Summer Graduate School
|Location:||Center for Mathematics of University of Notre Dame|
Show List of Lecturers
- Mark Johnson (University of Arkansas)
- Linquan Ma (Purdue University)
- Claudia Polini (University of Notre Dame)
- Javid Validashti (DePaul University)
Show List of Speakers
This summer school is held at the University of Notre Dame. Here is where one can find additonal information about the summer school, https://www3.nd.edu/~cmnd/programs/cmnd2019/graduate/index.html.
Linkage is a method for classifying ideals in local rings. Residual intersections is a generalization of linkage to the case where the two `linked' ideals need not have the same codimension. Residual intersections are ubiquitous: they play an important role in the study of blowups, branch and multiple point loci, secant varieties, and Gauss images; they appear naturally in intersection theory; and they have close connections with integral closures of ideals.
Commutative algebraists have long used the Frobenius or p-th power map to study commutative rings containing a finite field. The theory of tight closure and test ideals has widespread applications to the study of symbolic powers and to Briancon-Skoda type theorems for equi-characteristic rings.
Numerical conditions for the integral dependence of ideals and modules have a wealth of applications, not the least of which is in equisingularity theory. There is a long history of generalized criteria for integral dependence of ideals and modules based on variants of the Hilbert-Samuel and the Buchsbaum-Rim multiplicity that still require some remnants of finite length assumptions.
The Rees ring and the special fiber ring of an ideal arise in the process of blowing up a variety along a subvariety. Rees rings and special fiber rings also describe, respectively, the graphs and the images of rational maps between projective spaces. A difficult open problem in commutative algebra, algebraic geometry, elimination theory, and geometric modeling is to determine explicitly the equations defining graphs and images of rational maps.
The school will consist of the following four courses with exercise sessions plus a Macaulay2 workshop
- Linkage and residual intersections
- Characteristic p methods and applications
- Blowup algebras
- Multiplicity theory
- An introduction to Commutative Algebra by Atiyah and MacDonald
- Commutative Algebra with a view towards algebraic geometry by David Eisenbud - Chapters 2-13 and 17-21
For eligibility and how to apply, see the Summer Graduate Schools homepage
Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI.
associated graded rings
strongly F-regular rings
etale fundamental group
13A30 - Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13A35 - Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure [See also 13B22]
13B22 - Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)
13C40 - Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12]
13D40 - Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H10 - Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]
13H15 - Multiplicity theory and related topics [See also 14C17]
Jun 03, 2019
Jun 04, 2019
Jun 05, 2019
Jun 06, 2019
Jun 07, 2019
Jun 10, 2019
Jun 11, 2019
Jun 12, 2019
Jun 13, 2019
Jun 14, 2019