Titles and Abstracts

Bi-Lipschitz Geometry of Singularities

Let us consider a subspace $X$ of $R^n$ defined by polynomial equations and fix a point $p$ on $X$. To what does $X$ resemble near $p$?

When the implicit function theorem applies at $p$, the answer to the above question is clear!

What happens when $p$ is a singular point? A classical result ensures that $X$ is locally topologically conical: for any sufficiently small radius $\epsilon$, the intersection of $X$ with the ball of radius $\epsilon$ around $p$ is homeomorphic to the cone with vertex $p$ and base the intersection of $X$ with the sphere of radius $\epsilon$. Nevertheless, $X$ is generally not metrically conical: there exist parts of $X$ that contract faster than linearly as $\epsilon$ tends to $0$. A natural problem is then to construct classifications of germs up to local bi-Lipschitz homeomorphism.

I will give an introductory course on this very active subject, at the intersection of metric topology and algebraic geometry. There will be plenty of examples and illustrations.

Singularities and Surgeries

Singularities are concentrated topology. When an algebraic variety develops a singularity, some part of its topology is mercilessly crushed to a point. Sometimes, one can reconstruct what was crushed, which can give access to topologically interesting regions of the smoothed variety which one might not have noticed without the help of the singularity. Alternatively, passing between smoothings and resolutions, or different smoothings of a singularity can give useful surgery operations relating different manifolds. Moreover, allowing the equations of a variety to vary in a loop encircling a singular variety gives rise to monodromy diffeomorphisms of the variety; sometimes these diffeomorphisms are enough to generate the mapping class group, and sometimes the surgery operations allow you to give different descriptions of the same diffeomorphism, leading to relations in the mapping class group.

We will explore these circles of ideas through concrete examples. Specifically (roughly lecture-by-lecture):

• We will derive the chain relations in the mapping class groups of 2-dimensional manifolds and the associated 4-manifold surgery, and use this to construct an exotic 4-manifold.
• We will introduce rational blow-down and use this to construct another exotic 4-manifold (conjecturally diffeomorphic to the previous one).
• We will introduce techniques from almost toric geometry that allow you to draw 2-dimensional pictures to help visualize these surgeries.
• We will study some singularities that have several different smoothings, and look at the "wormhole surgeries" that you get by swapping them.

Lipschitz geometry of germs of complex surfaces

It has been known since the work of Tadeusz Mostowski in 1985 that the set of germs of complex surfaces up to bilipschitz equivalence is countable. Building on the work of Lev Birbrair, Walter Neumann and Anne Pichon on the bilipschitz classification of complex surface germs, we will describe how to explicitly construct an infinite number of germs of complex surfaces germs with isolated singularity that are pairwise homeomorphic (and in fact with same normalization up to isomorphism) but that are not pairwise bilipschitz equivalent.

Pinwheels in Ruled Surfaces and Rational Embedding Problems

$L_{p,q}$ pinwheels are Lagrangian CW-complexes which arise as vanishing cycles of certain quotient singularities. Much like usual smooth Lagrangians, they have standard Weinstein neighborhoods; the neighborhood of an $L_{p,q}$ pinwheel is diffeomorphic to a rational homology ball $B_{p,q}$ that has the lens space $L(p^2,pq-1)$ as a contact boundary. The simple CW-structure of the Lagrangian pinwheels makes them easy to construct via almost toric fibrations, where they appear as visible Lagrangians. I will discuss how to construct and obstruct a certain class of $L_{n,1}$ pinwheels in ruled surfaces, i.e. $S^2\times S^2$ and $ℂP^2\#\overline{ℂ}{P}^2$. The main tool on the obstructive side is the symplectic rational blow-up. I will then sketch some applications, mostly concerning compactifications of rational homology balls and related embedding problems, generalizing the classical case of embeddings of symplectic ellipsoids into symplectic balls. This is joint work with Johannes Hauber.

Classification of T-Singular Surfaces with Small $K^2/p_g$

In this talk, I will discuss recent work on the classification of complex surfaces with only T-singularities, ample dualizing sheaf, and $K^2\le 2p_g-4$. This work provides the first classification of singular surfaces allowing arbitrary values of $p_g$ and an unrestricted number of singularities. It also implies that the challenging Horikawa problem cannot be addressed through complex T-degenerations and proposes new questions regarding diffeomorphism types. This is joint work with Giancarlo Urzúa and Jaime Negrete.

Content

• When Can We Classify? Introduction to Geography of Surfaces: Overview of Gieseker moduli spaces of surfaces of general type $\left[\mathrm{Gie77}\right]$ and their KSBA compactifications $[\mathrm{Ale94},\mathrm{KSB88}]$.
• Why Should We Classify? Introduction to the Horikawa problem $\left[\mathrm{Hor76}\right]$ and an approach inspired by Manetti’s work $\left[\mathrm{Man01}\right]$.
• How Do We Classify? Summary of the main techniques used for classifying both smooth and singular complex surfaces with ample dualizing sheaf $[\mathrm{Hor76},\mathrm{CFPR24},\mathrm{ESU24},\mathrm{RU19},\mathrm{FRU23}]$. Presentation of our methods $\left[\mathrm{MNU24}\right]$ and discussion of the results (properties of the surfaces, connection with our motivation, geography results).

References

• $\left[\mathrm{Ale94}\right]$ V. Alexeev, Boundedness and K² for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810.
• $\left[\mathrm{CFPR24}\right]$ S. Coughlan, M. Franciosi, R. Pardini, and S. Rollenske, 2-Gorenstein stable surfaces with K²_X = 1 and χ(X) = 3, Preprint arXiv:2409.07854 (2024).
• $\left[\mathrm{ESU24}\right]$ J. Evans, A. Simonetti, and G. Urzúa, Tropical methods for stable Horikawa surfaces, Preprint arXiv:2405.02735 (2024).
• $\left[\mathrm{RU19}\right]$ J. Rana and G. Urzúa, Optimal bounds for T-singularities in stable surfaces, Adv. Math. 345 (2019), 814–844.
• $\left[\mathrm{FRU23}\right]$ F. Figueroa, J. Rana, and G. Urzúa, Optimal bounds for many T-singularities in stable surfaces, Preprint arXiv:2308.05624 (2023).
• $\left[\mathrm{Gie77}\right]$ D. Gieseker, Global moduli for surfaces of general type, Invent. Math. 43 (1977), no. 3, 233–282.
• $\left[\mathrm{Hor76}\right]$ E. Horikawa, Algebraic surfaces of general type with small C²₁. I, Ann. of Math. (2) 104 (1976), no. 2, 357–387.
• $\left[\mathrm{KSB88}\right]$ J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338.
• $\left[\mathrm{Man01}\right]$ M. Manetti, On the moduli space of diffeomorphic algebraic surfaces, Invent. Math. 143 (2001), no. 1, 29–76.
• $\left[\mathrm{MNU24}\right]$ V. Monreal, J. Negrete, and G. Urzúa, Classification of Horikawa surfaces with T-singularities, arXiv preprint arXiv:2410.02943 (2024).

Floer theoretic invariants of symplectic structure and singularities

One of the most successful algebraic invariants developed for symplectic structures is the Floer theory. In this talk I will introduce the open string version (Lagrangian Floer homology) and the closed string version (fixed point Floer homology) and explain their relation with singularity theories. Then I will introduce a result by Seidel that relates these two for the simplest type of singularity: Lefschetz fibration with one critical point, and my generalization of it for multiple critical points.