• 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: Eugenia Malinnikova, Norwegian University of Science and Technology
• Title: Quantitative unique continuation for elliptic PDEs and application (special colloquium)
• Date: 01/09/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

If a solution to a uniformly elliptic second order PDE with smooth coefficients vanishes on an open subset of a domain then it is zero on the whole domain. This is a classical result known as weak unique continuation. We will discuss stronger versions, including some recent quantitative results and outline applications to the study of eigenfunctions of Laplace-Beltrami operator on compact manifolds.

• 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: Alex Blumenthal, University of Maryland
• Title: Chaotic regimes for random dynamical systems (special colloquium)
• Date: 01/14/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

It is anticipated that chaotic regimes (characterized by, e.g., sensitivity with respect to initial conditions and loss of memory) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way. Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.

• Speaker: Chris Kottke, New College Florida
• Title: Compactification of monopole moduli spaces
• Date: 01/17/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

I will discuss joint work with Michael Singer and Karsten Fritzsch on compactifications of the moduli spaces $M_k$ of $\mathrm{SU}(2)$ magnetic monopoles on $\mathbf{R}^3$ . Via a geometric gluing procedure, we construct manifolds with corners compactifying the $M_k$ , the boundaries of which represent monopoles of charge $k$ decomposing into widely separated ‘monopole clusters' of lower charge. The hyperkahler metric on $M_k$ has a complete asymptotic expansion, the leading terms of which generalize the asymptotic metric discovered by Bielawski, Gibbons and Manton in the case that the monopoles are all widely separated. From the structure of the compactification, we are able to make partial progress toward proving Sen's conjecture for $L^2$ cohomology of the moduli spaces.

• Speaker: Daping , MSU
• Title: More on Scattering Diagram and Theta Functions
• Date: 01/17/2019
• Time: 3:00 PM - 4:00 PM
• Place: C117 Wells Hall

I will continue the discussion on scattering diagram and theta functions and relate them to the classical cluster theories. I will sketch Gross-Hacking-Keel-Kontsevich’s proofs of positive Laurent phenomenon, sign coherence, and a weak version of the cluster duality conjecture.

• Speaker: Zhe Zhang, MSU
• Title: Harmonic map flow in dimension two, I
• Date: 01/21/2019
• Time: 1:40 PM - 3:00 PM
• Place: C517 Wells Hall

Bubbling analysis due to Struwe.

• 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: Rebecca Winarski, University of Michigan
• Title: TBA
• Date: 01/24/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

TBA

• Speaker: Zhe Zhang, MSU
• Title: Harmonic map flow in dimension two, II
• Date: 01/28/2019
• Time: 1:40 PM - 3:00 PM
• Place: C517 Wells Hall

Bubble tree analysis and energy identity.

• Speaker: Jane Zimmerman, MSU
• Title: MTH 103A: Experiences from the classroom and future directions
• Date: 01/28/2019
• Time: 4:10 PM - 5:00 PM
• Place: C109 Wells Hall

No abstract available.