We consider convex optimization problems with stochastic order constraint. In these problems, utility functions play the role of Lagrange multipliers associated with the stochastic order constraint. We propose a penalty based method where the penalty function is defined on the space of utility functions. This is an ongoing project with Darinka Dentcheva. I will present some preliminary results.

In this talk, I will present recent progress of our research on the periodic‐parabolic logistic equation. First, I will give a brief review of the history of the logistic equation and the existing results. Then I will report our work on the periodic‐parabolic logistic equation with spatial and temporal degeneracies. More specifically, we examine the effects of various natural spatial and temporal degeneracies of the carrying capacity function on the long‐time dynamical behavior of the positive solution. Our analysis leads to a new eigenvalue problem for periodic‐parabolic operators over a varying cylinder and certain parabolic boundary blow‐up problems not known before. Our investigation shows that the temporal degeneracy causes a fundamental change of the dynamical behavior only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in the equation, the spatial degeneracy always induces fundamental changes of the dynamical behavior, though such changes differ significantly according to whether the temporal degeneracy is present or not.

Abstract: Resonances are important in the study of transient phenomena associated with the wave equation, especially in understanding the large time behavior of the solution to the wave equation when radiation losses are small. In this talk, I will present recent studies on the scattering resonances for photonic structures and Schrodinger operators. In particular, for a finite one dimensional photonic crystal with a defect, it is shown that the near bound-state resonances converge to the point spectrum of the infinite structure with an exponential rate. A similar result holds for the Schrodinger operator with a potential that is a low-energy well surrounded by a thick barrier. I will also discuss non near bound-state resonances for both cases briefly. This is a joint work with Prof. Fadil Santosa at the IMA.

Abstract: For some geophysical flows in which computational efficiency is of paramount importance, simplified mathematical models are central. For example, in climate modeling the Quasi-Geostrophic Equations (QGE), are commonly used in the numerical simulation of large scale wind-driven ocean circulations. Due to the requisite of long time integration in climate modeling, even for the simplified model, fast and accurate numerical algorithms are still desired. In this talk, we will pursue this direction and discuss efficient numerical approaches for QGE.

Abstract: Many basic equations in physics can be written in the form of conservation law. However, as it is common in the field of PDEs and SPDEs, classical or strong solutions do not exist in general and, on the other hand, weak solutions are not unique. The notion of kinetic formulation and kinetic solution turns out to be a very convenient tool to overcome these difficulties.

In this talk, I will present a well-posedness result for degenerate parabolic stochastic conservation laws and, in the case of hyperbolic stochastic conservation laws, discuss the hydrodynamic limit of an approximation in the sense of Bhatnagar-Gross-Krook (BGK-like for short).

Abstract: Identifying clusters of similar objects in data plays a significant role in a wide range of applications such as information retrieval, pattern recognition, computational biology, and image processing. A classical approach to clustering is to model a given data set as a graph whose nodes correspond to items in the data and edges indicate similarity between the corresponding endpoints, and then try to decompose the graph into dense subgraphs. Unfortunately, finding such a decomposition usually requires solving an intractable combinatorial optimization problem. In this talk, I will discuss a convex relaxation approach to identifying these dense subgraphs with surprising recovery properties: if the data is randomly sampled from a distribution of “clusterable” data, then the correct partition of the data into clusters can be exactly recovered from the optimal solution of the convex relaxation with high probability.

Abstract: Geometric singular perturbation theory is used to analyze systems of ordinary differential equations with much different time scales. Those kinds of systems are largely used in math biology. It is interesting to see what happens if noise being taken into consideration.

In this talk, we show the existence of random invariant manifolds and families of random invariant manifolds of a singular perturbation systems of ordinary di erential equations with uniformly small real noise. Then apply these to prove a random version of inclination theorem.

Abstract: One of the original motivations for the study of Stochastic PDEs traces it's origins to the study of turbulence. As such a better understanding of stochastic versions of the fundamental equations of fluid flow remain an important direction in applied analysis.

In this talk we discuss some recent results concerning the Stochastic Euler equations as well as some related issues concerning inviscid limits.

The periodic generalized Korteweg-de Vries equation (gKdV) can be interpreted as an infinite-dimensional Hamiltonian system. Some properties of finite-dimensional Hamiltonian dynamics can be extended to infinite dimensions; for example, the invariance of the Gibbs measure under the flow. In this talk we present invariance of the Gibbs measure for the (gauge-transformed) periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized.

In this talk we will discuss qualitative properties of positive solutions of general parabolic partial differential equations. Specifically, we will indicate how the shape of the domain influences the asymptotic properties of solutions of parabolic equations, that is, their behavior for large times. We show that regardless of the initial conditions, the functions in the omega-limit set have specific shapes, determined by the symmetry properties of the domain. As an application we prove that the shape of the domain simplifies the dynamics of solutions and it enables us to prove nontrivial convergence results. We will see that nonnegative, nontrivial equilibria which are not positive play an important role in these results. If time permits, we will discuss existence and non-existence results for such equilibria.

We discuss a numerical approach for simulating inertial manifolds for a class of stochastic evolutionary equations with multiplicative noise. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving the backward-forward stochastic system, and an ensemble of graphs representing the inertial manifold is consequently obtained. By a simple illustrative example, we then show the dimension of the original full system can be reduced through this technique.

In this talk we will discuss asymptotically asymmetric slowly non-dissipative reaction-diffusion equations, also known as grow-up equations, and recent developments with regards to the long-time behavior of solutions and the non-compact global attractor structure for these PDEs. I will present some of the background of this problem, and then introduce recent research utilizing Fučik spectra, nodal property methods, the time map, numerical continuation methods, and Conley index at infinity to determine both the bounded portion of the non-compact global attractor as well as the portion contained within the so-called Sphere at Infinity.

Additionally, we will discuss the transfinite heteroclinic structures which connect these two regions and why asymmetry adds significant complexity to the task of solving the connection problem for these heteroclinics. This talk is based on joint work with Kristen Moore (U. Mich.) and Juliette Hell (F.U. Berlin).

We consider ultra-short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. We will derive short-pulse equation (SPE) u_{xt} = u+1/6 (u^3)_{xx} in frequency band gaps. We will discuss the connection of SPE with the nonlinear Schrodinger equation, SPE in its characteristic coordinate and the transformation to sine-Gordon equation, and robustness of various solutions emanating from the sine-Gordon equation and their periodic generalizations. Finally, we will discuss the wellposedness of generalized Ostrovsky equation u_{xt} = u+1/6 (u^p)_{xx} with small initial data.

Loosely speaking, a canard is a special solution of a singularly perturbed system which closely follows an attracting slow manifold and then subsequently follows a repelling slow manifold for a long time. Canard orbits are ubiquitous in slow-fast systems where normal hyperbolicity is lost at a fold in the critical manifold, and often serve as intermediaries in parameter regimes connecting qualitatively different dynamical behavior. Techniques based on the blow-up method for singular points in the plane have proved useful for the study of canards, and have been extended in a variety of ways for higher dimensional systems.

In this talk, I'll briefly present the classical theory of planar canards and then go on to introduce recent results for a generalization called the torus canard. The torus canard was first discovered in the transition region from bursting to rapid spiking in a mathematical model of a cerebellar Purkinje cell, and has since been observed in many other models from the field of mathematical neuroscience. These and other applications of canard or generalized canard phenomena will be highlighted throughout the talk.

Conformally symplecti systems send a symplectic form into a multiple of itself. The conservation of this geometric structure provides identities that we use to prove "a-posteriori" theorems that show the existence of a solution close to an approximate solution. We also use the identities to obtain very efficient algorithms with small storage and operation counts.

It is a classical problem to study incidences between a finite number of points and a finite number of geometric objects. In this presentation, we see that continuous incidence theory can be used to derive a number of results in geometry, geometric measure theory, and analytic number theory.

The applications to geometry include a fractal variant of the regular value theorem. The applications to geometric measure theory include a generalization of Falconer distance problem in which we prove that a compact subset of R^d of sufficiently large Hausdorff dimension determines a positive proportion of all (k+1)-configurations described by certain restrictions.

The applications to Number Theory include counting integer lattice points neighborhoods of variable coefficient families of surfaces.