Georgia Institute of Technology College of Sciences

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.

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.

Ramsey theory studies the internal homogenity of mathematical structures (i.e. graphs, number sets), parts of which (subgraphs, number subsets) are arbitrarily coloured. Often, the sufficient object size implies the existence of a monochromatic sub-object. Combinatorial games are 2-player games of skill with perfect information. The theory of combinatorial games studies mostly the questions of existence of winning or drawing strategies. Let us consider an object that is studied by a particular Ramsey-type theorem. Assume two players alternately colour parts of this object by two colours and their goal is to create certain monochromatic sub-object. Then this is a combinatorial game. We focus on the minimum object size such that the appropriate Ramsey-type theorem holds, called "Ramsey number", and on the minimum object size such that the first player has a winning strategy in the corresponding combinatorial game, called "game number". In this talk, we investigate the "restricted Ramsey-type theorems". This means, we show the existence of first player's winning strategies, and we show that game numbers are surprisingly small, compared to Ramsey numbers. (This is joint work with Jarek Nesetril.)

In this work we study the ruin problem for a generalized Cramer-Lundberg reserve model with investments, under the modeling (volatility and claim intensity) uncertainty. We formulate the problem in terms of the newly developed theory on G-Expectation, initiated by S. Peng (2005). More precisely, we recast the problem as to determine the ruin probability under a G-expectation for a reserve process with a G-Compound Poisson type claim process, and perturbed by a G-Brownian motion. We show that the Lundberg bounds for a finite time ruin probability can still be obtained by an exponential $G$-martingale approach, and that the asymptotic behavior of the ruin, as the initial endowment tends to infinity, can be analyzed by the sample path large deviation approach in a G-expectation framework, with respect to the corresponding storage process. This is a joint work with Xin Wang.