• Speaker: Matthew Ballard, University of South Carolina
• Title: Exceptional collections: what they are and where to find them (special colloquium)
• Date: 01/07/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Analogous to orthonormal bases in linear algebra, exceptional collections in triangulated categories are the most atomic means of decomposition. In this talk, we will introduce exceptional collections drawing heavily on examples from noncommutative algebra, algebraic geometry and symplectic geometry. We will then address the question of where (and how) to find them.

• Speaker: Pavlo Pylyavskyy, University of Minnesota
• Title: Zamolodchikov periodicity and integrability (special colloquium)
• Date: 01/11/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin.

• Speaker: Wencai Liu, University of California, Irvine
• Title: Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators (special colloquium)
• Date: 01/21/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

The Harper's model is a tight-binding description of Bloch electrons on $\mathbb{Z}^2$ under a constant transverse magnetic field. In 1964, Mark Azbel predicted that both spectra and eigenfunctions of this model have self-similar hierarchical structure driven by the continued fraction expansion of the irrational magnetic flux. In 1976, the hierarchical structure of spectra was discovered numerically by Douglas Hofstadter, and was later observed in various experiments. The mathematical study of Harper's model led to the development of spectral theory of the almost Mathieu operator, with the solution of the Ten Martini Problem partially confirming the fractal structure of the spectrum. In this talk we will present necessary background and discuss the main ideas behind our confirmation (joint with S. Jitomirskaya) of Azbel's second prediction of the structure of the eigenfunctions. More precisely, we show that the eigenfunctions of the almost Mathieu operators in the localization regime, feature self-similarity governed by the continued fraction expansion of the frequency. These results also lead to the proof of sharp arithmetic transitions between pure point and singular continuous spectra, both in the frequency and the phase, as conjectured in 1994.

• Speaker: Wilfrid Gangbo, University of California, Los Angeles
• Title: A weaker notion of convexity for Lagrangians not depending solely on velocities and positions
• Date: 03/21/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

In dynamical systems, one often encounters actions $\mathcal{A}\equiv \int_{\Omega}L(x, v(x))\rho dx$ which depend only on $v$, the velocity of the system and on $\rho$ the distribution of the particles. In this case, it is well–understood that convexity of $L(x, \cdot)$ is the right notion to study variational problems. In this talk, we consider a weaker notion of convexity which seems appropriate when the action depends on other quantities such as electro–magnetic fields. Thanks to the introduction of a gauge, we will argue why our problem reduces to understanding the relaxation of a functional defined on the set of differential forms (Joint work with B. Dacorogna).

• Speaker: Emmy Murphy, Northwestern University
• Title: TBA
• Date: 04/18/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.