Graduate/Postdoc Topology and Singularities Seminar
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 Tommy Wong.
Mondays at 3:20 B139VV
|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)||Sections 2.3 and 2.4: Direct and inverse image and The adjunction triangle|
|Mar. 7 (M)||Section 2.5: Local Systems|
|Mar. 14 (M)||Section 3.1 and 3.2: Cohomological dimensions and the functor f!|
|Mar. 28 (M)||Sections 3.3 and 3.4: Poincare and Alexander duality and Vanishing results|
|Apr.. 4 (M)||Section 4.1: Constructible sheaves|
|Apr. 11 (M)||Section 4.2 and 4.3: Nearby and vanishing cycles and Characteristic varieties and cycles|
|Apr. 18 (M)||Section 5.1 and 5.2: Perverse sheaves and their properties|
|Apr. 25 (M)||Section 5.3: D-modules|
|May. 2 (M)||Section 5.4: Intersection cohomology|
If you would like to present a topic, please contact Tommy Wong.
(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.