Skip to Main Content

Advancing Research. Creating Connections.

The American Mathematical Society is dedicated to advancing research and connecting the diverse global mathematical community through publications, meetings and conferences, MathSciNet, professional services, advocacy, and awareness programs.

Learn more and join »

From Notices of the AMS

Figure: A purple cubic surface with three $A_2$ singularities.

Algebraic and Analytic Compactifications of Moduli Spaces

by Patricio Gallardo and Matt Kerr
Communicated by Steven Sam

The Tour Ahead

The basic objects of algebraic geometry, such as subvarieties of a projective space, are defined by polynomial equations. The seemingly innocuous observation that one can vary the coefficients of these equations leads at once to unexpectedly deep questions:

  • When are objects with distinct coefficients equivalent?
  • What types of geometric objects appear if those coefficients move "towards infinity"?
  • Can we make sense of "equivalence classes at infinity"?
Searching for answers leads to the discovery of moduli spaces and their compactifications, parametrizing equivalence classes of said objects.

The construction of compact moduli spaces and the study of their geometry amounts nowadays to a busy and central neighborhood of algebraic geometry. Any vibrant district in an old city, of course, has too many landmarks to visit, and the first job of a tour guide is to curate a selection of sites and routes — including multiple routes to the same site for the different perspectives they afford. Our tour today has three main stops: elliptic curves, Picard curves (together with "points on a line," their alter ego), and a brief panoramic glimpse of the general theory.

Read more »

Also in Notices
Algebraic Theory of Phase Retrieval
What is...a Graphical Design?
 

More from Notices »

 

News from the AMS

More News »

Feature Stories

More Feature Stories »

 

2022

JMM

28 September

September 28, 2022 – 4:00 to 5:00 PM Eastern

Webinar: AMS Travel Grant Funding for JMM

01 October

October 1-2, 2022 

Fall Eastern Sectional Meeting at University of Massachusetts-Amherst

12 October

October 12, 2022 

2022 Fall Grad School Fair - a virtual event

15 October

October 15-16, 2022

Fall Southeastern Sectional Meeting at University of Tennessee at Chattanooga

22 October

October 22-23, 2022

Fall Western Sectional Meeting at University of Utah, Salt Lake City, UT

2023

3 January

January 3, 2023

Workshop for Department Chairs and Leaders

04 January

January 4-7, 2023

2023 Joint Mathematics Meetings (JMM)

With fifteen partner organizations offering paper sessions, panels, workshops, student activities, and invited addresses, each day of JMM 2023 will provide an array of offerings — truly something for everyone.

Member Journals

Notices of the American Mathematical Society Bulletin of the American Mathematical Society

AMS Posters

Lathisms PosterOrder a free Lathisms poster for your classroom to help celebrate National Hispanic Heritage Month Sep 15 - Oct 15. This poster was developed in partnership with Lathisms, to showcase the contributions of Latinx and Hispanic mathematicians during Hispanic Heritage Month.