# Applied mathematics

Our subject spans a huge range, from proving theorems to making policy. Over the course of 2012-13, we will try to represent the diverse interests of our community in a series of workshops. In addition, we will have a working group on nonlinear problems, meeting every Thursday in Room 5417 at 2 PM during the term. A seminar series also takes place in Room 3209 on Fridays from 4 PM to 5 PM (see below for featured talks this fall).

All workshops will be held in the Science Center, Room 4102. Events are open to the scientific community, but we ask that you email its@gc.cuny.edu to register, so that we can provide the right amount of food and coffee (!).

Some funds are available to facilitate the participation of students and postdoctoral fellows from around the New York metropolitan area. Please email its@gc.cuny.edu if you would like to take advantage of this opportunity.

### Archive of Previous Symposia and Seminars

**November 8, 2013**

Nicholas Spizzirri, The Graduate Center CUNY. "Fourth-Order Time Stepping Schemes and the Kuramoto-Sivashinsky Equation."

**November 1, 2013**

Edward Farnum, Kean University. "Short Pulse Perturbation Theory."

**October 31, 2013**

Meijun Zhu, University of Oklahoma. "Reversed Hardy-Littlewood-Sobolev inequality."

**October 25, 2013**

Andrew Poje, College of Staten Island CUNY. "Observations of Ocean Turbulence at the Submesoscales: Classic Similarity Theory in GLAD Surface Drifter Data."

**October 24, 2013**

Meijun Zhu, University of Oklahoma. "Sharp Sobolev inequalities in global analysis."

**October 17, 2013**

Luca Capogna, Worcester Polytechnic Institute. "Smoothness of isometries between subRiemannian manifolds."

**October 11, 2013**

Vlad Vicol, Princeton University. "Long-time behavior of forced critical SQG."

**October 10, 2013**

Hui Wang, Rutgers University. "A semilinear singular Sturm-Liouville equation involving measure data."

**September 26, 2013**

Radoslaw Wojciechowski, York College CUNY. "Spectral properties and intrinsic metrics on infinite graphs."

**September 13, 2013**

R. Prashanth, Tata Institute, Bagalore, India. "Simplicity for Rayleigh quotient."

**September 5, 2013**

Ratnasingham Shivaji, University of North Carolina. "Uniqueness of Nonnegative Radial Solutions for Semipositone Problems on Exterior Domains."

May 2, 2013 -

**Finite-time blow-up vs. global regularity for equations of fluid motion**

Global well-posedness of an inviscid three-dimensional Pseudo-Hasegawa-Mima-Charney-Obukhov Model - Edriss S. Titi, Weizmann Institute of Science

Water waves and time-space resonances - Pierre Germain, Courant Institute, NYU

Asymptotic stability of mild solutions to the Navier-Stokes equations - Maria Schonbek, UC-Santa Cruz

Profile decompositions and the Navier-Stokes equations - Isabelle Gallagher, Université Paris-Diderot

Relative entropy applied to the stability of shocks for compressible fluids, and applications to asymptotic analysis - Alexis Vasseur, University of Texas

TBA - Peter Constantin, Princeton University

**25 April 25, 2013**-

**Topics in Numerical Analysis**

TBA

**March 14, 2013**-

**Pi-Day with Chern-Simons Theory**

Asymptotic behavior of solutions to the sigma_k-Yamabe

equation near isolated singularities - Zheng-Chao Han, Rutgers University

Uniqueness of topological solutions for a Chern-Simons model with two Higgs fields and two Gauge fields on a Torus - Jyotshana Prajapat, Petroleum Institute, Abu-Dhabi

Vortices in the Maxwell-Chern-Simons-Higgs Equations - Daniel Spirn, University of Minnessota

TBA - Gabriella Tarantello, University of Rome II

**February 28, 2013**-

**Perspective of the Ricci flow**

Problems in combinatorial and numerical Ricci flow - David Glickenstein, University of Arizona

Type II singularities of Ricci flow - Dan Knopf, University of Texas at Austin

Kahler-Ricci flow and birational surgery - Jian Song, Rutgers University

On normalized Ricci flow and smooth structures on 4-manifolds - Ioana Suvaina, Vanderbilt University

**December 6, 2012 - Symposium on Harmonic maps**

Almost complex surfaces in the product of two 3-spheres - John Bolton, Durham University, UK

A Harmonic Map Problem with Partial Free Boundary conditions - Fang-Hua Lin, Courant Institute (NYU), New-York

The analysis if conformal-minimal surfaces - Tristan Riviere, ETH, Switzerland

Harmonic maps into exceptional symmetric spaces - John C. Wood, University of Leeds, UK

**November 8, 2012 - Recent Progress in General Relativity**

The Geometry of Statics Spacetimes - Carla Cederbaum, Duke University

An isoperimetric concept for quasilocal mass - Gerhard Huisken, Max-Planck Institute for Gravitational Physics, Potsdam

TBA - Shadi Tahvildar-Zadeh, Rutgers University

Sharp Minkowski type inequality in the AdS-Schwarzschild space and the Penrose inequality for collapsing shells - Mu-Tao Wang, Columbia University, New-York

**April 27, 2012 - Seminar with Ionut Florescu - Stevens Institute of Technology, New Jersey**

Solving systems of PIDE's coming from regime switching jump diffusion models

In this talk we consider an underlying model where constant parameters are switching according to a continuous time Markov process. The times of switch are modeled using a Cox process. In addition the model features jumps. We examine the option pricing problem when the stock process follows this process and we find that a tightly coupled system of partially integro-differential equations needs to be solved. We exemplify the solution on several case studies. We also analyze two types of jump distributions the log double exponential due to Kou and a new distribution which we call a log normal mixture which seems to be useful in precisely modeling the jumps and distinguishing them from sampled variability.

**April 26, 2012 - Hyperbolic conservation laws and applications**

Nash equilibria for traffic flow on networks - Alberto Bressan, Penn State University

TVD fields for pairs of conservation laws and the p-system - Kris Jenssen, Penn State University

Using geometric singular perturbation theory to understand singular shocks - Barbara Lee Keyfitz, The Ohio State University

Existence results for the Euler equations of compressible fluids in one space dimension - Philippe G. LeFloch, Universite Paris VI (Pierre et Marie Curie), and CNRS

A partial hodograph transform at a sonic curve for the Euler system - Yuxi Zheng, Yeshiva University

**April 20, 2012 - Seminar with Peter Gordon - New Jersey Institute Technology, New Jersey - Local kinetics and self-similar dynamics of morphogen gradients**

Some aspects of pattern formation in developing embryos can be described by nonlinear reaction-diffusion equations. An important class of these models accounts for diffusion and degradation of a locally produced single chemical species and describe formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. At long times, solutions of such models approach a steady state in which the concentration decays with distance from the source of production. I will present our recent results that characterize the dynamics of this process. These results provide an explicit connection between the parameters of the problem and the time needed to reach a steady state value at a given position. I will also show that the long time behavior of such models, in certain cases, can be described in terms of very singular self-similar solutions. These solutions are associated with a limit of infinitely large signal production strength.

This is a joint work with: C. Muratov, S. Shvartsman, C. Sample and A.Berezhkovskii.

**March 30, 2012 - Seminar with Christina Mouser - William Paterson University, New Jersey - The Control of Frequency of a Conditional Oscillator Simultaneously Subjected to Multiple Oscillatory Inputs**

The gastric mill network of the crab Cancer borealis is an oscillatory network with frequency ~ 0.1 Hz. Oscillations in this network require neuromodulatory synaptic inputs as well as rhythmic inputs from the faster (~ 1 Hz) pyloric neural oscillator. We study how the frequency of the gastric mill network is determined when it receives rhythmic input from two different sources but where the timing of these inputs may differ. We find that over a certain range of the time difference one of the two rhythmic inputs plays no role what so ever in determining the network frequency while in another range, both inputs work together to determine the frequency. The existence and stability of periodic solutions to model sets of equations are obtained analytically using geometric singular perturbation theory. The results are validated through numerical simulations. Comparisons to experiments are also presented.

**March 23, 2012 - Seminar with Levent Kurt - The Graduate Center of CUNY - The Higher-Order Short Pulse Equation**

We derive an equation, the higher-order short pulse equation (HSPE), from the nonlinear wave equation to capture the dynamics of ultra-short solitons in cubic nonlinear media using both multiple scaling technique and re-normalization group. The multiple scaling derivation will be presented. The numerical solution of the HSPE as the exact one- and two-soliton solutions of the short pulse equation (SPE) being the initial conditions and its comparison to the numerical solutions of the SPE and original equation will also be demonstrated.

**March 22, 2012 - Aggregation models in biology**

Blowup in multidimensional aggregation equations - José Antonio Carrillo de la Plata, Universitat Autònoma de Barcelona

Nonlocal equations - Peter Constantin, Princeton University

Dispersal in Heterogeneous Landscapes - Yuan Lou, Ohio State University

Measured valued solutions for the Keller-Segel system - Juan J.L. Velázquez, Hausdorff Center for Mathematics, Bonn

**March 09, 2012 - Seminar with Keith Promislow - Michigan State University, East Lansing, MI - Network Formation and Ion Conduction in Ionomer Membranes**

Many important processes in the physical world can be described as a gradient (overdamped) flow of a variational energy. We present a broad formalism for the generation of new classes of higher-order variational energies with a physically motivated structure. In particular we reformulate the Cahn-Hilliard energy, which is well know to describe the surface area of mixtures, into a higher-order model of interfacial energy for mixtures of charged polymers (ionomers) with solvent. These materials are important as selectively conductive membrane separators in a wide variety of energy conversion devices, including polymer electrolyte membrane fuel cells, Lithium ion batteries, and dye sensitized solar cells.

Our reformulated energy, called the Functionalized Cahn-Hilliard (FCH) energy, captures elastrostatic interactions between the charged groups and the complex entropic effects generated by solvent-ion interactions, and allows us to unfold the bilayer and pore networks formed by the solvent phase imbibed into the polymer matrix. We discuss sharp interface reductions of the FCH energy, its gradient flows, and sharp interface reductions of the gradient flows that give rise to higher-order curvature driven flows. We also describe extensions to models that couple to ionic transport and as well as to multiphase models suitable to describe a wide range of membrane casting processes.

References

[1] N. Gavish, J. Jones, Z. Xu, A. Christlieb, K. Promislow, submitted to Polymers Special issue on Thin Membranes (2012).

[2] H. Zhang and K. Promislow, Critical points of Functionalized Lagrangians, Discrete and Continuous Dynamical Systems, A to appear.

[3] N. Gavish, G. Hayrapetyan, Y. Li, K. Promislow, Physica D, 240: 675-693 (2011).

[4] K. Promislow and B. Wetton, PEM fuel cells: A Mathematical Overview, Invited Review Paper to SIAM Applied Math. 70: 369-409 (2009)

**February 25, 2012 - Recent developments in minimal surfaces**

Constant mean curvature spheres in homogeneous three dimensional manifolds - William Meeks, University of Massachusetts

Dynamics and singularities of mean curvature flow - William Minicozzi, Johns Hopkins University

Maximal surfaces in the anti-de Sitter space and the universal Teichmüller space - Jean-Marc Schlenker, Universite Toulouse III

Polynomial Pick forms for affine spheres over the complex plane - Michael Wolf, Rice University

**February 17, 2012 - Seminar with Xing Zhong - New Jersey Institute Technology, New Jersey - Threshold Phenomena for Symmetric Decreasing Solutions of Reaction-Diffusion Equations**

We study the Cauchy problem for nonlinear reaction-diffusion equation (u_t = u_xx + f(u), u(x,0) = \phi (x), x \in R, t > 0), with different nonlinearities. By using energy functional and exponentially weighted functional, for symmetric decreasing initial conditions, we prove one-to-one relation between long time behavior of solution and limit value of energy. Then we study the threshold phenomena. This is a joint work with Cyrill Muratov

**October 27, 2011 - Recent advances in 3D Euler and Navier-Stokes equations**

The interplay between computation and analysis in the study of 3D incompressible flows - Tom Hou, California Institute of Technology

An alternative approach to regularity for the Navier-Stokes equations in critical spaces - Gabriel Koch, University of Sussex

Inviscid limit of the free boundary Navier-Stokes system - Nader Masmoudi, Courant Institute NYU

Drift diffusion equations with fractional diffusion and the surface quasi-geostrophic equation - Alexis Vasseur, University of Texas at Austin

**September 22, 2011 - Recent trends in nonlinear PDEs**

Saddle-shaped solutions to the scalar Ginzburg-Landau equations - Xavier Cabre, ICREA & Universitat Politecnica de Catalunya

Uniqueness and nondegeneracy of ground states for non-local equations in dimension one - Rupert Franck, Princeton University

Entire solutions of the Allen-Cahn equation - Changfeng Gui, University of Connecticut

An optimal partition problem for the Dirichlet eigenvalues - Fang-Hua Lin, Courant Institute NYU

**March 31, 2011 - Vortex dynamics & non-Equilibrium statistical mechanics**

Michael Kiessling (Rutgers University), David Dritschel (Univ. St. Andrews), Jeffrey Weiss (Univ. Colorado)

**March 09, 2011 - Keith Promislow - Michigan State University, East Lansing, MI - Network Formation and Ion Conduction in Ionomer Membranes**

Many important processes in the physical world can be described as a gradient (overdamped) flow of a variational energy. We present a broad formalism for the generation of new classes of higher-order variational energies with a physically motivated structure. In particular we reformulate the Cahn-Hilliard energy, which is well know to describe the surface area of mixtures, into a higher-order model of interfacial energy for mixtures of charged polymers (ionomers) with solvent. These materials are important as selectively conductive membrane separators in a wide variety of energy conversion devices, including polymer electrolyte membrane fuel cells, Lithium ion batteries, and dye sensitized solar cells. Our reformulated energy, called the Functionalized Cahn-Hilliard (FCH) energy, captures elastrostatic interactions between the charged groups and the complex entropic effects generated by solvent-ion interactions, and allows us to unfold the bilayer and pore networks formed by the solvent phase imbibed into the polymer matrix. We discuss sharp interface reductions of the FCH energy, its gradient flows, and sharp interface reductions of the gradient flows that give rise to higher-order curvature driven flows. We also describe extensions to models that couple to ionic transport and as well as to multiphase models suitable to describe a wide range of membrane casting processes.

**February 17, 2011 -**

**Seminar with Xing Zhong - New Jersey Institute Technology, New Jersey**-

**Threshold Phenomena for Symmetric Decreasing Solutions of Reaction-Diffusion Equations**

We study the Cauchy problem for nonlinear reaction-diffusion equation (u_t = u_xx + f(u), u(x,0) = \phi (x), x \in R, t > 0), with different nonlinearities. By using energy functional and exponentially weighted functional, for symmetric decreasing initial conditions, we prove one-to-one relation between long time behavior of solution and limit value of energy. Then we study the threshold phenomena. This is a joint work with Cyrill Muratov.

**Fall 2011 - Seminar with**

**Philippe LeFloch, Universite Paris VI - Pierre et Marie Curie -**

**Undercompressible shocks and moving phase boundaries**

**Fall 2011 - Seminar with Pam Cook, University of Delaware (with Lin Zhou, New York City College of TEchnology and Gareth McKinley, Massachusetts Institute of Technology) -**

**Complex (wormlike micellar) fluids: Shear banding and inertial effects**