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Colloquium
 Matthew Ballard, University of South Carolina
 Exceptional collections: what they are and where to find them (special colloquium)
 01/07/2019
 4:10 PM  5:00 PM
 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.

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Colloquium
 Eugenia Malinnikova, Norwegian University of Science and Technology
 Quantitative unique continuation for elliptic PDEs and application (special colloquium)
 01/09/2019
 4:10 PM  5:00 PM
 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 LaplaceBeltrami operator on compact manifolds.

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Colloquium
 Pavlo Pylyavskyy, University of Minnesota
 Zamolodchikov periodicity and integrability (special colloquium)
 01/11/2019
 4:10 PM  5:00 PM
 C304 Wells Hall
Tsystems 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, WronskianCasoratian duality in ODE, gauge/string theories, etc. Periodicity of certain Tsystems 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 Tsystems, 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 Tsystems 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 KazhdanLusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin.

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Colloquium
 Alex Blumenthal, University of Maryland
 Chaotic regimes for random dynamical systems (special colloquium)
 01/14/2019
 4:10 PM  5:00 PM
 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 volumepreserving 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 NavierStokes and 3D hyperviscous NavierStokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel PunshonSmith, Jinxin Xue and LaiSang Young.

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Informal Geometric Analysis Seminar
 Goncalo Oliveira, Universidade Federal Fluminense, Rio de Janeiro
 Gauge theory on AloffWallach spaces
 01/16/2019
 1:40 PM  3:00 PM
 C517 Wells Hall
I will describe joint work with Gavin Ball constructing and classifying G2instantons on AloffWallach spaces, which are the most interesting known examples of compact "nearlyparallel" G2manifolds.

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Geometry and Topology
 Chris Kottke, New College Florida
 Compactification of monopole moduli spaces
 01/17/2019
 2:00 PM  3:00 PM
 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.
Seminar in Cluster algebras
 Daping Weng, MSU
 More on Scattering Diagram and Theta Functions
 01/17/2019
 3:00 PM  4:00 PM
 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 GrossHackingKeelKontsevich’s proofs of positive Laurent phenomenon, sign coherence, and a weak version of the cluster duality conjecture.

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Informal Geometric Analysis Seminar
 Zhe Zhang, MSU
 Harmonic map flow in dimension two, I
 01/21/2019
 1:40 PM  3:00 PM
 C517 Wells Hall
Bubbling analysis due to Struwe.
Colloquium
 Wencai Liu, University of California, Irvine
 Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators (special colloquium)
 01/21/2019
 4:10 PM  5:00 PM
 C304 Wells Hall
The Harper's model is a tightbinding 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 selfsimilar 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 selfsimilarity 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.

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Colloquium
 Stavros Garoufalidis, Georgia Institute of Technology
 A brief history of quantum topology (special colloquium)
 01/23/2019
 10:20 AM  11:10 AM
 C304 Wells Hall
Quantum topology originated from Vaughan Jones's discovery of the Jones polynomial of a knot in 1985. I will explain the area and its interaction with mathematical physics, algebra, analysis, number theory and combinatorics.

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Geometry and Topology
 Rebecca Winarski, University of Michigan
 Solving the Twisted Rabbit Problem using trees
 01/24/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one?
After remaining open for 25 years, this problem was solved by BartholdiNekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier, and Margalit, we present an alternate solution using topology and geometric group theory that allows us to solve a more general problem.
Seminar in Cluster algebras
 Daping Weng, MSU
 More on Scattering Diagram and Theta Functions
 01/24/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
I will continue the discussion on scattering diagram and theta functions and relate them to the classical cluster theories. I will sketch GrossHackingKeelKontsevich’s proofs of positive Laurent phenomenon, sign coherence, and a weak version of the cluster duality conjecture.

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Informal Geometric Analysis Seminar
 Zhe Zhang, MSU
 Harmonic map flow in dimension two, II
 01/28/2019
 1:40 PM  3:00 PM
 C517 Wells Hall
Bubble tree analysis and energy identity.
Student Algebra & Combinatorics
 Nick Ovenhouse, MSU
 Total Positivity
 01/28/2019
 3:00 PM  4:00 PM
 C517 Wells Hall
A matrix is "totally positive" if all of its minors are positive. We will discuss combinatorial models of total positivity using weighted graphs, as well as some criteria and characterizations for checking total positivity. If there is time, we will also discuss total positivity in the Grassmannian manifold, along with combinatorial models.

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