Home /  Graphical Models in Algebraic Combinatorics (SLMath)

Summer Graduate School

Graphical Models in Algebraic Combinatorics (SLMath) June 23, 2025 - July 03, 2025
Parent Program: --
Location: SLMath: Eisenbud Auditorium, Atrium
Organizers Christian Gaetz (University of California, Berkeley), David Keating (University of Illinois at Urbana-Champaign), Melissa Sherman-Bennett (University of California, Davis), LEAD Anna Weigandt (University of Minnesota)
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Description
1121 image
<p>A plane partition and an hourglass plabic graph</p>

This school will introduce students to a range of powerful combinatorial tools used to understand algebraic objects ranging from the homogeneous coordinate ring of the Grassmannian to symmetric functions.   The summer school will center around two main lecture series "Webs and Plabic Graphs" and "Vertex Models and Applications".   While the exact applications differ, both courses will center on graphical models for algebraic problems closely related to Grassmannian and its generalizations.  This school will be accessible to a wide range of students.  Students will leave the school with a solid grasp of the combinatorics of webs, plabic graphs, and the six-vertex model, an understanding of their algebraic applications, and a taste of current research directions.

School Structure

There will be two lectures each day, as well as two problem sessions. 

Prerequisites

(1) Abstract Algebra by Dummit and Foote, 3rd edition

(a) Chapter 1, Introduction to Groups 
(b) Section 4.1, Group Actions and Permutation Representations 
(c) Sections 7.1-7.4, Introduction to Rings 
(d) Chapter 11, Vector Spaces 

(2) Section 2 of Gillespie’s “Variations on a theme of Schubert calculus” 

We also recommend that students read the following papers, to gain exposure to the main topics of the summer school before the lectures start. Large portions of these papers will be mentioned during the lectures (without proof), so they are not strictly prerequisite material. 

Suggested Reading: 
(1) “How the Alternating Sign Matrix Conjecture was Solved,” by David Bressoud and James Propp
(2) Claudio Procesi, “Lie Groups”, p. 241-246
(3) Expository paper on the Grassmannian and its homogeneous coordinate ring by Drew A. Hudec

Recommended Preparatory Reading

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • Grassmannian

  • plabic graphs

  • cluster algebras

  • webs

  • vertex models

  • Schubert calculus

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Schedule, Notes/Handouts & Videos
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Jun 23, 2025
Monday
09:00 AM - 09:15 AM
  Welcome to SLMath
09:15 AM - 10:30 AM
  Lecture A
Melissa Sherman-Bennett (University of California, Davis)
10:30 AM - 11:00 AM
  Break
11:00 AM - 12:15 PM
  Exercise Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
Anna Weigandt (University of Minnesota)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:45 PM
  Exercise Session
Jun 24, 2025
Tuesday
09:15 AM - 10:30 AM
  Lecture A
Christian Gaetz (University of California, Berkeley)
10:15 AM - 10:30 AM
  Morning Break
11:00 AM - 12:15 PM
  Exercise Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
David Keating (University of Illinois at Urbana-Champaign)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:45 PM
  Exercise Session
Jun 25, 2025
Wednesday
09:15 AM - 10:15 AM
  Lecture A
Melissa Sherman-Bennett (University of California, Davis)
10:15 AM - 10:30 AM
  Morning Break
10:30 AM - 11:30 AM
  Exercise Session
11:30 AM - 02:00 PM
  Picnic Lunch
02:00 PM - 03:00 PM
  Lecture B
David Keating (University of Illinois at Urbana-Champaign)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Exercise Session
Jun 26, 2025
Thursday
09:15 AM - 10:30 AM
  Lecture A
Christian Gaetz (University of California, Berkeley)
10:30 AM - 11:00 AM
  Morning Break
11:00 AM - 12:15 PM
  Exercise Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
Anna Weigandt (University of Minnesota)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:45 PM
  Exercise Session
Jun 27, 2025
Friday
09:15 AM - 10:30 AM
  Lecture A
Melissa Sherman-Bennett (University of California, Davis)
10:30 AM - 11:00 AM
  Morning Break
11:00 AM - 12:15 PM
  Exercise Session
12:15 PM - 12:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
David Keating (University of Illinois at Urbana-Champaign)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:45 PM
  Exercise Session
Jun 30, 2025
Monday
09:15 AM - 10:30 AM
  Lecture A
Christian Gaetz (University of California, Berkeley)
10:30 AM - 11:00 AM
  Morning Break
11:00 AM - 12:15 PM
  Exercise Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
Anna Weigandt (University of Minnesota)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:45 PM
  Exercise Session
Jul 01, 2025
Tuesday
09:15 AM - 10:30 AM
  Lecture A
Melissa Sherman-Bennett (University of California, Davis)
10:30 AM - 11:00 AM
  Morning Break
11:00 AM - 12:15 PM
  Exercise Session
12:15 PM - 12:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
David Keating (University of Illinois at Urbana-Champaign)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Exercise Session
Jul 02, 2025
Wednesday
09:15 AM - 10:30 AM
  Lecture A
Christian Gaetz (University of California, Berkeley)
10:30 AM - 11:00 AM
  Morning Break
11:00 AM - 12:15 PM
  Exercise Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
Anna Weigandt (University of Minnesota)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:45 PM
  Exercise Session
Jul 03, 2025
Thursday
09:15 AM - 10:30 AM
  Lecture A
Melissa Sherman-Bennett (University of California, Davis)
10:30 AM - 10:45 AM
  Morning Break
10:45 AM - 11:15 AM
  Participant Survey
11:15 AM - 12:15 PM
  Exercise Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 03:00 PM
  Lecture B
David Keating (University of Illinois at Urbana-Champaign)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Exercise Session
04:30 PM - 04:45 PM
  Closing Remarks