Georgia Institute of Technology College of Sciences

Cathepsins are enzymes that can cleave collagen and elastin, major structural proteins of tissue and organs, and participate in tissue-destructive disease progression seen in osteoporosis, arthritis, atherosclerosis, and cancer metastasis. Detection of mature cathepsins and quantification of specific activity have proven difficult due to instability of the mature, active enzyme extracellularly, which has led to them being overlooked in a number of diseases. During this seminar, Dr. Platt will discuss the important development of a reliable, sensitive method to detect the activity of mature cathepsins K, L, S, and V. Then he will focus on their progress towards developing a comprehensive computational model of cathepsin-mediated degradation of extracellular matrix, based on systems of ordinary differential equations. From the computational model and experimental results, a general assumption of inertness between familial enzymes was shown to be invalid as it failed to account for the interaction of these proteases among themselves and within their microenvironment. A consequence of this was significant overestimation of total degradative potential in multiple cathepsin reaction systems. After refining the system to capture the cathepsin interactive dynamics and match the experimental degradation results, novel mechanisms of cathepsin degradation and inactivation were revealed and suggest new ways to inhibit their activity for therapeutic benefit.

In this talk we will outline proof due to Plameneveskaya and Van-Horn Morris that every virtually overtwisted contact structure on L(p,1) has a unique Stein filling. We will give a much simplified proof of this result. In addition, we will talk about classifying Stein fillings of ($L(p,q), \xi_{std})$ using only mapping class group basics.

At this time, in late September, 2011 the Dow Jones Industrial Average has just suffered its worst week since October, 2008; the Standard & Poor 500 Average just completed its worst week in the past five years; and financial markets worldwide under severe stress. We think it is timely to look at aspects of the role played by "financial engineering" (also known as "mathematical finance" or "quantitative finance") in the genesis of the on-going crisis. In this talk, we examine several structured investment vehicles (SIVs) devised by financial engineers and sold worldwide to many "investors". It will be seen that these SIVs were doomed from inception. In light of these results, we are dismayed by the mathematical models propagated over the past decade by financial ``engineers'' and ``experts'' in structured finance, and it heightens our fears about the durability of the on-going worldwide financial crisis.

In work on the Riemann zeta function, it is of interest to evaluate certain integrals involving the characteristic polynomials of N x N unitary matrices and to derive asymptotic expansions of these integrals as N -> \infty. In this talk, I will obtain exact formulas for several of these integrals, and relate these results to conjectures about the distribution of the zeros of the Riemann zeta function on the critical line. I will also explain how these results are related to multivariate statistical analysis and to the hypergeometric functions of Hermitian matrix argument.

I will discuss the Thurston norm for fibered hyperbolic 3-manifolds.

The secondary polytope of a point configuration A is a polytope whose faces are in bijection with regular subdivions of A, e.g. the secondary polytope of the vertices of polygon is an associahedron. The resultant of a tuple of point configurations A_1, A_2, ..., A_k in Z^n is the set of coefficients for which the polynomials with supports A_1, A_2, ..., A_k have a common root with no zero coordinates over complex numbers, e.g. when each A_1 is a standard simplex and k = n+1, the resultant is defined by a determinant. The Newton polytope of a polynomial is the convex hull of the exponents, e.g. the Newton polytope of the determinant is the perfect matching polytope. In this talk, I will explain the close connection between secondary polytopes and Newton polytopes of resultants, using tropical geometry, based on joint work with Anders Jensen.