
Topology @ TUM group seminar
Workshop February 2025
In February 2025 our group is organizing a Workshop on Gray Products and Lax Constructions in Raitenhaslach.Winter semester 2024/25
In the winter semester 2024/25 we are giving informal internal talks to get to know each other better, given how many new group members we are.Oct 31 | Pelle Steffens | Introduction to derived smooth geometry
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Nov 7 | Walker Stern | Frobenius objects in span categories
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Nov 14 | no seminar due to Simons Collaboration meeting |
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Nov 21 | Jackson van Dyke | Anomalies and projective field theories
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Nov 28 | organization meeting |
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Dec 5 | Anja Švraka | Factorization algerbas and additivity
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Jan 23 |
Yang Kidon Yang | Skeins and factorization homology
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Feb 7 | Will Stewart | Relative field theories
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External guests
Sep 10 | Lorenzo Riva (Notre Dame) | A higher category of Rozansky-Witten modelsOne of the connections between mathematical physics and algebraic topology is that between quantum field theories (QFTs) and functorial topological quantum field theories (TQFTs): given a sufficiently structured family of n-dimensional QFTs (say sigma models) studied using path integral techniques, one hopes to construct a symmetric monoidal n-category C such that the functorial TQFTs associated to the fully dualizable objects of C recover the original QFTs. In this talk we will go over such a construction for the Rozansky-Witten models, a family of QFTs parametrized by holomorphic symplectic manifolds studied by Rozansky and Witten in ’97. In particular we will discuss a general method for producing (infty, n+1)-categories whose first n categorical layers are given by some form of spans in an infty-category and the last layer is given by local systems on those spans with a “push-pull” composition formula resembling the Fourier-Makai transform. This construction can in particular be applied to the 3-dimensional Rozansky-Witten models; doing so we recover computations of Rozansky-Witten for the associated TFTs and results of Brunner-Carqueville-Fragkos-Roggenkamp on the affine fragment of Rozansky-Witten. |
Aug 8 | Lory Aintablian (MPIM Bonn) | Lie theory in tangent categoriesThe infinitesimal counterpart of a Lie group(oid) is its Lie algebra(oid). I will show that the differentiation procedure works in any category with an abstract tangent structure in the sense of Rosick{'y}. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category $\mathcal{C}$ with a scalar $R$-multiplication, where $R \in \mathcal{C}$ is a ring object. This procedure recovers infinite-dimensional Lie theory. This is joint work with Christian Blohmann. |
Aug 8 | Luuk Stehouwer (Dalhousie) | Dagger 2-categories
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July 23 | Quoc Ho (Hong Kong) | HOMFLY-PT homology and the categorical trace of Hecke categoriesMany interesting TQFTs can be realized as sigma models. When coupled with a sheaf theory, sigma models can be used in algebraic geometry to construct TQFTs valued in (higher) categories. With the advent of higher categorical algebra developed by Lurie and others, this perspective has been employed to great success in geometric representation theory, as exemplified, for instance, by the work of Ben-Zvi, Gaitsgory, Nadler, and others in the various forms of the geometric Langlands program. In this talk, I will demonstrate how the study of categorified link invariants can also benefit from this point of view, using the theory of Soergel bimodules as an example. In particular, I will describe the value of the corresponding TQFT on the circle and relate it to the category of character sheaves, yielding, as a consequence, a relation between the HOMFLY-PT link homology theory and coherent sheaves on the Hilbert schemes of points on $\mathbb{C}^2$ as conjectured Gorsky--Negut--Rasmussen. This is joint work with Penghui Li. The talk will be informal, requiring no background in link homology, algebraic geometry, or representation theory. |
July 15 | Julian Holstein | Categorifying Donaldson-Thomas invariantsTo the singularities of a hypersurface one may associate increasingly sophisticated invariants: Milnor numbers, sheaves of vanishing cycles and categories of Matrix factorizations. It is an interesting problem to globalize these locally defined invariants. The Milnor numbers associated to the moduli spaces of suitable sheaves on a Calabi-Yau 3-fold leads to the highly influential Donaldson-Thomas invariants. Brav-Bussi-Dupont-Joyce-Szendroi have globalized the sheaf of vanishing cycles to construct a categorification of Donaldson-Thomas invariants (obstructed by some orientation data). I will take about globalizing higher categorical invariants. This depends on the study of (-1)-shifted symplectic structures on derived schemes, but I will not expect the audience to know what those words mean. This is work in progress, joint with B. Hennion and M. Robalo. |
May 23 | Jonte Gödicke (Hamburg) | Rigid Hall monoidal structuresIn the study of Topological Field Theories in dimension 3, rigid tensor categories play a fundamental role. Commonly studied examples of these arise from finite groups through a linearization construction. This same construction, however, can yield more intriguing examples known as Hall monoidal structures, derived from any 2-Segal space. These structures, for instance, are anticipated to have connections with quantum groups. In this talk I will classify those 2-Segal spaces that induce rigid Hall monoidal structures. For this I will utilize a 2-categorical formulation of rigidity to translate questions about the rigidity of these monoidal structures to questions about homotopy coherent algebra in span categories. |