## Seminars and Colloquia by Series

Monday, March 24, 2014 - 11:00 , Location: Skiles 005 , Professor Michael Li , Univeristy of Alberta , , Organizer: Yingfei Yi
Many complex models from science and engineering can be studied in the framework of coupled systems of differential equations on networks. A network is given by a directed graph. A local system is defined on  each vertex, and directed edges represent couplings among vertex  systems. Questions such as stability in the large, synchronization,  and complexity in terms of dynamic clusters are of interest. A more  recent approach is to investigate the connections between network  topology and dynamical behaviours. I will present some recent results  on the construction of global Lyapunov functions for coupled systems  on networks using a graph theoretic approach, and show how such  a construction can help us to establish global behaviours of compelx  models.
Monday, February 24, 2014 - 11:00 , Location: Skiles 005 , Aynur Bulut , Univ. of Michigan , Organizer: Rafael de la Llave
In this talk we will discuss recent work, obtained in collaboration with Jean Bourgain, on new global well-posedness results along Gibbs measure evolutions for the radial nonlinear wave and Schr\"odinger equations posed on the unit ball in two and three dimensional Euclidean space, with Dirichlet boundary conditions. We consider initial data chosen according to a Gaussian random process associated to the Gibbs measures which arise from the Hamiltonian structure of the equations, and results are obtained almost surely with respect to these probability measures. In particular, this renders the initial value problem supercritical in the sense that there is no suitable local well-posedness theory for the corresponding deterministic problem, and our results therefore rely essentially on the probabilistic structure of the problem. Our analysis is based on the study of convergence properties of solutions. Essential ingredients include probabilistic a priori bounds, delicate estimates on fine frequency interactions, as well as the use of invariance properties of the Gibbs measure to extend the relevant bounds to arbitrarily long time intervals.
Monday, February 17, 2014 - 11:00 , Location: Skiles 006 , Chunhua Shan , School of Mathematics, Georgia Institute of Technology , Organizer:
In 1994, Dumortier, Roussarie and Rousseau launched a program aiming at proving the ﬁniteness part of Hilbert’s 16th problem for the quadratic system. For the program, 121 graphics need to be proved to have ﬁnite cyclicity. In this presentation, I will show that 4 families of HH-graphics with a triple nilpotent singularity of saddle or elliptic type have finite cyclicity. Finishing the proof of the cyclicity of these 4 families of HH-graphics represents one important step towards the proof of the finiteness part of Hilbert’s 16th problem for quadratic systems. This is a joint work with Professor Christiane Rousseau and Professor Huaiping Zhu.
Friday, February 7, 2014 - 15:00 , Location: Skiles 06 , Alex Haro , Univ. of Barcelona , Organizer: Rafael de la Llave
This talk is devoted to quasi-periodic Schrödinger operators beyond theAlmost Mathieu, with more general potentials and interactions. The  linksbetween the spectral properties of these operators and the dynamicalproperties of the associated quasi-periodic linear skew-products rule thegame. In particular, we present a Thouless formula  and some consequencesof Aubry duality. This is a joint work with Joaquim Puig~
Monday, February 3, 2014 - 11:00 , Location: Skiles05 , Marta Canadell , Univ. of Barcelona , Organizer: Rafael de la Llave
We present a KAM-like theorem for the existence of quasi-periodic tori with a prescribed Diophantine rotation for a discrete family of dynamical system. The theorem is stated in an a posteriori format, so it can be used to validate numerical computations. The method of proof provides an efficient algorithm for computing quasi-periodic tori. We also present implementations of the algorithm, illustrating them throught several examples and observing different mechanisms of breakdown of qp invariant tori. This is a joint work with Alex Haro.
Wednesday, January 22, 2014 - 11:00 , Location: Skiles 006 , Renato Calleja , IIMAS UNAM , Organizer: Rafael de la Llave
We present a numerical  study of the dynamics of a state-dependent delay equation with two state dependent delays that are linear in the state. In particular, we study some of the the dynamical behavior driven by the existence of two-parameter families of invariant tori. A formal normal form analysis predicts the existence of torus bifurcations and the appearance of a two parameter family of stable invariant tori. We investigate the dynamics on the torus thought a Poincaré section. We find some boundaries of Arnold tongues and indications of loss of normal hyperbolicity for this stable family. This is joint work with A. R. Humphries and B. Krauskopf.
Friday, January 17, 2014 - 11:05 , Location: Skiles 005 , , University of British Columbia, Vancouver. , Organizer:
(Joint work with A. Avila and S. Jitomirskaya). An analytic, complex, one-frequency cocycle is given by a pair $(\alpha,A)$ where $A(x)$ is an analytic and 1-periodic function that maps from the torus $\mathbb(R) / \mathbb(Z)$ to the complex $d\times d$ matrices and $\alpha \in [0,1]$ is a frequency. The pair is interpreted as the map $(\alpha,A)\,:\, (x,v) \mapsto (x+\alpha), A(x) v$. Associated to the iterates of this map are (averaged) Lyapunov exponents $L_k(\alpha,A)$ and an Osceledets filtration. We prove joint-continuity in $(\alpha,A)$ of the Lyapunov exponents at irrational frequencies $\alpha$, give a criterion for domination and prove that for a dense open subset of cocycles, the Osceledets filtration comes from a dominated splitting which is an analogue to the Bochi-Viana Theorem.
Tuesday, January 7, 2014 - 15:05 , Location: Skiles 005 , Xifeng Su , Beijing Normal University , Organizer:
We consider the semi-linear elliptic PDE driven by the fractional Laplacian: \begin{equation*}\left\{%\begin{array}{ll}    (-\Delta)^s u=f(x,u) & \hbox{in $\Omega$,} \\    u=0 &  \hbox{in $\mathbb{R}^n\backslash\Omega$.} \\\end{array}% \right.\end{equation*}An $L^{\infty}$ regularity result is given, using De Giorgi-Stampacchia iteration  method.By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais-Smale condition without Ambrosetti-Rabinowitz condition for non-local elliptic equations is proved. Two non-trivial solutions are given under some weak hypotheses. Non-local elliptic equations with concave-convex nonlinearities are also studied, and existence of at least six solutions are obtained. Moreover, a global result of Ambrosetti-Brezis-Cerami type is given, which  shows that the effect of  the parameter $\lambda$ in the nonlinear term changes considerably the nonexistence, existence and multiplicity of solutions.
Monday, January 6, 2014 - 11:00 , Location: Skiles 005 , James Meiss* , Department of Applied Mathematics, University of Colorado, Boulder , Organizer:
Synchronization of coupled oscillators, such as grandfather clocks or metronomes, has been much studied using the approximation of strong damping in which case the dynamics of each reduces to a phase on a limit cycle. This gives rise to the famous Kuramoto model.  In contrast, when the oscillators are Hamiltonian both the amplitude and phase of each oscillator are dynamically important. A model in which all-to-all coupling is assumed, called the Hamiltonian Mean Field (HMF) model, was introduced by Ruffo and his colleagues. As for the Kuramoto model, there is a coupling strength threshold above which an incoherent state loses stability and the oscillators synchronize. We study the case when the moments of inertia and coupling strengths of the oscillators are heterogeneous. We show that finite size fluctuations can greatly modify the synchronization threshold by inducing correlations between the momentum and parameters of the rotors. For unimodal parameter distributions, we find an analytical expression for the modified critical coupling strength in terms of statistical properties of the parameter distributions and confirm our results with numerical simulations. We find numerically that these effects disappear for strongly bimodal parameter distributions. *This work is in collaboration with Juan G. Restrepo.
Monday, November 25, 2013 - 16:00 , Location: Skiles 05 , Rodrigo Trevino , Cornell Univ./Tel Aviv Univ. , Organizer: Rafael de la Llave
The three objects in the title come together in the study of ergodic properties of geodesic flows on flat surfaces. I will go over how these three things are intimately related, state some classical results about the unique ergodicity of translation flows and present new results which generalize much of the classical theory and also apply to non-compact (infinite genus) surfaces.