# Difference between revisions of "Graduate/Postdoc Topology and Singularities Seminar"

## Fall 2021 / Spring 2022

This year the seminar will continue to be on Zoom, hosted by Laurentiu Maxim and Botong Wang. The meeting information is below. We meet on Mondays, at 10am. Seminar announcements will be sent to the "singularities" mailing list. To subscribe, please send an email at: join-singularities@lists.wisc.edu (or, if you get an error message, just email maxim@math.wisc.edu). We plan to have all talks recorded.

The seminar is targeted at junior mathematicians with an interest in the topological study of singularities. We ask the speakers to take this fact into consideration when preparing their talks.

Topic: Topology and Singularities Seminar

Meeting ID: 997 5395 0856 Passcode: singular

Video recordings of lectures are available at: https://uwmadison.box.com/s/i7kcnfd992qxdky2bat053l3qjjaa8s7

date speaker title
Sept 10-11

TIBAR60 Conference

Sept 20 Botong Wang (UW-Madison) Perverse sheaves on varieties with large fundamental group and the Singer-Hopf conjecture
Sept 27 Rodolfo Aguilar Aguilar (Sofia, Bulgaria) Arrangements, fundamental groups and homology planes
October 4 Yongqiang Liu (USTC, China) $L^2$-type invariants and cohomology jump loci for smooth complex quasi-projective varieties
October 25 Matthias Zach (Kaiserslautern) Vanishing cycles on singular spaces with orbit decomposition
Nov 1 Eva Elduque (Universidad Autónoma de Madrid) Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules
Nov 8 Brian Hepler (UW-Madison) Vanishing Cycles for Irregular Local Systems
Nov 15 Xiping Zhang (Tongji University, Shanghai) On Milnor Classes of Determinantal Hypersurfaces
Nov 22 Asvin Gothandaraman (UW-Madison) On topological invariants of the space of irreducible polynomials in many variables
Nov 29 Moisés Herradón Cueto (LSU) Rigidity of difference equations
Dec 6 Jacob Matherne (Bonn) Equivariant log-concavity, hyperplane arrangements, and representation stability
Dec 13 Luca Di Cerbo (U Florida) $L^2$-type invariants and graph-like manifolds
Winter Break
February 7 Alex Suciu (Northeastern) Alexander invariants of algebraic models for spaces
February 21 Yilong Zhang (Ohio State) Monodromy on Vanishing Cycles
March 7 Daniel Bath (KU Leuven) Logarithmic Comparison Theorems for Hyperplane Arrangements, Twisted or Otherwise
March 21 Joana Cirici (Barcelona) Hodge-de Rham numbers of almost complex 4-manifolds
April 4 Jose Rodriguez (Madison) Maximum likelihood degrees --- From statistics to singularities and the computations in between
April 18 Minyoung Jeon (Ohio State) Mather classes of Schubert varieties via small resolutions
May 2 Alex Hof (Madison) Consistency of Milnor Fibers for Deformations of Arbitrary-Dimensional Hypersurface Singularities

## Abstracts

### Botong Wang

The Singer-Hopf conjecture predicts that for a compact aspherical manifold $X$ of real dimension 2$n$, the Euler characteristic of $X$ has sign $(-1)^n$. In a joint work with Yongqiang Liu and Laurentiu Maxim, we made a stronger conjecture in the complex algebraic setting that if a smooth complex projective variety is aspherical, then all perverse sheaves on the variety has nonnegative Euler characteristics. We confirm this conjecture in a special case, when the fundamental group admits a faithful, semisimple, cohomologically rigid representation. Our approach is to construct a complex variation of Hodge structure on the variety and relate the nonnegativity of Euler characteristics with certain curvature conditions on the period domain. Joint work with Donu Arapura.

### Rodolfo Aguilar Aguilar

We will show some examples of homology planes: smooth, affine complex surfaces with trivial reduced integral homology, of log-general type which have an infinite fundamental group. These could be the first examples where such infinitude is shown, they arise as partial compactifications of the complement of an arrangement of lines in the complex projective plane.

### Yongqiang Liu

Let $X$ be a smooth complex quasi-projective variety with a fixed epimorphism $\nu:\pi_1(X)\to \mathbb{Z}$. In this paper, we consider the asymptotic behavior of invariants such as Betti numbers with all possible field coefficients and the order of the torsion subgroup of singular homology associated to $\nu$, known as the $L^2$-type invariants. For degree 1, we give concrete formulas to compute these limits by geometric information of $X$ when $\nu$ is orbifold effective. The proof relies on the theory of cohomology jump loci. We give a detailed study about the degree 1 cohomology jump loci of $X$ with arbitrary algebraically closed field coefficients. As an application, when $X$ is a hyperplane arrangement complement, a combinatorial upper bound is given for the number of parallel positive dimension components in degree 1 cohomology jump loci with complex coefficients. Another application is that we give a positive answer to a question posed by Denham and Suciu: for any prime number $p>1$, there exists a central hyperplane arrangement such that its Milnor fiber has non-trivial $p$-torsion in homology and $p$ does not divide the number of hyperplanes in the arrangement. Joint work with Fengling Li.

### Matthias Zach

The vanishing cycles of a holomorphic function $f$ on a singular space $(X,0) \subset (\mathbb{C}^n,0)$ capture the (reduced) cohomology of the nearby Milnor fiber $M_{f|X}(0)$. Classical results by Milnor, Hamm, Le, and Greuel treat the case of functions with isolated singularity on a smooth space $(X,0)$, but little is known in general when $(X,0)$ is singular. We exhibit techniques to explicitly compute the vanishing cohomology of smooth Milnor fibers for isolated complete intersection singularities $f|(X,0)$, in case $(X,0)$ has a Whitney regular decomposition into orbits of some Lie group action. Time permitting, we will also discuss the implications for singular Milnor fibers and non-isolated singularities.

### Eva Elduque

In previous work jointly with Geske, Herradón Cueto, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this talk, we will talk about the solution to this question, as well as some consequences and explicit computations. Joint work with Moisés Herradón Cueto.

### Brian Hepler

Motivated by the recent results of D'Agnolo-Kashiwara in dimension one, We give a generalization of the notion of vanishing cycles to the setting of enhanced ind-sheaves on an arbitrary complex manifold X. Specifically, we show that there are two distinct (but Verdier-dual) functors that deserve the name of “irregular” vanishing cycles associated an arbitrary holomorphic function f on X. Loosely, these functors capture the two distinct ways in which an irregular local system on the complement of the hypersurface V(f) can be extended across that hypersurface.

From this perspective, we give an (conjectural) interpretation of the enhanced perverse vanishing cycles object in terms of two distinct (but Verdier-dual) notions from the theory of Stokes-filtered local systems with poles along a divisor D: as the sheaf of sections with greater than rapid decay along D, and as the sheaf of sections with moderate growth along D. This is joint work with Andreas Hohl and Claude Sabbah.

### Xiping Zhang

The local Milnor fibration of a complex hypersurfaces is one of the essential objects in singularity theory, and the topological invariants assigned to them (the local Milnor numbers and Lê numbers) are important invariants of study. In this talk we will discuss these invariants on determinantal hypersurfaces, from the angle of their characteristic classes: the Milnor class and the Lê class. We provide an explanation of the Lê class for polarized hypersurfaces, and show that this definition matches the involution formula between Milnor class and Lê class given in [Callejas-Bedregal-Morgado-Seade 2014]. Then we give a new approach to the computation of these local invariants, mainly on building a connection between the local Euler obstructions and the Milnor numbers for these hypersurfaces. This is a joint work with Terence Gaffney.

### Asvin Gothandaraman

In joint work with Andy O'Desky, we study the combinatorics of polynomial factorization and use it to compute some topological invariants of the space of irreducible polynomials in many variables. This involves a generalization of the ring of symmetric functions and the associated combinatorics of partitions. The methods also apply verbatim to other graded monoid spaces such as symmetric quotients of spaces and allow us to give quick proofs of older results in the literature and generalizations to the setting of irreducible polynomials.

The Deligne-Simpson problem simply asks if, given n conjugacy classes of matrices, we can find a matrix in each class such that their product is the identity. This problem quickly turns into a problem about local systems: is there a local system on the n-punctured sphere with given local monodromy around each puncture? When is this local system unique up to isomorphism? These questions have given rise to a lot of interesting mathematics, and one can take "local systems" to mean l-adic sheaves, locally constant sheaves or D-modules, for example. I will talk about this beautiful theory in the D-module setting, and then I will discuss how to come up with such a theory for difference equations instead of differential equations, what parts translate, what parts don't, and what parts we can hope will translate. This is joint work in progress with Daniel Sage.

### Jacob Matherne

In 2012, June Huh proved that the Betti numbers of the complement of a complex hyperplane arrangement form a log-concave sequence. I will talk about a notion called equivariant log-concavity which takes into account the situation when the arrangement has symmetries, so that the cohomology of the complement becomes a representation of the symmetry group. For the braid arrangement (where the complement is the configuration space of a collection of points in the plane and the symmetric group permutes the points), we prove equivariant log-concavity in an infinite number of cases using the theory of representation stability. If time permits, I will mention equivariant log-concavity results and conjectures for related objects coming from hyperplane arrangements/matroids. This is joint work with Dane Miyata, Nick Proudfoot, and Eric Ramos.

### Luca Di Cerbo

In this talk, I will discuss the Singer conjecture for graph-like manifolds in higher dimensions.

### Alex Suciu

Let $A$ be a connected, commutative differential graded algebra over a field $\mathbb{k}$ of characteristic $0$, such that $A^1$ is finite-dimensional. To such an object we associate a module $\mathfrak{B}(A)$ over the symmetric algebra of the $\mathbb{k}$-dual of $A^1$, which we call the Alexander invariant of $A$. We use a modified BGG correspondence to find a presentation for this module, and to connect it to the holonomy Lie algebra and the resonance varieties of $A$. When the cdga $A$ models a space $X$ in an appropriate way, we relate the Alexander invariant of $A$ to that of $X$, generalizing known results in the case when $X$ is $1$-formal. I will illustrate the general theory with some examples arising from the study of complex algebraic varieties and their fundamental groups.

### Yilong Zhang

Let $X$ be a smooth projective variety, and $Y$ be a smooth hyperplane section. As $Y$ moves in the set of all smooth hyperplane sections of $X$, Schnell showed that the monodromy on the vanishing cohomology on $Y$ can recover the middle dimensional cohomology of $X$ in rational coefficient. When $X$ is a smooth hypersurface in $P^4$, we'll improve the result by only considering monodromy on a single vanishing cycle on $Y$. The proof is based on a degeneration argument reducing to the degree 3 case, i.e., when $X$ is a cubic threefold. We'll also study the compactification of the locus of vanishing cycles and explore its relationship to ADE singularities on singular hyperplane sections.

### Daniel Bath

In the 1990s Terao and Yuzvinsky conjectured that central, reduced hyperplane arrangements satisfy the Logarithmic Comparison Theorem. That is, the logarithmic de Rham complex computes the cohomology of the complement with constant coefficients. We prove this conjecture as consequence of a larger result: central, reduced hyperplane arrangements satisfy the Twisted Logarithmic Comparison Theorem, subject to mild restrictions on the weights defining the twist. This asserts that the twisted logarithmic de Rham complex computes the cohomology of the complement with coefficients the rank one local system corresponding to the twist. Our restrictions are "mild" because all such rank one local systems admit weights, and hence a twisted logarithmic de Rham complex, satisfying our restrictions.

### Joana Cirici

I will introduce a Frölicher-type spectral sequence that is valid for all almost complex manifolds, yielding a natural Dolbeault cohomology theory for non-integrable structures. As an application, I will focus on the case of almost complex 4-manifolds, showing how the Frölicher-type spectral sequence gives rise to Hodge-de Rham numbers with very special properties. This is joint work with Scott Wilson.

### Jose Rodriguez

Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Given a model and sample data, the aim is to find a point in the model to "best explain" the data. An important class of models are algebraic and have maximum likelihood estimates realized as a point in an algebraic variety called the likelihood correspondence. In this talk, I will review MLE, introduce computational algebraic methods for describing a zero dimensional variety containing the estimate, and say how algebraic topology can be used to determine this variety's degree. No background in statistics will be assumed and numerous examples will be provided.

### Minyoung Jeon

The Chern-Mather class is a characteristic class, generalizing the Chern class of a tangent bundle of a nonsingular variety to a singular variety. It uses the Nash-blowup for a singular variety instead of the tangent bundle. In this talk, we consider Schubert varieties, known as singular varieties in most cases, in the even orthogonal Grassmannians and discuss the work computing the Chern-Mather class of the Schubert varieties by the use of the small resolution of Sankaran and Vanchinathan with Jones’ technique. We also describe the Kazhdan-Lusztig class of Schubert varieties in Lagrangian Grassmannians, as an analogous result if time permitted.

### Alex Hof

Given a holomorphic family of function germs defining hypersurface singularities, we can ask whether the Milnor fiber varies consistently; in the isolated case, it is well-known that the answer is always yes (in the sense that the family defines a fibration above the complement of the discriminant), and this allows us to obtain a distinguished basis of vanishing cycles for a singularity by perturbing it slightly. In the non-isolated case, this is not always true, and there has long been interest in finding conditions under which this kind of consistency does occur. We give a powerful algebraic condition which is sufficient (and possibly necessary) for this purpose - namely, that the analogous statement will hold so long as the critical locus of the family, considered as an analytic scheme, is flat over the parameter space.

## Fall 2020 / Spring 2021

This year the seminar is on Zoom, hosted by Laurentiu Maxim and Botong Wang. The meeting information is below. We meet on Mondays, at 10am. Seminar announcements will be sent to the "singularities" mailing list. To subscribe, please send an email at: join-singularities@lists.wisc.edu (or, if you get an error message, just email maxim@math.wisc.edu). We plan to have all talks recorded.

The seminar is targeted at junior mathematicians with an interest in the topological study of singularities. We ask the speakers to take this fact into consideration when preparing their talks.

Topic: Topology and Singularities Seminar

Meeting ID: 923 4871 0211

Passcode: 752425

Video recordings of all talks can be found at the url: https://uwmadison.box.com/v/SingularitiesElduque

## Abstracts

### Eva Elduque

Mixed Hodge structures on Alexander modules

Abstract: Let ƒ : U → C be an algebraic map from a smooth complex connected algebraic variety U to the punctured complex line C. Using ƒ to pull back the exponential map C → C, one obtains an infinite cyclic cover Uƒ of the variety U. The homology groups of this infinite cyclic cover, which are endowed with Z-actions by deck transformations, determine the family of Alexander modules associated to the map ƒ. In this talk, we will discuss how to equip the torsion part of the Alexander modules (with respect to the Z-actions) with canonical mixed Hodge structures. Since Uƒ is not an algebraic variety in general, these mixed Hodge structures cannot be obtained from Deligne's theory. The resulting mixed Hodge structures on Alexander modules have some desirable properties. For example, the covering space map Uƒ → U induces morphisms of mixed Hodge structures in homology, where the homology of U is equipped with Deligne's mixed Hodge structure. We will explore several consequences/applications of this fact, regarding weights and semisimplicity. We will also compare the mixed Hodge structures on Alexander modules to other well studied mixed Hodge structures in the literature, including the limit mixed Hodge structure on the generic fiber of ƒ. Joint work with C. Geske, M. Herradón Cueto, L. Maxim, and B. Wang.

### Yongqiang Liu

A question of Bobadilla-Kollár for the abelian variety case

Abstract: In their 2012 paper, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this talk, we answer positively the integral homology version of their question in the case of abelian varieties. This is based on a joint work with Laurentiu Maxim and Botong Wang (arXiv:2006.09295).

### Brian Hepler

Sabbah-Mochizuki-Kedlaya's Hukuhara-Levelt-Turrittin Theorem and Deligne's Stokes Structures

Abstract: In the first of two expository talks, we will discuss the solution(s) to the Riemann-Hilbert correspondence for holonomic D-modules in dimension one, primarily motivating the construction of the target category of objects that represent "solutions to ODEs with irregular singularities." Our first prototype of a solution in the local analytic case, the Stokes-filtered local system, is due to Deligne and is based in the asymptotic theory of differential equations.

Irregular Perverse Sheaves in Dimension One

Abstract: In the second of two expository talks on the irregular Riemann-Hilbert correspondence, we describe Deligne's solution via Stokes-filtered local systems in dimension one, and more generally the Abelian category of such objects as a schematic for irregular perversity. With this intuition, we then describe the much-more general language of enhanced ind-sheaves and the solution of Kashiwara-D'Agnolo in terms of these "simpler" objects. Time permitting, we also describe the equivalent characterization of irregular perversity of Kuwagaki by way of irregularly constructible complexes of sheaves of modules over a finite Novikov ring.

### Sebastián Olano

On the nonnegativity of stringy Hodge numbers

Abstract: Stringy Hodge numbers are a generalization of the usual Hodge numbers of a smooth projective variety. Batyrev introduced them to formulate the topological mirror symmetry test for singular Calabi-Yau varieties. These numbers are defined on a wider class of projective varieties with mild singularities, which are studied in birational geometry. In contrast to the usual Hodge numbers, stringy Hodge numbers are not defined via a cohomology theory. Consequently, Batyrev conjectured that they are nonnegative. This nonnegativity represents a numerical constraint on the exceptional divisor of a resolution of singularities, and thus, it is of intrinsic interest in birational geometry. In this talk, I will present positive results towards Batyrev’s conjecture.

### Jeremy Usatine

Stringy invariants and toric Artin stacks

Abstract: Stringy Hodge numbers are certain generalizations, to the singular setting, of Hodge numbers. Unlike usual Hodge numbers, stringy Hodge numbers are not defined as dimensions of cohomology groups. Nonetheless, an open conjecture of Batyrev's predicts that stringy Hodge numbers are nonnegative. In the special case of varieties with only quotient singularities, Yasuda proved Batyrev's conjecture by showing that the stringy Hodge numbers are given by orbifold cohomology. For more general singularities, a similar cohomological interpretation remains elusive. I will discuss a conjectural framework, proven in the toric case, that relates stringy Hodge numbers to motivic integration for Artin stacks, and I will explain how this framework applies to the search for a cohomological interpretation for stringy Hodge numbers. This talk is based on joint work with Matthew Satriano.

### Lei Wu

Constructibility of Log de Rham Complexes for Lattices of Regular Holonomic D-modules

Abstract: In the classical Hodge theory, the de Rham complex is quasi-isomorphic to the C-constant sheaf on a complex manifold X. Fixing a normal crossing divisor on X, one can construct the logarithmic (log) de Rham complex. Grothendieck comparison says that the log de Rham complex is quasi-isomorphic to the perverse sheaf given by the maximal extension of the constant sheaf on the complement of the divisor. Deligne then extended the comparison to the case for Deligne lattices associated to complex local systems on the complement of the divisor and obtained the so-called Grothendieck-Deligne comparison which leads to the construction of Riemann-Hilbert Correspondence for regular holonomic D-modules. In the log category, one can construct lattices for all regular holonomic D-modules. In this talk, I will discuss the log de Rham complexes for lattices of regular holonomic D-modules and prove their constructibility in general by using relative D-modules. If time allows, I will talk about some open questions about Riemann-Hilbert Correspondence in the log category as well as for relative D-modules.

Bounds for the non-quasihomogeneity degree of a hypersurface singularity in low dimension

Abstract: In his celebrated 1971 paper, K. Saito proved that a hypersurface singularity is quasi-homogeneous if and only if its Milnor, μ, and Tjurina, τ, numbers coincide. After that, one can define the non-quasihomogeneity degree of a hypersurface singularity as μ-τ. In this talk, we will focus on studying optimal bounds for the non-quasihomogeneity degree of the type Cμ, where C<1 is a rational number. Our main motivation to this topic is the following question posed by Dimca and Greuel in 2017: Is it true that for any plane curve singularity μ/τ<4/3?

In this talk I will present a complete answer to this question by using techniques of surface singularities. I will show how these techniques allow us to fit the Dimca and Greuel question as part of the general problem of finding optimal bounds for the non-quasihomogeneity degree of the previous type. As a consequence, we can link the problem of studying optimal bounds for the non-quasihomogeneity with an old standing conjecture posed by Durfee in 1978.

### Mihai Tibăr

On some polar degree conjectures

Abstract: Dolgachev (2000) initiated the study of "Cremona polar transformations", introduced the invariant Pol(V) for projective hypersurfaces V, and classified the homaloidal plane curves, i.e. plane projective curves C with Pol(C)=1. I'll discuss here the proof of Dolgachev's conjecture by Dimca and Papadima (2003), their conjecture on the classification of homaloidal hypersurfaces with isolated singularities proved by Huh (2014), and Huh's conjecture on the classification of hypersurfaces with isolated singularities and Pol(V) =2, proved recently.

Moderately discontinuous algebraic topology

Abstract: I will explain an Algebraic Topology which captures metric information on the degeneration of links to singular points of subanalytic germs as the radius decreases. We have provided the foundations for homology and homotopy, and most of the usual theorems of the topological world have appropriate versions (Seifert-Van Kampen, Hurewicz comparison, relative homology, Mayer-Vietoris, metric homotopy invariance, finite generation,...) I will present the first applications of the theory. (Joint work with S. Heinze, M. Pe Pereira, E. Sampaio.)

### Avi Steiner

Vanishing criteria for tautological systems

Abstract: Tautological systems are vast generalizations of A-hypergeometric systems to the case of an arbitrary reductive algebraic group. Much of the interest in such systems has come from their application to period integrals of Calabi--Yau hypersurfaces. As with A-hypergeometric systems, part of the input data is a parameter $\beta$. I will discuss joint work with P. Görlach, T. Reichelt, C. Sevenheck, and U. Walther discussing criteria which bounds the number of parameters $\beta$ which give a non-trivial tautological system.

### Irma Pallarés Torres

The Brasselet-Schürmann-Yokura conjecture for rational homology manifolds

Abstract: The Brasselet-Schürmann-Yokura conjecture is a conjecture of characteristic classes of singular spaces formulated by J. P. Brasselet, J. Schürmann, and S. Yokura. The conjecture predicts the equality between the Hodge L-class and the Goresky-MacPherson L-class for compact complex algebraic varieties that are rational homology manifolds. In this talk, I will explain two proofs of the conjecture. The first, via classical Hodge theory for projective varieties. This is a joint work with J. Fernández de Bobadilla. The second, using the theory of mixed Hodge modules for general compact algebraic varieties. This is a joint work with J. Fernández de Bobadilla and M. Saito.

### Manuel González Villa

On a quadratic form associated with the nilpotent part of the monodromy of a curve

Abstract: Joint work with Lilia Alanís-López, Enrique Artal Bartolo, Christian Bonatti, Xavier Gómez-Mont, and Pablo Portilla Cuadrado. We study the nilpotent part N of certain pseudo-periodic automorphisms of surfaces appearing in singularity theory. We associate a quadratic form Q defined on the first (relative to the boundary) homology group of the Milnor fiber F of any germ analytic curve on a normal surface. Using the twist formula and techniques from mapping class group theory, we prove that the form Q obtained after killing ker N is definite positive, and that its restriction to the absolute homology group of F is even whenever the Nielsen-Thurston graph of the monodromy automorphism is a tree. The form Q is computable in terms of the Nielsen-Thurston or the dual graph of the semistable reduction, as illustrated with several examples. Numerical invariants associated to Q are able to distinguish plane curve singularities with different topological types but the same spectral pairs or Seifert form. Finally, we discuss a generic linear germ defined on a superisolated surface with not smooth ambient space.

### Feng Hao

Holomorphic 1-forms, smoothness of morphisms to abelian varieties and some linearity properties

Abstract: In this talk I will discuss some results on how the existence of nowhere vanishing holomorphic 1-forms on a smooth complex projective variety X affect the singularities of morphisms from X to (simple) abelian varieties. Also, I will discuss some results on the linearity of the set of holomorphic 1-forms admitting zeros, which is related to the study of singularities of Albanese map. This is a joint work with Yajnaseni Dutta and Yongqiang Liu.

### Alex Suciu

Ab-exact extensions and Milnor fibrations of arrangements

Abstract: We study the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as split extensions with trivial monodromy in first homology with appropriate coefficients. We use these techniques to gain further understanding of the Milnor fibration of the complement of a hyperplane arrangement.

### Rares Rasdeaconu

Moduli spaces of stable, rank 1, torsion free sheaves on real curves

Abstract: The counting of rational curves representing primitive homology classes on complex or real K3 surfaces is governed by the Yau-Zaslow formula and its real analog, respectively. A natural approach to extend such formulae to the non-primitive case requires the computation of the Euler characteristic of moduli spaces of stable, rank one sheaves on curves which are possibly reducible and non-reduced. The recent developments in this direction will be presented (joint work with V. Kharlamov).

Positive factorizations of monodromies on links of isolated complex surface singularities

Abstract: We present a generalization of a classical result concerning smooth germs of surfaces, by showing that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs are a product of right-handed Dehn twists. We explore some consequences of this theorem in the realm of mapping class groups of surfaces. If time permits, we will talk about the higher-dimensional counterpart of the result and how it possibly provides an obstruction to the smoothability of isolated singularities.

Minimal Exponents and a conjecture of Teissier

Abstract: The minimal exponent, defined by M. Saito, is an invariant of hypersurface singularities which is a refinement of the log canonical threshold. One expects singularities to worsen when intersecting with a smooth hypersurface, and Teissier conjectured in the 80's an inequality describing how much the minimal exponent can decrease in this process. I will describe joint work with Mircea Mustaţă (building on work he has done with Eva Elduque) in which we prove this conjecture.

### Daniel Bath

Towards a Logarithmic Comparison Theorem for Quasi-Free Divisors

Abstract: One way to compute the cohomology of the complement of a hypersurface {f=0} is to compute the cohomology of the de Rham complex of meromorphic forms with poles of arbitrary order along f. Sitting inside this complex is the logarithmic de Rham complex, consisting of certain forms with poles of order at most one. It is a long standing question to find necessary and sufficient conditions on the hypersurface ensuring that the natural inclusion of complexes is a quasi-isomorphism, that is, that f satisfies the Logarithmic Comparison Theorem. The modern approach utilizes D-module techniques and the "best" sufficient conditions require assuming that (among other things) the singular locus of f is, morally, as large as possible. To relax this assumption, we introduce a new variant of the Logarithmic Comparison Theorem where: logarithmic derivations are replaced with a submodule closed under Lie brackets; the logarithmic de Rham complex is replaced with a new de Rham complex. I will discuss our proposed strategy for proving this Quasi-Free Logarithmic Comparison Theorem which employs curious Bernstein--Sato polynomial-type constructions. Work in progress; joint with Luis Narvaez-Macarro and Francisco Castro-Jimenez.

### Dominik Wrazidlo

Intersection spaces and Poincaré completion

Abstract: In this survey talk, I will present a two-step method that assigns rational Poincaré duality spaces to certain compact oriented pseudomanifolds by modifying only a neighborhood of the singular set. The first step of the method consists of the intersection space construction, which was invented by Banagl, and was later extended by Agustín and Fernández de Bobadilla to a larger class of pseudomanifolds of arbitrary stratification depth. I will explain work in progress about the study of the rational homotopy type of intersection spaces. The second step of the method is a joint work with T. Essig, in which we extend intersection spaces to rational Poincaré duality spaces in certain cases by constructing a fundamental class. We achieve such a "Poincaré completion" by attaching a finite number of cells to the intersection space, which generalizes work of Klimczak. The focus of this talk lies on a survey of main ideas, examples, and possible future directions of research.

### Jörg Schürmann

(Equivariant) characteristic classes of singular toric varieties

Abstract: In this survey talk, we discuss for singular toric varieties different (equivariant) characteristic classes like Chern, Todd, L and Hirzebruch classes, with applications to (weighted) lattice points counting and Euler-MacLaurin type formulae for lattice polytopes. This is about joint work with Laurentiu Maxim.

### Jose Ignacio Cogolludo

Homological invariants of even Artin kernels

Abstract: Even Artin groups form a special family generalizing right-angle Artin groups that are associated with even-labeled graphs. Some of their properties as groups can be described in terms of their defining graph. In this talk we will discuss different properties associated with the homology of cocyclic subgroups and other finiteness conditions. This is a joint work in progress with C. Martínez-Pérez and R. Blasco-García.

Alexander modules and Mellin transform

Abstract: I will talk about the study of Alexander modules of algebraic varieties using Gabber and Loeser's Mellin transform. The main strength of this approach is that it allows the application of the full machinery of the theory of perverse sheaves, and even mixed Hodge modules. We obtain new results about the structure of Alexander modules, especially about their torsion part and, in the multivariable case, their artinian submodules. It also yields a mixed Hodge structure on the maximal artinian submodules of the Alexander modules. This is based on joint work with Eva Elduque, Christian Geske, Laurentiu Maxim and Botong Wang.

### Owen Barrett

The derived category of the abelian category of constructible sheaves

Abstract: Nori proved in 2002 that given a complex algebraic variety $$X$$, the bounded derived category of the abelian category of constructible sheaves on $$X$$ is equivalent to the usual triangulated category $$D(X)$$ of bounded constructible complexes on $$X$$. He moreover showed that given any constructible sheaf $$\mathcal F$$ on $$\mathbb{A}^n$$, there is an injection $$\mathcal F\hookrightarrow\mathcal G$$ with $$\mathcal G$$ constructible and $$H^i(\mathbb{A}^n,\mathcal G)=0$$ for $$i>0$$.

In this talk, I’ll discuss how to extend Nori’s theorem to the case of a variety over an algebraically closed field of positive characteristic, with Betti constructible sheaves replaced by $$\ell$$-adic sheaves. This is the case $$p=0$$ of the general problem which asks whether the bounded derived category of $$p$$-perverse sheaves is equivalent to $$D(X)$$, resolved affirmatively for the middle perversity by Beilinson.

### Ruijie Yang

Decomposition theorem for semisimple local systems

Abstract: In this talk, I would like to present a new and geometric proof of Sabbah's Decomposition Theorem, which asserts that any semisimple local system remains semisimple after taking the direct image under proper algebraic maps.

Sabbah's proof is D-module-theoretic in nature and relies on his theory of polarizable twistor D-modules, which generalizes Saito's theory of polarizable Hodge modules. Instead, we combine the topological approach developed by De Cataldo-Migliorini and Simpson's theory of mixed twistor structures to give the new proof. Along the way, we obtain new results about the cohomology of semisimple local systems. Joint work with Chuanhao Wei.

## Fall 2018

The Seminar meets at 10.30 to 11:30 on Fridays in Van Vleck 901.

date speaker title
Oct 5 Fenglin Li "Hasse principle and u-invariant"
Oct 26 Fenglin Li "Hasse principle and u-invariant (II)"
Nov 2 José Rodríguez "Maximum likelihood degree"

### Abstracts

#### Nov 2: José Rodríguez

Maximum likelihood degree

In statistics, point estimation uses sample data to calculate the "best estimate" of an unknown population parameter. For example, the sample average can be used to estimate the population mean. While there are many different point estimators, some of the most common ones are the maximum likelihood estimator (MLE), method of moments, and generalized method of moments (GMM).

In algebraic statistics statistical models are studied through the lens of algebra, geometry, and combinatorics. From model selection to inference, this interdisciplinary field has seen applications in a wide range of statistical procedures. In this talk, I will review maximum likelihood estimation and the maximum likelihood degree (ML degree) for discrete models. In particular, I will discuss how the ML degree gives a measure of algebraic complexity of the point estimate for MLE and how we can compute it using tools from topology and geometry. If time permits I will also discuss how we can use maximum likelihood degrees to study singularities.

## Fall 2017

The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.

date speaker title
Oct 4 Eva Elduque "Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (I)"
Oct 11 Eva Elduque "Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (II)"
Oct 18 Sebastian Baader "Dehn twist length in mapping class groups"
Oct 25 Cancelled
Nov 1 Christian Geske "Algebraic Intersection Spaces (I)"
Nov 8 Christian Geske "Algebraic Intersection Spaces (II)"
Nov 15 Laurentiu Maxim "Stratified Morse Theory: an overview (I)"
Nov 22 Thanksgiving break
Nov 29 Laurentiu Maxim "Stratified Morse Theory: an overview (II)"
December 6 Alexandra Kjuchukova "Singular branched covers of four-manifolds and applications"
December 13 TBD "TBA"

## Spring 2017

Fridays at 11:00 VV901

The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.

date speaker title
Jan 27 Christian Geske "Intersection Spaces and Equivariant Moore Approximation I"
Feb 3 Christian Geske "Intersection Spaces and Equivariant Moore Approximation II"
Feb 10 Sashka "The Wirtinger Number of a knot equals its bridge number I"
Feb 17 Sashka "The Wirtinger Number of a knot equals its bridge number II"
Feb 24 Christian Geske "Intersection Spaces and Equivariant Moore Approximation III"
Mar 3 Manuel Gonzalez Villa "Multiplier ideals of irreducible plane curve singularities"

## Fall 2016

Wednesdays at 14:30 VV901

The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.

date speaker title
Sept. 14 (W) Laurentiu Maxim "Alexander-type invariants of hypersurface complements"
Sept. 21 (W) Botong Wang "Cohomology jump loci"
Sept. 28 (W) Alexandra Kjuchukova "On the Bridge Number vs Meridional Rank Conjecture"
Oct 5 (W) Manuel Gonzalez Villa "Introduction to Newton polyhedra"
Oct 12 (W) Manuel Gonzalez Villa "More on Newton polyhedra"
Oct 26 (W) Christian Geske "Intersection Spaces"
Nov 2 (W) Christian Geske "Intersection Spaces Continued"
Nov 9 (W) CANCELLED
Nov 16 (W) Eva Elduque "Braids and the fundamental group of plane curve complements"
Nov 30 (W) Laurentiu Maxim "Novikov homology of hypersurface complements"
Dec 7 (W) CANCELLED
Dec 14 (W) Eva Elduque Specialty Exam: "Twisted Alexander invariants of plane curve complements"

## Spring 2016

Mondays at 3:20 B139VV

The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.

The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.

date speaker title
Feb. 8 (M) Christian Geske Section 1.1 and 1.2: Category of complexes and Homotopical category
Feb. 15 (M) Eva Elduque Sections 1.3 and 1.4: Derived category and derived functors
Feb. 22 (M) Botong Wang Sections 2.1 and 2.2: Generalities on Sheaves and Derived tensor products
Feb. 29 (M) Christian Geske Hypercohomology and Holomorphic Differential Forms on Analytic Varieties
Mar. 7 (M) Eva Elduque Section 2.3: Direct and inverse image
Mar. 14 (M) Cancelled
Mar. 28 (M) Cancelled
Apr. 4 (M) Cancelled
Apr. 11 (M) Christian Geske Section 2.3 cont.
Apr. 18 (M) Cancelled
Apr. 25 (M) Cancelled
May. 2 (M) Cancelled

## Abstracts

(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).

This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).

The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.

Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.

## Fall 2015

Thursdays 4pm in B139VV

date speaker title
Sept. 24 (Th) KaiHo (Tommy) Wong Twisted Alexander Invariant for Knots and Plane Curves
Oct. 1 (Th) Alexandra (Sashka) Kjuchukova Linking numbers and branched covers I
Oct. 8 (Th) Alexandra (Sashka) Kjuchukova Linking numbers and branched covers II
Oct. 15 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture I
Oct. 22 (Th) Yun Su (Suky) Pretalk Higher-order degrees of hypersurface complements., Survey on Alexander polynomial for plane curves.
Oct. 29 (Th) Yun Su (Suky) Aftertalk Higher-order degrees of hypersurface complements.
Nov. 5 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture II
Nov. 12 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture III
Nov. 19 (Th) Eva Elduque Stiefel-Whitney classes
Dec. 3 (Th) Eva Elduque Grass-mania!
Dec. 10 (Th) KaiHo (Tommy) Wong Pretalk Milnor Fiber of Complex Hyperplane Arrangements

## Abstracts

### Th, Sep 24: Tommy

Twisted Alexander Invariant of Knots and Plane Curves.

I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.

### Th, Oct 1 and 8: Sashka

Linking numbers and branched coverings I and II

Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves.

In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.

### Th, Oct 15, Nov 5 and Nov 12: Manuel

On poles of zeta functions and monodromy conjecture I and II

Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.

### Th, Nov 19: Eva

Stiefel-Whitney classes

Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.

### Th, Dec 3: Eva

Grass-mania!

In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.

This talk is for the most part self contained, so it doesn't matter if you missed the previous one.

### Th, Dec 10: Tommy

A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!

I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.

## Spring 2014

We continue with Professor Alex Suciu's work.

## Fall 2014

We follow Professor Alex Suciu's work this semester.

But we will not meet at a regular basis.

## Spring 2014

We meet on Tuesdays 3:30-4:25pm in room B211.

date speaker title
Feb. 25 (Tue) Yongqiang Liu Monodromy Decomposition I
Mar. 4 (Tue) Yongqiang Liu Monodromy Decomposition II
Mar. 25 (Tue) KaiHo Wong Conjecture of lower bounds of Alexander polynomial
Apr. 8 (Tue) Yongqiang Liu Nearby Cycles and Alexander Modules

## Fall 2013

We are learning Hodge Theory this semester and will be following three books:

1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II

2. Peters, Steenbrink, Mixed Hodge Structures

We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.

date speaker title
Sep. 18 (Wed) KaiHo Wong Discussions on book material
Sep. 25 (Wed) Yongqiang Liu Milnor Fibration at infinity of polynomial map
Oct. 9 (Wed) KaiHo Wong Discussions on book material
Oct. 16 (Wed) Yongqiang Liu Polynomial singularities
Nov. 13 (Wed) KaiHo Wong Discussions on book material

## Spring 2013

date speaker title
Feb. 6 (Wed) Jeff Poskin Toric Varieties III
Feb.13 (Wed) Yongqiang Liu Intersection Alexander Module
Feb.20 (Wed) Yun Su (Suky) How do singularities change shape and view of objects?
Feb.27 (Wed) KaiHo Wong Fundamental groups of plane curves complements
Mar.20 (Wed) Jörg Schürmann (University of Münster, Germany) Characteristic classes of singular toric varieties
Apr. 3 (Wed) KaiHo Wong Fundamental groups of plane curves complements II
Apr.10 (Wed) Yongqiang Liu Milnor fiber of local function germ
Apr.17 (Wed) 2:45pm-3:45pm (Note the different time) KaiHo Wong Formula of Alexander polynomials of plane curves

## Abstracts

### Wed, 2/27: Tommy

Fundamental groups of plane curves complements

I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed.

## Fall 2012

date speaker title
Sept. 18 (Tue) KaiHo Wong Organization and Milnor fibration and Milnor Fiber
Sept. 25 (Tue) KaiHo Wong Algebraic links and exotic spheres
Oct. 4 (Thu) Yun Su (Suky) Alexander polynomial of complex algebraic curve (Note the different day but same time and location)
Oct. 11 (Thu) Yongqiang Liu Sheaves and Hypercohomology
Oct. 18 (Thu) Jeff Poskin Toric Varieties II
Nov. 1 (Thu) Yongqiang Liu Mixed Hodge Structure
Nov. 15 (Thu) KaiHo Wong Euler characteristics of hypersurfaces with isolated singularities
Nov. 29 (Thu) Markus Banagl, University of Heidelberg High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres

## Abstracts

### Thu, 10/4: Suky

Alexander polynomial of complex algebraic curve

I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. From the definition, it is clear that Alexander polynomial is an topological invariant for curves. I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. Calculations of some examples will be provided.