Graduate/Postdoc Topology and Singularities Seminar: Difference between revisions
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'''Constructibility of Log de Rham Complexes for Lattices of Regular Holonomic D-modules'''
'''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
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 , I will talk about some open questions about Riemann-Hilbert Correspondence in the log category as well as for relative D-modules.
===Patricio Almirón Cuadros===
===Patricio Almirón Cuadros===
Revision as of 21:59, 29 October 2020
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: email@example.com (or, if you get an error message, just email firstname.lastname@example.org). 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
Video recordings of all talks can be found at the url: https://uwmadison.box.com/v/SingularitiesElduque
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.
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).
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.
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.
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.
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.
Patricio Almirón Cuadros
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.
The Seminar meets at 10.30 to 11:30 on Fridays in Van Vleck 901.
|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"|
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.
The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.
|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"|
|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"|
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.
|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"|
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.
|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"|
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.
|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|
If you would like to present a topic, please contact Eva Elduque or Christian Geske.
(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.
Thursdays 4pm in B139VV
|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|
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
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
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.
We continue with Professor Alex Suciu's work.
We follow Professor Alex Suciu's work this semester.
But we will not meet at a regular basis.
We meet on Tuesdays 3:30-4:25pm in room B211.
|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|
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.
|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|
|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|
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.
|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|
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.