The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a kind of four-dimensional decomposition of the Milnor fiber of the associated singularity. The aim of this course is to explain the structure of a proof of this conjecture, obtained in collaboration with Maria Angelica Cueto and Dmitry Stepanov. lt uses a combination of toric, tropical and logarithmic geometry. [-]

The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a kind of four-dimensional decomposition of the Milnor fiber of the associated singularity. The aim of this course is to explain the structure of a proof of this conjecture, obtained in collaboration with Maria Angelica Cueto and Dmitry Stepanov. lt uses a combination of toric, tropical and logarithmic geometry. [-]

The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a kind of four-dimensional decomposition of the Milnor fiber of the associated singularity. The aim of this course is to explain the structure of a proof of this conjecture, obtained in collaboration with Maria Angelica Cueto and Dmitry Stepanov. lt uses a combination of toric, tropical and logarithmic geometry. [-]

The splice type singularities introduced in 2001 by Neumann and Wahl provide the largest class known so far of links of isolated complete intersection surface singularities which are integral homology spheres. These singularities are determined up to equisingularity by particular kinds of decorated trees, called splice diagrams. Neumann and Wahl formulated the so-called Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a kind of four-dimensional decomposition of the Milnor fiber of the associated singularity. The aim of this course is to explain the structure of a proof of this conjecture, obtained in collaboration with Maria Angelica Cueto and Dmitry Stepanov. lt uses a combination of toric, tropical and logarithmic geometry. [-]

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.[-]

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.[-]