Exponential networks and the life of partitions

• Johannes Walcher (Heidelberg University)

Room: 2nd Floor, Seminar Room 2L8 Following a quick summary of exponential networks, I will describe in some detail the explicit correspondence, obtained in collaboration with Banerjee, Romo and Senghaas, between torus fixed points of the Hilbert scheme of points in the plane and anomaly-free finite webs attached to the quadratically framed pair of pants.

Some aspects of exponential networks (Online)

• Mauricio Romo (SIMIS/Fudan University)

Room: 2nd Floor, Seminar Room 2L8 I will describe exponential networks and its uses in defining (some version of) Donaldson-Thomas invariants and BPS quivers. If time allows I will present some other interpretation of these invariants as Euler characteristic of certain families of special Lagrangians.

(Partial) abelianization and higher Teichmüller spaces

• Clarence Kineider (MPI Leipzig)

Room: 2nd Floor, Seminar Room 2L8 Higher Teichmüller spaces are connected components in the space of representations from a surface group into a higher rank Lie group. The first examples of these are Hitchin components for split real Lie groups. I will give an overview of the known examples of higher Teichmüller spaces, via the notion of Theta-positivity introduced by Guichard-Wienhard to generalize Fock-Goncharov and Lusztig total positivity. I will then present how spectral networks offer a convenient set of tools to study those higher Teichmüller spaces, in particular Fock-Goncharov-like coordinate systems on them.

Spectral networks as screening contours (Online)

• Andrew Neitzke (Yale University)

Room: 2nd Floor, Seminar Room 2L8 I will describe a role of spectral networks in 2-dimensional conformal field theory: they can be used as "screening contours" in a new construction of Virasoro conformal blocks from branched double covers. The key new point is that, when three exponentiated screening contours end on a branch point of the cover, they cancel an unwanted singularity of the conformal block there. The talk is intended to be self-contained: in particular I will explain what Virasoro conformal blocks are. This is a report of joint work with Qianyu Hao.

Spectral networks and Betti Lagrangians (Online)

• Nick Nho

Room: 2nd Floor, Seminar Room 2L8 In this talk I will discuss the interface of spectral network theory and real contact symplectic topology. I begin with an introduction to weave theory, which allows one to construct exact Lagrangians from totally degenerate spectral networks (otherwise known as N-graphs). I then broaden the classical notion of a meromorphic spectral curve to the wider setting of Betti Lagrangians and formulate a corresponding concept of generic spectral networks. We give lots of explicit examples—BNR spectral curves, ADE singularities, and N-triangle configurations. Finally, I outline how spectral networks admit both pseudoholomorphic and categorical characterisations: the former through Floer theory, the latter through the partially wrapped Fukaya category theory, leading to a concrete understanding of the Gaiotto–Moore–Neitzke non-abelianisation functor. I will close by sketching potential applications.

Thimbles and resummation to networks

• Cambell Wheeler (IHES)

Room: 2nd Floor, Seminar Room 2L8 I will discuss how parametric resurgence recovers many of the basic properties of networks and some examples that arise from q-hypergeometric functions, which play an important role in Chern-Simons theory and open topological strings. I will explain how to construct these networks from some basic building blocks coming from the Faddeev quantum dilogarithm and the geometry of thimbles in an appropriate homology theory. This is based on joint work with Andersen-Fantini-Kontsevich.

Alternating diagrams and non-commutative cluster Lagrangians

• Alexander Goncharov (Yale University)

Room: 2nd Floor, Seminar Room 2L8 Given a bipartite graph G on a possibly punctured surface S, there is a (non-commutative) cluster Poisson variety X(G,S). It depends only on the equivalence class of G under certain elementary transformations. A threefold M which bounds the surface S with filled punctures gives rise to a Lagrangian in the generic symplectic fibers of X(G,S). I will explain that it carries a natural non-commutative cluster symplectic structure. The construction requires a 3d generalization of bipartite surface graphs. The talk reflects joint work with Maxim Kontsevich.

Discussion session Room: 2nd Floor, Seminar Room 2L8