Graduate/Postdoc Topology and Singularities Seminar

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

If you would like to present a topic, please contact Tommy Wong. The seminar meets on Thursdays at 4pm (note the recent change of time) in B139VV.

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) TBA TBA
Nov. 19 (Th) Eva Elduque TBA
Dec. 3 (Th) Eva Elduque TBA
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 and Nov 5: 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.

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


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


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.