**Lecturer: ** Lauren Williams
(office Evans Hall 913, e-mail `williams@math.berkeley.edu`)

**Office Hours:** by appointment

The classical theory of total positivity studied totally positive matrices, matrices with all minors positive. The theory was pioneered by Gantmacher, Krein, Schoenberg, Whitney, and others in the first half of the 20th century, and subsequently generalized in the setting of Lie theory by Lusztig in the 1990's, who had observed that his canonical bases had surprising positivity properties. Meanwhile in 2000, Fomin and Zelevinsky introduced cluster algebas, a class of combinatorially defined commutative rings, which provide a unifying structure for phenomena in total positivity, and turned out to be connected to a host of other fields, including quiver representations, Teichmuller theory, Poisson geometry, etc. In this course I'll survey these developments, with a particular focus on the totally nonnegative Grassmannian, whose combinatorial structure was first developed by Postnikov. If time permits I may discuss recent developments in the field, including positroids, amplituhedra, KP solitons, and Newton-Okounkov bodies.

I will assume that people have some familiarity with combinatorics. Familiarity with root systems would also be helpful. I will not assume prior knowledge of cluster algebras.

References for total positivity:

- Karlin, Total positivity, Volume I, Stanford University Press, Stanford, CA 1968.
- Lusztig, Total positivity in reductive groups, Lie theory and geometry: in honor of B. Kostant, Progress in Math. 123, Birkhauser, 1994, 531--568.
- Fomin and Zelevinsky, Total positivity: tests and parametrizations.

References for cluster algebras:

- Fomin, Williams, and Zelevinsky, Introduction to cluster algebras. Chapters 1-3.
- Williams, Cluster algebras: an introduction.
- Fomin and Zelevinsky, Cluster algebras: Notes for the CDM-03 conferences, International Press, 2004.
- Fomin and Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497--529.
- Fomin and Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63--121.
- Berenstein, Fomin and Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1--52.
- Fomin and Zelevinsky, Cluster algebras IV: Coefficients, Compos. Math. 143 (2007), 112--164.
- Gekhtman, Shapiro, and Vainshtein, Cluster algebras and Poisson geometry, Amer. Math. Soc, 2010.
- Cluster Algebras Portal

References for Grassmannians:

- Postnikov, Total Positivity, Grassmannians, and networks.
- Morales (writeup of lecture notes by Postnikov), Lectures on the positive Grassmannian.
- Postnikov, Speyer, Williams Matching polytopes, toric geometry, and the non-negative part of the Grassmannian.

Other topics:

- Arkani-Hamed and Trnka, The Amplituhedron.
- Karp and Williams, The m=1 amplituhedron and cyclic hyperplane arrangements.
- Ardila, Rincon, Williams, Positively oriented matroids are realizable.
- Kodama, Williams, KP solitons, total positivity, and cluster algebras.