Georgia Institute of Technology College of Sciences

In this talk we will present some recent results about the matrix representation of the multiplication operator in terms of a basis of either orthogonal polynomials (OPUC) or orthogonal Laurent polynomials (OLPUC) with respect to a nontrivial probability measure supported on the unit circle. These are the so called GGT and CMV matrices.When spectral linear transformations of the measure are introduced, we will find the GGT and CMV matrices associated with the new sequences of OPUC and OLPUC, respectively. A connection with the QR factorization of such matrices will be stated. A conjecture about the generator system of such spectral transformations will be discussed.Finally, the Lax pair for the GGT and CMV matrices associated with some special time-depending deformations of the measure will be analyzed. In particular, we will study the Schur flow, which is characterized by a complex semidiscrete modified KdV equation and where a discrete analogue of the Miura transformation appears. Some open problems for time-depending deformations related to spectral linear transformations will be stated.This is a joint work with K. Castillo (Universidad Carlos III de Madrid) and L. Garza (Universidad Autonoma de Tamaulipas, Mexico).

Couette flows are shear flows with a linear velocity profile. Known by Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time, for L^2 vorticity. It is interesting to know if the perturbed Euler flow near Couette tends to a nearby shear flow. Such problems of nonlinear inviscid damping also appear for other stable flows and are important to understand the appearance of coherent structures in 2D turbulence. With Chongchun Zeng, we constructed non-parallel steady flows arbitrarily near Couette flows in H^s (s<3/2) norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H^s (s<3/2) norm. We also showed that in (vorticity) H^s (s>3/2) neighborhood of Couette flows, the only steady structures (including travelling waves) are stable shear flows. This suggests that the long time dynamics near Couette flows in (vorticity) H^s (s>3/2) space might be simpler. Similar results will also be discussed for the problem of nonlinear Landau damping in 1D electrostatic plasmas.

Tissue remodeling involves the activation of proteases, enzymes capable of degrading the structural proteins of tissue and organs. The implications of the activation of these enzymes span all organ systems and therefore, many different disease pathologies, including cancer metastasis. This occurs when local proteolysis of the structural extracellular matrix allows for malignant cells to break free from the primary tumor and spread to other tissues. Mathematical models add value to this experimental system by explaining phenomena difficult to test at the wet lab bench and to make sense of complex interactions among the proteases or the intracellular signaling changes leading to their expression. The papain family of cysteine proteases, the cathepsins, is an understudied class of powerful collagenases and elastases implicated in extracellular matrix degradation that are secreted by macrophages and cancer cells and shown to be active in the slightly acidic tumor microenvironment. Due to the tight regulatory mechanisms of cathepsin activity and their instability outside of those defined spaces, detection of the active enzyme is difficult to precisely quantify, and therefore challenging to target therapeutically. Using valid assumptions that consider these complex interactions we are developing and validating a system of ordinary differential equations to calculate the concentrations of mature, active cathepsins in biological spaces. The system of reactions considers four enzymes (cathepsins B, K, L, and S, the most studied cathepsins with reaction rates available), three substrates (collagen IV, collagen I, and elastin) and one inhibitor (cystatin C) and comprise more than 30 differential equations with over 50 specified rate constants. Along with the mathematical model development, we have been developing new ways to quantify proteolytic activity to provide further inputs. This predictive model will be a useful tool in identifying the time scale and culprits of proteolytic breakdown leading to cancer metastasis and angiogenesis in malignant tumors.