# Mathematical Sciences Research Institute

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1. # Summer Graduate SchoolChip Firing and Tropical Curves

LEAD Matthew Baker (Georgia Institute of Technology), David Jensen (University of Kentucky), Sam Payne (Yale University)

Tropical geometry uses a combination of techniques from algebraic geometry, combinatorics, and convex polyhedral geometry to study degenerations of algebraic varieties; the simplest tropical objects are tropical curves, which one can think of as "shadows" of algebraic curves.  Linear equivalence of divisors on an abstract tropical curve is determined by a simple but rich combinatorial process called "chip firing", which was discovered independently in the discrete setting by physicists and graph theorists.  From a pedagogical point of view, one can view tropical curves as a combinatorial model for the highly analogous but more abstract theory of algebraic curves, but there is in fact much more to the story than this: one can use tropical curves and chip firing to prove theorems in algebraic geometry and number theory.  This field is relatively new, so participants will have the opportunity to start from scratch and still get a glimpse of the cutting edge in this active research area.

Jul 25, 2016
Monday
 09:15 AM - 09:30 AM Welcome to MSRI 09:30 AM - 10:30 AM Lecture Matthew Baker (Georgia Institute of Technology) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture Matthew Baker (Georgia Institute of Technology) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM TA Session 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Jul 26, 2016
Tuesday
 09:30 AM - 10:30 AM Lecture David Jensen (University of Kentucky) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture David Jensen (University of Kentucky) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM TA Session 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Jul 27, 2016
Wednesday
 09:30 AM - 10:30 AM Lecture Sam Payne (Yale University) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture Sam Payne (Yale University) 11:30 AM - 02:00 PM BBQ at Tilden Park 02:00 PM - 03:00 PM TA Session 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Jul 28, 2016
Thursday
 09:30 AM - 10:30 AM Lecture Matthew Baker (Georgia Institute of Technology) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture Matthew Baker (Georgia Institute of Technology) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM TA Session 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Jul 29, 2016
Friday
 09:30 AM - 10:30 AM Lecture David Jensen (University of Kentucky) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture David Jensen (University of Kentucky) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM TA Session 03:30 PM - 04:30 PM Problem Sets
Aug 01, 2016
Monday
 09:30 AM - 10:30 AM Lecture Sam Payne (Yale University) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture Sam Payne (Yale University) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM TA Session 03:30 PM - 04:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Aug 02, 2016
Tuesday
 09:30 AM - 10:30 AM Lecture Matthew Baker (Georgia Institute of Technology) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture Matthew Baker (Georgia Institute of Technology) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM TA Session 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Aug 03, 2016
Wednesday
 09:30 AM - 10:30 AM Lecture David Jensen (University of Kentucky) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture David Jensen (University of Kentucky) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM TA Session 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Aug 04, 2016
Thursday
 09:30 AM - 10:30 AM Lecture Sam Payne (Yale University) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture Sam Payne (Yale University) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Lecture Matthew Baker (Georgia Institute of Technology) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Problem Sets
Aug 05, 2016
Friday
 09:30 AM - 10:30 AM Lecture Yoav Len (Fields Institute for Research in Mathematical Sciences) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Lecture Dhruv Ranganathan (Massachusetts Institute of Technology) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Lecture David Jensen (University of Kentucky) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Lecture Sam Payne (Yale University)
2. # Summer Graduate SchoolElectronic Structure Theory

LEAD Lin Lin (University of California, Berkeley), Jianfeng Lu (Duke University), James Sethian (University of California, Berkeley)

Ab initio or first principle electronic structure theories, particularly represented by Kohn-Sham density functional theory (KS-DFT), have been developed into workhorse tools with a wide range of scientific applications in chemistry, physics, materials science, biology etc. What is needed are new techniques that greatly extend the applicability and versatility of these approaches. At the core, many of the challenges that need to be addressed are essentially mathematical. The purpose of the workshop is to provide graduate students a self-contained introduction to electronic structure theory, with particular emphasis on frontier topics in aspects of applied analysis and numerical methods.

Jul 18, 2016
Monday
 08:00 AM - 05:30 PM Electronic Structure Theory Lin Lin (University of California, Berkeley)
Jul 19, 2016
Tuesday
 08:00 AM - 05:30 PM Electronic Structure Theory Jianfeng Lu (Duke University)
Jul 20, 2016
Wednesday
 08:00 AM - 05:00 PM Electronic Structure Theory Lin Lin (University of California, Berkeley)
Jul 21, 2016
Thursday
 08:00 AM - 05:30 PM Electronic Structure Theory Jianfeng Lu (Duke University)
Jul 22, 2016
Friday
 08:00 AM - 05:30 PM Electronic Structure Theory Lin Lin (University of California, Berkeley)
Jul 25, 2016
Monday
 All Day Electronic Structure Theory Lin Lin (University of California, Berkeley), Jianfeng Lu (Duke University)
Jul 26, 2016
Tuesday
 All Day Electronic Structure Theory Lin Lin (University of California, Berkeley), Jianfeng Lu (Duke University)
Jul 27, 2016
Wednesday
 All Day Electronic Structure Theory Lin Lin (University of California, Berkeley), Jianfeng Lu (Duke University)
Jul 28, 2016
Thursday
 All Day Electronic Structure Theory Lin Lin (University of California, Berkeley), Jianfeng Lu (Duke University)
Jul 29, 2016
Friday
 All Day Electronic Structure Theory Lin Lin (University of California, Berkeley), Jianfeng Lu (Duke University)
3. # Summer Graduate SchoolAn Introduction to Character Theory and the McKay Conjecture

Robert Guralnick (University of Southern California), Pham Tiep (University of Arizona)

Character Theory of Finite Groups provides one of the most powerful tools to study groups. In this course we will give a gentle introduction to basic results in the Character Theory, as well as some of the main conjectures in Group Representation Theory, with particular emphasis on the McKay Conjecture.

Group Photo

Jul 11, 2016
Monday
 08:40 AM - 08:55 AM Introduction to MSRI 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea Break 03:45 PM - 05:00 PM Discussion session 2
Jul 12, 2016
Tuesday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch break 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea break 03:45 PM - 05:00 PM Discussion session 2
Jul 13, 2016
Wednesday
 08:45 AM - 10:00 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:00 AM - 10:15 AM Break 10:15 AM - 11:30 AM Lecture Gabriel Navarro (University of Valencia) 11:30 AM - 02:00 PM BBQ organized by MSRI 02:00 PM - 05:00 PM Free afternoon
Jul 14, 2016
Thursday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch break 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea break 03:45 PM - 05:00 PM Discussion session 2
Jul 15, 2016
Friday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch break 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea break 03:45 PM - 05:00 PM Discussion session 2
Jul 18, 2016
Monday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch break 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea break 03:45 PM - 05:00 PM Discussion session 2
Jul 19, 2016
Tuesday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Pham Tiep (University of Arizona) 12:00 PM - 02:00 PM Lunch break 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea break 03:45 PM - 05:00 PM Discussion session 2
Jul 20, 2016
Wednesday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch 02:00 PM - 05:00 PM Free afternoon
Jul 21, 2016
Thursday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch break 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea break 03:45 PM - 05:00 PM Discussion session 2
Jul 22, 2016
Friday
 09:00 AM - 10:15 AM Lecture I. Martin Isaacs (University of Wisconsin-Madison) 10:15 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Gabriel Navarro (University of Valencia) 12:00 PM - 02:00 PM Lunch break 02:00 PM - 03:15 PM Discussion session 1 03:15 PM - 03:45 PM Tea break 03:45 PM - 05:00 PM Discussion session 2
4. # Summer Graduate SchoolHarmonic Analysis and Elliptic Equations on real Euclidean Spaces and on Rough Sets

LEAD Steven Hofmann (University of Missouri), Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT))

The goal of the workshop is to present harmonic analysis techniques in $R^n$ (the flat" setting), and then to show how those techniques extend to much rougher settings, with application to the theory of elliptic equations. Thus, the subject matter of the workshop will introduce the students to an active, current research area:  the interface between harmonic analysis, elliptic PDE, and geometric measure theory.

Group Photo

Jun 13, 2016
Monday
 09:00 AM - 09:15 AM Introduction to MSRI 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
Jun 14, 2016
Tuesday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
Jun 15, 2016
Wednesday
 09:15 AM - 10:15 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:15 AM - 10:30 AM Break 10:30 AM - 11:30 AM Lecture Steven Hofmann (University of Missouri) 11:30 AM - 02:00 PM BBQ Lunch 02:00 PM - 02:45 PM Free time to work on problems 02:45 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 04:45 PM Discussion Session II
Jun 16, 2016
Thursday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
Jun 17, 2016
Friday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
Jun 20, 2016
Monday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
Jun 21, 2016
Tuesday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
Jun 22, 2016
Wednesday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:30 PM - 05:00 PM Discussion Session II
Jun 23, 2016
Thursday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Lecture - Steven Hofmann (University of Missouri) 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
Jun 24, 2016
Friday
 09:15 AM - 10:30 AM Lecture Jose Maria Martell (Instituto de Ciencias Matematicas (ICMAT)) 10:30 AM - 10:45 AM Break 10:45 AM - 12:00 PM Lecture Steven Hofmann (University of Missouri) 12:00 PM - 01:30 PM Lunch 01:30 PM - 02:30 PM Free time to work on problems 02:30 PM - 03:30 PM Discussion Session I 03:30 PM - 04:00 PM Tea Break 04:00 PM - 05:00 PM Discussion Session II
5. # MSRI-UPMSRI-UP 2016: Sandpile Groups

Federico Ardila (San Francisco State University), Duane Cooper (Morehouse College), Maria Mercedes Franco (Queensborough Community College (CUNY)), Luis Garcia Puente (Sam Houston State University), Herbert Medina (Loyola Marymount University), LEAD Suzanne Weekes (Worcester Polytechnic Institute)

The MSRI-UP summer program is designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences. Due to funding restrictions, only U.S. citizens and permanent residents are eligible to apply and the program cannot accept foreign students regardless of funding. The academic portion of the 2016 program will be led by Prof. Luis Garcia-Puente of Sam Houston State University.

Jul 22, 2016
Friday
 All Day Sandpile Group Presentations 09:20 AM - 10:05 AM On the Accessibility Numbers in the Sandpile Monoid of a Graph Ernest Castorena (University of California, Berkeley), Dominika Palinko (University of California, Berkeley), Cecily Santiago (University of Minnesota Twin Cities) 10:05 AM - 10:50 AM The Sandpile Group of a Thick Cycle Graph Diane Christine Alar (San Francisco State University), Jonathan Celaya (Rice University), Micah Henson (Spelman College) 11:00 AM - 11:45 AM On the Sandpile Group of a Circulant Graph Anna Comito (Wesleyan University), Jennifer Garcia (University of California, Berkeley), Justin Rivera (Wentworth Institute of Technology) 01:00 PM - 01:45 PM Combinatorial and Algebraic Properties of Generalized Crown Graphs Carlos Agrinsoni Santiago (University of Puerto Rico, Cayey), Angel Burr (University of Arizona), Ruben Hurtado (California State Polytechnic University) 01:45 PM - 02:30 PM The Avalanche Polynomial of the Complete Bipartite Graph Karlie Elliott (Humboldt State University), Drisana Mosaphir (Harvard University), Maleek Richardson (North Carolina Agricultural and Technical State University) 02:40 PM - 03:25 PM Bijections Between the Recurrent Sandpiles of an Eulerian Digraph and Its Reverse Tafari James (Haverford College), Casandra Monroe (Princeton University), Ricardo Rojas-Echenique (Reed College)
6. # WorkshopGeometric Flows in Riemannian and Complex Geometry

Tobias Colding (Massachusetts Institute of Technology), LEAD John Lott (University of California, Berkeley), Natasa Sesum (Rutgers University)

The workshop will concentrate on parabolic methods in both Riemannian and complex geometry. The topics will include

- Ricci flow. Analytic questions about Ricci flow in three dimensions. Possible applications of Ricci flow to 4-manifold topology. Ricci flow in higher dimensions under curvature assumptions.

- Kähler-Ricci Flow. Applications to the Kähler-Einstein problem. Connections to the minimal model program. Study of Kähler-Ricci solitons and limits of Kähler-Ricci flow.

- Mean curvature flow. Singularity analysis. Generic mean curvature flow.

- Other geometric flows such as Calabi flow and pluriclosed flow.

May 02, 2016
Monday
 09:15 AM - 09:30 AM Welcome 09:30 AM - 10:30 AM Fully nonlinear flows with surgery Simon Brendle (Stanford University) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Ricci flow from metrics with isolated conical singularities Felix Schulze (University College) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Singularities of mean curvature flow Tom Ilmanen (ETH Zürich) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Manifolds with almost nonnegative curvature operator Burkhard Wilking (Westfälische Wilhelms-Universität Münster)
May 03, 2016
Tuesday
 09:30 AM - 10:30 AM Hermitian curvature flows Gang Tian (Princeton University) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Asymptotic rigidity of noncompact shrinking gradient Ricci solitons Brett Kotschwar (Arizona State University) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Flow by the Gauss curvature and its power Lei Ni (University of California, San Diego) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Hypersurfaces of Low Entropy Lu Wang (University of Wisconsin-Madison) 04:30 PM - 06:20 PM Reception
May 04, 2016
Wednesday
 09:30 AM - 10:30 AM Ricci flow through singularities Bruce Kleiner (New York University, Courant Institute) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM The Ricci flow on the sphere with marked points Jian Song (Rutgers University)
May 05, 2016
Thursday
 09:30 AM - 10:30 AM Convergence of weak Kaehler-Ricci flows on minimal models of positive Kodaira dimension Vincent Guedj (Institut de Mathématiques de Toulouse) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Convergence of Ricci flows with bounded scalar curvature Richard Bamler (University of California, Berkeley) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Ancient solutions, their asymptotics and uniqueness Natasa Sesum (Rutgers University) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM The Chern-Ricci flow Ben Weinkove (Northwestern University)
May 06, 2016
Friday
 09:30 AM - 10:30 AM Global solutions of the Teichmueller harmonic map flow Peter Topping (University of Warwick) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM A gap theorem and some uniform estimates for Ricci flows on homogeneous spaces Miles Simon (Otto-von-Guericke-Universität Magdeburg) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Smoothing properties and uniqueness of the weak Kaehler-Ricci flow Eleonora Di Nezza (Imperial College, London) 03:00 PM - 03:30 PM Tea
7. # WorkshopHot Topics: Cluster algebras and wall-crossing

LEAD Mark Gross (University of Cambridge), Paul Hacking (University of Massachusetts, Amherst), Sean Keel (University of Texas), Lauren Williams (University of California, Berkeley)

Cluster algebras were introduced in 2001 by Fomin and Zelevinsky to capture the combinatorics of canonical bases and total positivity in semisimple Lie groups. Since then they have revealed a rich combinatorial and group-theoretic structure, and have had significant impact beyond these initial subjects, including string theory, algebraic geometry, and mirror symmetry. Recently Gross, Hacking, Keel and Kontsevich released a preprint introducing mirror symmetry techniques into the subject which resolved several long-standing conjectures, including the construction of canonical bases for cluster algebras and positivity of the Laurent phenomenon. This preprint reformulates the basic construction of cluster algebras in terms of scattering diagrams (or wall-crossing structures). This leads to the proofs of the conjectures and to new constructions of elements of cluster algebras. But fundamentally they provide a new tool for thinking about cluster algebras.

The workshop will bring together many of the different users of cluster algebras to achieve a synthesis of these new techniques with many of the different aspects of the subject. There will be lecture series on the new techniques, and other lecture series on connections with Lie theory, quiver representation theory, mirror symmetry, string theory, and stability conditions.

Mar 28, 2016
Monday
 09:45 AM - 10:00 AM Welcome 10:00 AM - 11:00 AM SYZ mirror symmetry in the complement of a divisor and regular functions on the mirror Denis Auroux (University of California, Berkeley) 11:00 AM - 11:30 AM Break 11:30 AM - 12:30 PM Cluster algebras and cluster varieties Lauren Williams (University of California, Berkeley) 12:30 PM - 02:30 PM Lunch 02:30 PM - 03:30 PM Theta Functions for Log Calabi-Yau manifolds I -II Sean Keel (University of Texas) 03:30 PM - 04:00 PM Tea 04:00 PM - 05:00 PM Scattering diagrams, broken lines and theta functions Mark Gross (University of Cambridge)
Mar 29, 2016
Tuesday
 10:00 AM - 11:00 AM Examples of cluster varieties and their scattering diagrams. Paul Hacking (University of Massachusetts, Amherst) 11:00 AM - 11:30 AM Break 11:30 AM - 12:30 PM Scattering diagrams, broken lines and theta functions Mark Gross (University of Cambridge) 12:00 PM - 02:30 PM Lunch 02:30 PM - 03:30 PM Theta Functions for Log Calabi-Yau manifolds I -II Sean Keel (University of Texas) 03:30 PM - 04:00 PM Tea 04:00 PM - 05:00 PM Hitchin systems, spectral networks and noncommutative clusters Andrew Neitzke (University of Texas) 05:00 PM - 06:20 PM Reception
Mar 30, 2016
Wednesday
 10:00 AM - 11:00 AM Scattering diagrams from stability conditions Tom Sutherland (Università di Pavia) 11:00 AM - 11:30 AM Break 11:30 AM - 12:30 PM Donaldson-Thomas transformations for moduli spaces of G-local systems on surfaces Linhui Shen (Northwestern University)
Mar 31, 2016
Thursday
 10:00 AM - 11:00 AM Donaldson-Thomas transformations for moduli spaces of G-local systems on surfaces Linhui Shen (Northwestern University) 11:00 AM - 11:30 AM Break 11:30 AM - 12:30 PM Mirror symmetry for homogeneous spaces Clelia Pech (University of Kent at Canterbury) 12:30 PM - 02:30 PM Lunch 02:30 PM - 03:30 PM Cluster duality and mirror symmetry for the Grassmannian Lauren Williams (University of California, Berkeley) 03:30 PM - 04:00 PM Tea 04:00 PM - 05:00 PM Cluster Algebras and Exact Lagrangian Surfaces Harold Williams (University of Texas)
Apr 01, 2016
Friday
 10:00 AM - 11:00 AM Wall structures in mirror symmetry Bernd Siebert (Universität Hamburg) 11:00 AM - 11:30 AM Break 11:30 AM - 12:30 PM Mirror symmetry for homogeneous spaces Clelia Pech (University of Kent at Canterbury) 12:30 PM - 02:30 PM Lunch 02:30 PM - 03:30 PM Generic bases are dual semicanonical bases for unipotent cells Christof Geiss (UNAM - Universidad Nacional Autonoma de Mexico) 03:30 PM - 04:00 PM Tea 04:00 PM - 05:00 PM Stability conditions and cluster varieties Tom Sutherland (Università di Pavia)
8. # WorkshopKähler Geometry, Einstein Metrics, and Generalizations

Olivier Biquard (École Normale Supérieure), Simon Donaldson (Imperial College, London), Gang Tian (Princeton University), LEAD Jeff Viaclovsky (University of Wisconsin-Madison)

The workshop will integrate elements from complex differential geometry with Einstein metrics and their generalizations. The topics will include

- Existence of Kähler-Einstein metrics and extremal Kähler metrics. Notions of stability in algebraic geometry such as Chow stability, K-stability, b-stability, and polytope stability. Kähler-Einstein metrics with conical singularities along a divisor.

- Calabi-Yau metrics and collapsed limit spaces. Connections with physics and mirror symmetry.

- Einstein metrics and their moduli spaces, ε-regularity, noncompact examples such as ALE, ALF, and Poincaré-Einstein metrics. Generalizations of the Einstein condition, such as Bach-flat metrics and Ricci solitons.

- Sasaki-Einstein metrics and metrics with special holonomy. New examples and classification problems.

Mar 21, 2016
Monday
 09:15 AM - 09:30 AM Welcome 09:30 AM - 10:30 AM Holomorphic fibrations on Calabi-Yau manifolds and collapsing Valentino Tosatti (Northwestern University) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Foliations with positive slopes and stability of cotangent bundle Mihai Paun (Korea Institute for Advanced Study (KIAS)) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Tian's properness conjectures Yanir Rubinstein (University of Maryland) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Kähler-Einstein metrics and higher alpha-invariants Heather Macbeth (Massachusetts Institute of Technology)
Mar 22, 2016
Tuesday
 09:30 AM - 10:30 AM Manifolds of bounded Ricci curvature and the codimension $4$ conjecture Jeff Cheeger (New York University, Courant Institute) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Disconnecting the G_2 moduli space Johannes Nordstrom (University of Bath) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Infinite dimensional geometries on the space of Kahler metrics and applications to canonical metrics Tamas Darvas (University of Maryland) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Kähler Ricci flow on Fano manifold Bing Wang (University of Wisconsin-Madison) 04:30 PM - 06:20 PM Reception
Mar 23, 2016
Wednesday
 09:30 AM - 10:30 AM Compactness of conformally compact Einstein manifolds in dimension 4 Sun-Yung Chang (Princeton University) 10:30 AM - 11:00 AM Tea 11:00 AM - 12:00 PM Asymptotically Conical Calabi-Yau manifolds Ronan Conlon (Florida International University)
Mar 24, 2016
Thursday
 09:30 AM - 10:30 AM L^2 Curvature Bounds on Manifolds with Bounded Ricci Curvature Aaron Naber (Northwestern University) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM New Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 4-folds Dominic Joyce (University of Oxford) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Duality between the pseudoeffective and the movable cone on a projective manifold David Witt Nystrom (Chalmers University of Technology) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Gromov-Hausdorff collapse of Calabi-Yau manifolds Mark Gross (University of Cambridge)
Mar 25, 2016
Friday
 09:30 AM - 10:30 AM ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo (State University of New York, Stony Brook) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Stability of weak solutions of the complex Monge-Ampere equation on compact Hermitian manifolds Slawomir Kolodziej (Jagiellonian University) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM The Hilbert--Einstein functional and the Sasaki-Futaki invariant Eveline Legendre (Université de Toulouse III (Paul Sabatier)) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Fano-Ricci limit spaces and spectral convergence Akito Futaki (University of Tokyo)
9. # WorkshopCritical Issues in Mathematics Education 2016: Observing, Evaluating and Improving Mathematics Teaching from the Early Grades through the University

Hyman Bass (University of Michigan), Michael Driskill (Math for America ), LEAD Mark Hoover (University of Michigan), LEAD Deborah Hughes Hallett (University of Arizona), Danny Martin (University of Illinois at Chicago), Miriam Sherin (Northwestern University)

The 2016 CIME workshop focuses directly on the teaching of mathematics at the university and precollege levels. Teaching is not easy to examine in disciplined ways because it is so familiar and seems so obvious.  Although teaching shapes students’ opportunities to learn, what teachers are actually doing is difficult to observe and describe. This impedes work on improving teaching.

This workshop will offer the opportunity to study and talk closely about mathematics teaching through close observation and discussion of video tapes in a setting that will bring together professionals with a range of perspectives, knowledge, experience, and orientations. The goal of the workshop is to develop language and methods for describing, analyzing and evaluating what can be seen in the classroom, with the ultimate goal of helping us shape and improve teaching — our own and more broadly.

Four questions structure the highly interactive design of the workshop:

1. What skills are needed for observing teaching in ways that inform improvement efforts? What is involved in observing teaching? What is the teacher saying and doing? What are students saying and doing? What is the mathematics at play? What else is happening? And what do these imply for teaching?
2. How can the practice and use of observation be structured in order to improve mathematics teaching? What approaches are available? What are their strengths and weaknesses?
3. Observation-based assessment of teaching: Why, what, and how? What are the risks?
4. How can we develop and sustain a cross-professional community that observes and evaluates teaching in such a way that different communities communicate with and learn from each other to support a cycle of improvement in the teaching of mathematics at all levels?

The workshop will provide a library of videos of mathematics teaching for study. In addition, participants are encouraged to submit a short video clip of their own teaching, together with a brief background commentary.  These videos will provide a central text for our collective work on discussing and assessing mathematics teaching.

Group Photo

Feb 10, 2016
Wednesday
 03:30 PM - 04:00 PM Registration 04:00 PM - 04:30 PM Welcome and framing the workshop David Eisenbud (MSRI - Mathematical Sciences Research Institute), Miriam Sherin (Northwestern University) 04:30 PM - 05:30 PM When words get in your eyes: On challenges of investigating mathematics-in-teaching and on the importance of paying attention to words Anna Sfard (The University of Haifa) 05:30 PM - 06:30 PM Supporting collegiate mathematics teaching: Current realities Sarah Eichhorn (University of California, Irvine), Deborah Hughes Hallett (University of Arizona), Robert Indik (University of Arizona), Mara Landers (Los Medanos College), Scott Peterson (Oregon State University) 06:30 PM - 07:30 PM Reception
Feb 11, 2016
Thursday
 08:30 AM - 09:45 AM (How) Can we “see” the work of teaching mathematics? Deborah Ball (University of Michigan) 09:45 AM - 10:45 AM Seeing the math in teaching Roger Howe (Yale University), Lindsey Mann (University of Michigan) 10:45 AM - 11:15 AM Break 11:15 AM - 12:15 PM Attending to student thinking and their interactions when working in small groups Chris Rasmussen (San Diego State University) 12:15 PM - 01:00 PM Lunch 01:00 PM - 02:30 PM Observing practices that support learners’ identity development and participation in mathematics classrooms Lawrence Clark (University of Maryland), Imani Goffney (University of Maryland), Whitney Johnson (Morgan State University), Danny Martin (University of Illinois at Chicago) 02:30 PM - 02:45 PM Break 02:45 PM - 03:45 PM Parallel session 1.a: Mathematical micro-identities: Students’ positioning and learning during mathematics lessons Marcy Wood (University of Arizona) 02:45 PM - 03:45 PM Parallel session 1.b: Watching mathematicians at work: Mathematical knowledge for teaching calculus and the practice of examining student work Natasha Speer (University of Maine) 02:45 PM - 03:45 PM Parallel session 1.c: Using video to develop pre-service teachers’ professional vision of ambitious mathematics instruction Elizabeth van Es (University of California, Irvine) 02:45 PM - 03:45 PM Parallel session 1.d: Teaching and learning mathematical practices in the early elementary grades Hyman Bass (University of Michigan), Sarah Selling (University of Michigan) 03:45 PM - 04:15 PM Tea Break 04:15 PM - 05:15 PM Designing video clubs for teacher learning Miriam Sherin (Northwestern University) 05:15 PM - 06:00 PM The practice and use of observation in powerful professional development: The teaching-for-robust-understanding (TRU) framework Alan Schoenfeld (University of California, Berkeley)
Feb 12, 2016
Friday
 08:30 AM - 08:40 AM Shifting focus from learning to observe to using observing to learn Mark Hoover (University of Michigan) 08:40 AM - 09:45 AM Structuring the practice and use of observation to improve developmental mathematics courses in a mathematics department Dawn Berk (University of Delaware), James Hiebert (University of Delaware) 09:45 AM - 10:45 AM Preparing secondary mathematics teachers to facilitate video clubs Michael Driskill (Math for America ), Kristen Smith (Math for America) 10:45 AM - 11:15 AM Break 11:15 AM - 12:15 PM Parallel session 2.a: Advancing quality teaching: Using video to support professional development in the Community College Pathways network Ann Edwards (Carnegie Foundation for the Advancement of Teaching) 11:15 AM - 12:15 PM Parallel session 2.b: Teaching Triads: Enhancing teaching through the use of Cognitive Coaching Scott Peterson (Oregon State University) 11:15 AM - 12:15 PM Parallel session 2.c: Key ideas for effective professional development: Purposeful classroom observation and non judgmental data collection Hyman Bass (University of Michigan), Akihiko Takahashi (DePaul University) 12:15 PM - 01:00 PM Lunch 01:00 PM - 02:00 PM Parallel session 3.d: Leading mathematics observations: An observational tool for providing feedback Nicole Garcia (University of Michigan), Meghan Shaughnessy (University of Michigan) 01:00 PM - 02:00 PM Parallel session 3.a: Reflecting on one’s own classroom video to enhance classroom interactions William Day (Math for America DC), Julia Penn (Math for America DC) 01:00 PM - 02:00 PM Parallel session 3.b: Lessons learned from Lesson Study Travis Lemon (Math for America - Utah) 01:00 PM - 02:00 PM Parallel session 3.c: Observing across communities: Learning to observe undergraduate mathematics classes by supervising student teachers Cody Patterson (University of Arizona) 02:00 PM - 03:00 PM Observing and talking about teaching: Departmental leadership Deborah Hughes Hallett (University of Arizona), Wayne Raskind (Wayne State University), Ayse Sahin (Wright State University), Douglas Ulmer (University of Arizona) 03:00 PM - 03:30 PM Tea Break 03:30 PM - 04:30 PM A conversation about practical next steps Deborah Ball (University of Michigan) 04:30 PM - 05:00 PM Closing session: Reflections on ideas discussed at the workshop James Hiebert (University of Delaware), Anna Sfard (The University of Haifa)
10. # WorkshopIntroductory Workshop: Modern Riemannian Geometry

LEAD Tobias Colding (Massachusetts Institute of Technology), John Lott (University of California, Berkeley), Jeff Viaclovsky (University of Wisconsin-Madison)

The week will be devoted to an introduction to modern techniques in Riemannian geometry. This is intended to help graduate students and younger researchers get a headstart, in order to increase their participation during the main semester programs and research lectures. To increase outreach, the week will focus on Riemannian geometry and should be largely accessible. Some minicourses on topics of recent interest will be included. The workshop will also have semi-expository lectures dealing with aspects of spaces with curvature bounded from below, since such spaces will occur throughout the semester. We expect that many Berkeley mathematicians and students will participate in the introductory workshop.

Jan 18, 2016
Monday
 09:15 AM - 09:30 AM Welcome 09:30 AM - 10:30 AM Manifolds with lower sectional curvature bounds and Alexandrov geometry Karsten Grove (University of Notre Dame) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Comparison geometry for Ricci curvature I Guofang Wei (University of California, Santa Barbara) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Ricci flat spaces and metrics with special or exceptional holonomy I Mark Haskins (Imperial College, London) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Aspects of Einstein metrics on 4-manifolds I Michael Anderson (State University of New York, Stony Brook)
Jan 19, 2016
Tuesday
 09:30 AM - 10:30 AM Mean curvature flow Tobias Colding (Massachusetts Institute of Technology) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Positively and non-negatively curved manifolds and (torus) symmetries Catherine Searle (Wichita State University) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Metric measure spaces satisfying Ricci curvature lower bounds I Andrea Mondino (University of Warwick) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Semiconcave functions in Alexandrov geometry Vitali Kapovitch (University of Toronto) 04:30 PM - 06:20 PM Reception
Jan 20, 2016
Wednesday
 09:30 AM - 10:30 AM A panoramic glimpse of nonnegative curvature Karsten Grove (University of Notre Dame) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Comparison geometry for Ricci curvature II Guofang Wei (University of California, Santa Barbara)
Jan 21, 2016
Thursday
 09:30 AM - 10:30 AM Ricci flat spaces and metrics with special or exceptional holonomy II Mark Haskins (Imperial College, London) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Aspects of Einstein metrics on 4-manifolds II Michael Anderson (State University of New York, Stony Brook) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM The maximal symmetry rank conjecture for nonnegative curvature Catherine Searle (Wichita State University) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Minimal submanifolds and lower curvature bounds I Richard Schoen (University of California, Irvine)
Jan 22, 2016
Friday
 09:30 AM - 10:30 AM Minimal submanifolds and lower curvature bounds II Richard Schoen (University of California, Irvine) 10:30 AM - 11:00 AM Break 11:00 AM - 12:00 PM Mean curvature flow Tobias Colding (Massachusetts Institute of Technology) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Metric measure spaces satisfying Ricci curvature lower bounds II Andrea Mondino (University of Warwick) 03:00 PM - 03:30 PM Tea 03:30 PM - 04:30 PM Measure-metric boundary and Liouville theorem in Alexandrov geometry Vitali Kapovitch (University of Toronto)
11. # WorkshopConnections for Women: Differential Geometry

Christine Breiner (Fordham University), LEAD Natasa Sesum (Rutgers University)

The purpose of this meeting is to help junior female researchers to become familiar with the focus topics of the main MSRI program, and also for the junior researchers to have an opportunity to get acquainted with more senior women researchers in differential geometry.

This workshop is open to all mathematicians.

Jan 14, 2016
Thursday
 09:15 AM - 09:30 AM Welcome 09:30 AM - 10:30 AM Topology of Closed Hypersurfaces of Small Entropy Lu Wang (University of Wisconsin-Madison) 10:30 AM - 11:00 AM Tea Break 11:00 AM - 12:00 PM Ancient solutions to geometric flows Panagiota Daskalopoulos (Columbia University) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Monge-Ampère equations on quasi-projective varieties Eleonora Di Nezza (Imperial College, London) 03:00 PM - 03:30 PM Tea Break 03:30 PM - 04:30 PM Sliced Filling Volumes and Intrinsic Flat Convergence Christina Sormani (CUNY, Graduate Center) 04:30 PM - 05:30 PM Panel Discussion 06:30 PM - 08:30 PM Dinner at the Taste of Himalayas
Jan 15, 2016
Friday
 09:30 AM - 10:30 AM Finite total $Q$-curvature on a locally conformally flat manifold Yi Wang (Johns Hopkins University) 10:30 AM - 11:00 AM Tea Break 11:00 AM - 12:00 PM Convergence of Manifolds and Metric Spaces with Boundary Raquel Perales (UNAM - Universidad Nacional Autonoma de Mexico) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Singularity Formation of the Yang-Mills Flow Casey Kelleher (University of California, Irvine) 03:00 PM - 03:30 PM Tea Break 03:30 PM - 04:30 PM Local Isoperimetric Constant Estimate for Integral Ricci Curvature Guofang Wei (University of California, Santa Barbara)