Teaching





 

Vienna (VDS) Summer School (Sep 16-22, 2018)


Summer 2018

NYC College of Technology:

 

 


Spring 2018

NYC College of Technology:

 


Spring 2017


KULeuven:
    GOB63a: Topics in Commutative Algebra
(room E200 01.209)  
Tentative syllabus
Course description (I still may make some changes to it)
Lecture Notes (in progress, so make sure to open to most recent version)
Extra-credit assignment I
Final
    Group I (June 13) is in 200B.00.05
    Group II (June 23) is in 200D.06.32

 CUNY:  I am on sabbatical leave for the entire 2017 calendar year



Fall 2016


NYC College of Technology:

 

 

Summer 2016


NYC College of Technology:

Spring 2016


NYC College of Technology:

 


Fall 2015


NYC College of Technology:

CUNY Graducate Center:

About LaTeX:


 

Summer 2015


NYC College of Technology:

 

 

 

 

 

 

 


Spring 2015


NYC College of Technology:

 

 

Summer 2014


NYC College of Technology:

 

 

 


Spring 2014


NYC College of Technology:

 

 

 

 

 


Fall 2013

NYC College of Technology:
Summer 2013


NYC College of Technology:

 

 

 

 

 


Spring 2013


NYC College of Technology:

 


Fall 2012

NYC College of Technology:
CUNY Graducate Center:

 

 

About LaTeX:

Summer 2012


NYC College of Technology:

 

 


Spring 2012

NYC College of Technology:



CUNY Graducate Center:

 



Fall 2011

NYC College of Technology:


CUNY Graducate Center:

The model theory seminar in the Fall of 2011 will not have a one unified theme. Instead we will continue several topics introduced in previous semesters. We will begin with a four week introduction/tutorial of types in Peano Arithmetic with applications to the classification of elementary pairs. There will also be a connection to the theory of Abstract Elementary Classes.

 

 



Spring 2011


NYC College of Technology:
CUNY Graducate Center:

 

About LaTeX:






Summer 2010 at KULeuven


      Chromatic products in algebra and geometry 

Ultraproducts of rings, and variant constructions, called chromatic products, can be used to prove some deep theorems in (pure) algebra and algebraic geometry. One such application, which I have already lectured about in Leuven a few years back, is tight closure theory in characteristic zero. In this lecture series, I will describe further applications, obtained by similar principles.

Lecture I (Mon. June 21, 14:00) "Ultrarings and uniform bounds"

      (Ia) Definition of ultrarings:  algebraic, sheaf-theoretic, and geometric constructions;
      (Ib) Lefschetz hulls as adjoints of the forgetful functor: flat embeddings of rings inside ultrarings,
      (Ic) Application 1: uniform Artin Approximation
      (Id) Application 2: uniform bounds in algebra for linear properties
      (Ie) (optional) Application 3: tight closure in characteristic zero
      (If) (optional) Application 4: big Cohen-Macaulay algebras in characteristic zero

Lecture II (Wed. June 23, 14:00) "Protoproducts and homological conjectures"

(Ic) (continuation) uniform Artin Approximation
(IIa) Protoproducts as subrings of ultrarings
(IIb) Application 1: uniform bounds for the etale protograde
(IIc) (optional) Ax-Kochen-Ershov principle and transfer in mixed characteristic
(IId) (optional) Application 2: Asymptotic homological conjectures (example: the Monomial Conjecture)

Lecture III "Cataproducts and classification of singularities"

(IIIa) Cataproducts and catapowers: definition and properties
(IIIb) Similarity relation on germs of singularities.
(IIIc) Application 1: Borel classifications and the jet metric.
(IIId) Application 2: ring properties via uniform arithmetic.
(IIIe) (optional) Application 3: Asymptotic homological conjectures (example: the New Improved Intersection Conjecture)

Lecture IV "Ultra-Frobenius and rational singularities"

(IVa) The action of the ultra-Frobenius on ultra-cohomology
(IVb) F-singularities versus "birational" singularities
(IVc) Application 1: quotient singularities are rational
(IVd) Application 2: vanishing theorems a la Kodaira and Kawata-Viehweg

Additional reading material from "The use of ultraproducts in commutative algebra", Hans Schoutens, Lecture Notes in Mathematics 1999, Springer (2010)



NYC College of Technology:

Fall 2010

Fall 2009-Spring 2010 (on leave)

Spring 2009

 
Fall 2008

Spring 2008

Fall 2007


Spring2007

Fall 2006

Spring 2006 (on leave)


Fall 2005



CUNY Graducate Center:

Fall 2010


Fall 2008



Spring 2008

Fall 2007

Fall 2006
Spring 2006 (on leave)

Fall 2005