Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on a weighted graph with no loopscalled a splice diagram. In this talk, I will report on joint work with Patrick Popescu-Pampu and Dmitry Stepanov (arXiv: 2108.05912) that sheds new light on these singularities via tropical methods, reproving some of Neumann and Wahl'searlier results on these singularities, and showings that splice type surface singularities are Newton non-degenerate in the sense of Khovanskii.[-]

We compare the topological Milnor fibration and the motivic Milnor fibreby introducing a common extension : the complete Milnor fibration. This extension is constructed using either logarithmic geometry or an oriented (multi)graph construction, for a complex regular function with only normal crossings. The comparison uses quotients by the action of the group of positive real numbers. We study moreover how this model changes under blowings-up. Joint work with J.-B. Campesato and A. Parusinski.[-]

I am going to define toroidal crossing singularities and toroidal crossing varieties and explain how to produce them in large quantities by subdividing lattice polytopes. I will then explain the statement of a global smoothing theorem proved jointly with Felten and Filip. The theorem follows the tradition of well-known theorems by Friedman, Kawamata-Namikawa and Gross-Siebert. In order to apply a variant of the theorem to construct (conjecturally all) projective Fano manifolds with non-empty anticanonical divisor, Corti and Petracci discovered the necessity to allow for particular singular log structures that are known by the inspiring name 'admissible'. I will explain the beautiful classical geometric curve-in-surface geometry that underlies this notion and hint at why we believe that we can feed these singular log structures into the smoothing theorem in order to produce all 98 Fano threefolds with very ample anticanonical class by a single method.[-]

Donaldson-Thomas invariants are numerical invariants associated to Calabi-Yau varieties. They can be obtained by glueing singularity invariants from local models of a suitable moduli space endowed with a (-1)-shifted symplectic structure.
By studying the moduli of such local models, we will explain how to recover Brav-Bussi-Dupont-Joyce-Szendroi's perverse sheaf categorifying the DT-invariants, as well as a strategy for glueing more evolved singularity invariants, such as matrix factorizations.
This is joint work with M. Robalo and J. Holstein.[-]

Moduli spaces of Higgs bundles for Langlands dual groups are conjecturally related by a form of mirror symmetry. For SL_n and PGL_n, Hausel and Thaddeus conjectured a topological mirror symmetry given by an equality of (twisted orbifold) Hodge numbers, which was proven by Groechenig-Wyss-Ziegler and later by Maulik-Shen. We lift this to an isomorphism of Voevodsky motives, and thus in particular an equality of (twisted orbifold) rational Chow groups. Our method is based on Maulik and Shen's approach to the Hausel-Thaddeus conjecture, as well as showing certain motives are abelian, in order to use conservativity of the Betti realisation on abelian motives. The same idea also enables us to prove a motivic chi-independence result. If there is time, I will explain how motivic nearby cycles can be used to specialise these results to positive characteristic. This is joint work with Simon Pepin Lehalleur.[-]

Reidemeister torsion defines an element in the determinant line of a finite CW complex. I will explain its family version which allows one to define a volume form on a mapping stack whose source has a simple homotopy type. One family of examples is given by character stacks of finite CW complexes: for surfaces one recovers the symplectic volume form while for 3-manifolds one obtains orientation data necessary to define cohomological DT invariants. Another family of examples is given by the volume form on the derived loop space related to the Todd class. This is a report on work joint with Florian Naef.[-]

Cluster algebras without coefficients admit (additive) categorifications given by certain 2-Calabi-Yau triangulated categories. Relative Calabi-Yau structures in the sense of Toën and Brav-Dyckerhoff appear when one tries to extend the theory to cluster algebras with coefficients. In the context of cluster algebras associated with marked surfaces, Merlin Christ has constructed the corresponding categorifications. We will report on a slightly different approach starting from an ice quiver with potential. The talk is based on work by Yilin Wu in his thesis and more recent joint work with him.[-]