Geometry, Topology, Group Actions, and Singularities in the Americas

Sergei Gukov: Quivers, Curve Counting, and Fermionic Sums in Topology

Abstract: The representation theory of quantum groups is closely related to the representation theory of vertex algebras. In this relation, generally referred to as the Kazhdan-Lusztig correspondence, fermionic forms provide a natural link between the two sides relating graded dimensions of VOA representations to the braiding data. Fermionic forms were extensively studied in rational VOAs during 80s and early 90s, and then in logarithmic VOAs from mid-90s and early 2000s till present day. More recently, the same structure was independently discovered in a completely different kind of representation theory that has to do with enumerative geometry and curve counting, namely in quiver representations. In this talk, I will review both lines of development and establish a precise link between them. One benefit of building a bridge between quivers and vertex algebras is that it sheds a new light on fermionic forms and provides a simple general method of writing the fermionic forms in much larger families than traditional methods allow.

R. Paul Horja: A Categorical Description of Discriminants

Abstract: I will explain a proposal for the B-side categories appearing in toric homological mirror symmetry along the strata of the generalized discriminant locus due to Aspinwall, Plesser and Wang. A conjectural construction of the web of associated spherical functors and some K-theoretic supporting evidence will be discussed. This is joint work with Ludmil Katzarkov.

Kyoung-Seog Lee: Logarithmic Transformations and Invariants of Elliptic Surfaces

Abstract: Logarithmic transformation is an important analytic operation introduced by Kodaira in the 1960s. One can obtain an elliptic surface with multiple fibers by performing logarithmic transformations on an elliptic surface without multiple fibers. In this talk, I will discuss how certain algebro-geometric invariants of elliptic surfaces are changed under logarithmic transformations.

Anatoly Libgober: Braid Monodromy and Divisibility of Alexander Polynomials of Real Singular Curves

Abstract: The talk reviews the role of Alexander polynomials in algebraic geometry. Also it describes symmetries of the braid monodromy decomposition for a class of plane curves defined over reals including particularly interesting case of the real curves with no real points. As a consequence, talk describes new divisibility relations for Alexander invariants of such curves.

Ernesto Lupercio: Quantum Toric Geometry

Abstract: In this mini-course, I will introduce a generalization of toric geometry called Quantum Toric Geometry. This generalizes toric geometry in the way the quantum torus generalizes the usual torus. Quantum toric varieties admit moduli, and their moduli spaces will be the subject of my second talk. This is joint work with Katzarkov, Meersseman, and Verjovsky.

Alfredo Nájera: Deformation Theory for Finite Cluster Complexes

Abstract: Cluster complexes are a certain class of simplicial complexes that naturally arise in the theory of cluster algebras. They codify a wealth of fundamental information about cluster varieties. The purpose of this talk is to elaborate on a geometric relationship between cluster varieties and cluster complexes. In vague words this relationship is the following: cluster varieties of finite cluster type with universal coefficients can be obtained via a torus action on a certain Hilbert scheme constructed from the cluster complex. Time permitting I will elaborate on how to generalize this approach beyond the cluster setting. This is based on a joint project with Nathan Ilten and Hipolito Treffinger.

Andras Nemethi: Multivariable Poincaré Series of Complex Singularities

Abstract: For any analytic complex isolated singularity we define a multivariable Poincaré series associated with its analytic type. In the plane curve case we relate it with the embedded topological type, in the surface case with a multivariable combinatorial `zeta function' computed from the resolution graph. For both analytic and topological versions we analyse the asymptotic behaviour of their coefficients: in his way we recover the geometric genus of the germ, respectively the Seiberg-Witten invariant of the link.

Alfonso Ruiz: Decomposition of Definable Sets in Algebraically Closed Fields

Bernardo Uribe: Pontrjagin Duality on Multiplicative Gerbes

Abstract: I will talk about a proposal for the first step towards a generalization of the Twisted Drinfeld Double to continuous groups. This is joint work with Jaider Blanco and Konrad Waldorf.

Alberto Verjovsky: Hilbert-Blumenthal and Bianchi Quaternionic Orbifolds

Abstract: In this talk, I will explain recent results that generalize to the Hamilton quaternions and quaternion algebras over number fields of the classical Hilbert-Blumental varieties and their cusps. This is a work in progress with Adrián Zenteno from CIMAT, Guanajuato, Mexico.

Miguel A. Xicoténcatl: On Mapping Class Groups of Non-Orientable Surfaces

Abstract: The Nielsen realization problem asks whether finite subgroups of the mapping class group Mod(Sg) can act on the surface Sg. A complete proof of this fact was given in the '80s in the case of orientable surfaces. In this talk we show that Nielsen realization holds for non-orientable surfaces Ng with marked points, i.e. every finite subgroup of the mapping class group Mod(Ng, k) can be lifted to a subgroup of Diff(Ng, k). In contrast, we show the natural projection from Diff(Ng) to Mod (Ng) does not admit a section for large g. We also study the p-periodicity of Mod(Ng, k) and classify the normalizers of cyclic subgroups of prime order p. As an application we determine the p-primary component of the Farrell cohomology of Mod(Ng, k) in some cases. This is joint work with N. Colin.

Tony Yue Yu: Mirror Structure Constants via Non-Archimedean Analytic Disks

Abstract: For any smooth affine log Calabi-Yau variety U, we construct the structure constants of the mirror algebra to U via counts of non-archimedean analytic disks in the skeleton of the Berkovich analytification of U. This generalizes our previous construction with extra toric assumptions. The technique is based on an analytic modification of the target space as well as the theory of skeletal curves. Consequently, we deduce the positivity and integrality of the mirror structure constants. If time permits, I will discuss further generalizations and virtual fundamental classes. Joint work with S. Keel.