formal, implying that all ordinary Massey products vanish. It was shown recently that this is not in general true for the ABC-version. I will explain some context and history of these operations and survey results about when they vanish and when they do not.

Luuk Stehower
Higher Hermitian structures and unitarity in functorial field theory
Unitary functorial field theories require higher-categorical analogues of Hilbert spaces, but it is not established how to define “higher Hilbert space”. I will describe a framework which organizes several distinct notions of "finite-dimensional 2-Hilbert space" that already exist in the literature. The basic idea is that "G-Hermitian pairings" arise as G-fixed points for subgroups G of O(2) acting on the 2-category of 2-vector spaces, and so positivity is imposed by selecting suitable fixed-point data. This recovers, in a uniform way, C*-categories, categories with vector space valued inner product, and Baez 2-Hilbert spaces for specific choices of G. I will end with a proposed inductive definition of n-Hilbert space. This is joint work with Giovanni Ferrer, Lukas Müller, and David Penneys.

Roberto Tellez-Dominguez
T_2-Duality: Using Lie 3-groups to lift T-Duality and S-Duality
Recent work by Nikolaus-Waldorf, Kim-Sämann and Waldorf has led to a description of T-Duality in terms of Lie 2-groups and adjusted connections. Crucially, this approach provides not just a definition of T-Dual backgrounds, but also an explicit description of how to construct such backgrounds that generalizes the Buscher rules to a topologically non-trivial setting. In this talk I will discuss a similar construction, inspired by M-theory, which uses Lie 3-groups and adjusted connections to describe a topology-changing transformation of 3-form potentials which we call T_2-Duality. This duality admits a dimensional reduction to T-Duality of 2-form potentials, and is equivariant under a natural action of the S-Duality group. This is based on arXiv:2505.13368 and ongoing work with Gianni Gagliardo and Christian Sämann.

Bernardo Uribe
Topological Invariants of Magnetic Symmetries
In this talk I will present the definition of the magnetic equivariant K-theory and I will show how the coefficients of this theory are determined. Furthermore, I will show how the group of twisting for this theory could be understood.The talk will start with a motivation from the condensed matter world to address these mathematical issues.

Milena Weiershausen
Morita equivalence of shifted symplectic Lie n-groupoids
Symplectic manifolds are important structures in physics, but the category of smooth manifolds does not contain all colimits, which means that in particular not all quotients can be resolved. Because of this, we have to consider higher constructions like Lie groupoids and eventually Lie n-groupoids. Thus, it is natural to ask whether we can define symplectic structures on these higher objects as well. While some concepts of symplectic Lie groupoids have existed for a while, the general theory of m-shifted symplectic Lie n-groupoids for all n has a nice notion of Morita equivalence. In this talk, I will give an overview of shifted symplectic structures based on the work by Miquel Cueca and Chenchang Zhu https://arxiv.org/abs/2112.01417 and explain how they are preserved under Morita equivalence based on my work https://arxiv.org/abs/2505.24018.

Mayuko Yamashita
Duality in equivariant topological modular forms
I will explain an ongoing project with David Gepner on duality in equivariant TMF and its significance in Segal-Stolz-Teichner program. The mathematical content is regarded as a categorification of Fourier transforms, and also thought of as the elliptic-cohomology version of the T-duality in K-theory. Via Segal-Stolz-Teichner, it is tied to the “dual symmetry” arising from the gauging of categorical symmetries in quantum field theories.

Mahmoud Zeinalian
Toledo-Tong’s Todd Class as a Conjugacy Invariant for Subgroups of Biholomorphisms
I will describe how Toledo and Tong’s construction of a Cech parametrix for the delbar operator on a complex manifold can be adapted to define a conjugacy invariant for any subgroup G of the biholomorphism group of an n-dimensional complex vector space. This invariant takes values in degree n group cohomology of G with coefficients in holomorphic n-forms, and it provides an obstruction to conjugating G into a subgroup of affine transformations. Furthermore, there are two methods for obtaining this invariant. In fact, there is a framework for a Hirzebruch-Riemann-Roch theorem in the setting of infinite analytic prestacks, developed using Toledo and Tong’s techniques, which encompasses the above application. This is joint work with Cheyne Glass and Thomas Tradler.

Chenchang Zhu
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