Talk_id  Date  Speaker  Title 
17497

Monday 1/7 4:10 PM

Matthew Ballard, University of South Carolina

Exceptional collections: what they are and where to find them (special 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.

17495

Wednesday 1/9 4:10 PM

Eugenia Malinnikova, Norwegian University of Science and Technology

Quantitative unique continuation for elliptic PDEs and application (special 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.

17501

Friday 1/11 4:10 PM

Pavlo Pylyavskyy, University of Minnesota

Zamolodchikov periodicity and integrability (special 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.

17500

Monday 1/14 4:10 PM

Alex Blumenthal, University of Maryland

Chaotic regimes for random dynamical systems (special 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.

17496

Monday 1/21 4:10 PM

Wencai Liu, University of California, Irvine

Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators (special 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.

17511

Wednesday 1/23 10:20 AM

Stavros Garoufalidis, Georgia Institute of Technology

A brief history of quantum topology (special 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.

14368

Thursday 3/21 4:10 PM

Wilfrid Gangbo, University of California, Los Angeles

A weaker notion of convexity for Lagrangians not depending solely on velocities and positions
 Wilfrid Gangbo, University of California, Los Angeles
 A weaker notion of convexity for Lagrangians not depending solely on velocities and positions
 03/21/2019
 4:10 PM  5:00 PM
 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).

14365

Thursday 3/28 4:10 PM

Ken Ono, Emory University

Jensen–Polya Program for the Riemann Hypothesis and Related Problems
 Ken Ono, Emory University
 Jensen–Polya Program for the Riemann Hypothesis and Related Problems
 03/28/2019
 4:10 PM  5:00 PM
 C304 Wells Hall
In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xifunction. This hyperbolicity had only been proved for degrees $d=1,2,3$. We prove the hyperbolicity of all (but possibly finitely many) the Jensen polynomials of every degree $d$. Moreover, we establish the outright hyperbolicity for all degrees $d< 10^{26}$. These results follow from an unconditional proof of the "derivative aspect" GUE distribution for zeros. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.

15385

Thursday 4/18 4:10 PM

Emmy Murphy, Northwestern University

Flexibility in contact and symplectic geometry
 Emmy Murphy, Northwestern University
 Flexibility in contact and symplectic geometry
 04/18/2019
 4:10 PM  5:00 PM
 C304 Wells Hall
We discuss a number of $h$principle phenomena which were recently discovered in the field of contact and symplectic geometry. In generality, an $h$principle is a method for constructing global solutions to underdetermined PDEs on manifolds by systematically localizing boundary conditions. In symplectic and contact geometry, these strategies typically are well suited for general constructions and partial classifications. Some of the results we discuss are the characterization of smooth manifolds admitting contact structures, high dimensional overtwistedness, the symplectic classification of flexible Stein manifolds, and the construction of exotic Lagrangians in $C^n$.

14350

Thursday 4/25 3:10 PM

André Neves, University of Chicago

Recent progress on existence of minimal surfaces
 André Neves, University of Chicago
 Recent progress on existence of minimal surfaces
 04/25/2019
 3:10 PM  4:00 PM
 C304 Wells Hall
A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of MarquesNeves and Song. I will survey the history of the problem and the several contributions made.
