|Professor: Louis H Kauffman, Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago|
|Title:Graphical Invariants of Knots and Links|
|Abstract:This talk is joint work with Qingying Deng. We generalize the signed Tutte polynomial relationship with the Kauffman bracket model of the Jones polynomial to a new polynomial defined on signed cyclic graphs (graphs with signed edges and cyclic orders at the vertices) and show how these graphs are codings for checkerboard colorable virtual links. We show how virtualization of classical links corresponds to simple operations on the planar signed cyclic graphs. We relate our new polynomial invariant to both the bracket polynomials for virtual knots and links and to the Bollobas-Riordan polynomial.|
|Professor Yanlai Chen , Dept of Mathematics, University of Massachusetts, Dartmouth|
|Title:Ultra-efficient Reduced Basis Method and Its Integration with Uncertainty Quantification|
|Abstract: Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RBM solution is maintained through a rigorous a posteriori error estimator whose efficient development is critical and involves fast eigensolvers. After giving a brief introduction of the RBM/RCM, this talk will show our recent work on significantly delaying the curse of dimensionality for uncertainty quantification, and new fast algorithms for speeding up the offline portion of the RBM/RCM by around 6-fold.|
|Professor Seokchan Kim, Dept of Mathematics, ChangWon National University, Changwon, Korea|
|Title:FEM to compute Numerical Solution of PDEs with Corner Singularities using SIF|
|Abstract: We consider the Poisson equation with homogeneous Dirichlet boundary condition defined on non convex polygonal domain with one re-entrant corner. Solution of such equation has singular behavior near that re-entrant corner and can be expressed as a sum of the regular part and the singular part. The coefficient of the singular part is called ‘the Stress Intensity Factor. The talk is to introduce a new method to obtain an accurate numerical solution for the Poisson Equation with corner singularities using the Stress Intensity Factor.|
|Ching-Shan Chou, Department of Mathematics, The Ohio State University|
|Title:Cell signaling, cell morphogenesis and cell-cell communication|
|Abstract: Cell-to-cell communication is fundamental to biological processes which require cells to coordinate their functions. In this talk, we will present the first computer simulations of the yeast mating process, which is a model system for investigating proper cell-to-cell communication. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the alpha-factor protease Bar1, and the regulation of sensing sensitivity.|
|Valery Romanovski, Center for Applied Mathematics and Theoretical Physics|
|Title: Some algebraic tools for investigation of systems of ODEs|
|Abstract: We give an introduction to algorithms of the elimination theory and methods for solving polynomial systems and show how they can be used for the qualitative investigation of autonomous systems of ordinary differential equations. We then apply them to study the May-Leonard system which models some ecological and chemical processes.|
|Luan Hoang, Department of Mathematics, Texas Tech University|
|Title: Non-Darcy flows in heterogeneous porous media|
|Abstract: The most common equation to describe fluid flows in porous media is the Darcy law. However, this linear equation is not valid in many situations, particularly, when the Reynolds number is large or very small.
In the first part of this talk, we survey the Forchheimer models and their generalizations for compressible fluids in heterogeneous porous media. The Forchheimer coefficients in this case are functions of the spatial variables. We derive a parabolic equation for the pressure which is both singular/degenerate in the spatial variables, and degenerate in the pressure’s gradient.
In the second part, we model different flow regimes, namely, pre-Darcy, Darcy and post-Darcy, which may be present in different portions of a porous medium. To study these complex flows, we use a single equation of motion to unify all three regimes. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure’s gradient is either small or large.
We estimate the pressure and its gradient for all time in terms of the initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on the initial and boundary data, and the structural stability for the equations are established.