Georgia Institute of Technology College of Sciences

We derive at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a selection of exponential Levy models, restricting our attention to asset-price models whose log returns structure is a Levy process. We consider two main problems. First, we consider very general Levy models that are in the domain of attraction of a stable random variable. Under some relatively minor assumptions, we give first-order at-the-money call-price and implied volatility asymptotics. In the case where our Levy process has Brownian component, we discover new orders of convergence by showing that the rate of convergence can be of the form t^{1/\alpha} \ell( t ) where \ell is a slowly varying function and \alpha \in (1,2). We also give an example of a Levy model which exhibits this new type of behavior where \ell is not asymptotically constant. In the case of a Levy process with Brownian component, we find that the order of convergence of the call price is \sqrt{t}. Second, we investigate the CGMY process whose call-price asymptotics are known to third order. Previously, measure transformation and technical estimation methods were the only tools available for proving the order of convergence. We give a new method that relies on the Lipton-Lewis formula, guaranteeing that we can estimate the call-price asymptotics using only the characteristic function of the Levy process. While this method does not provide a less technical approach, it is novel and is promising for obtaining second-order call-price asymptotics for at-the-money options for a more general class of Levy processes.

Under the operation of connected sum, the set of knots in the 3-sphere forms a monoid. Modulo an equivalence relation called concordance, this monoid becomes a group called the knot concordance group. We will consider various algebraic methods -- both classical and modern -- for better understanding the structure of this group.

It is known that certain medium, for example electromagnetic field and Bose Einstein condensate, has positive speed of sound. It is observed that if the medium is in its equilibrium state, then an invading subsonic particle will slow down due to friction; and the speed of a supersonic particle will slow down to the speed of sound and the medium will radiate. This is called Cherenkov radiation. It has been widely discussed in physical literature, and applied in experiments. In this talk I will present some rigorous mathematical results. Joint works with Juerg Froehlich, Israel Michael Sigal, Avy Soffer, Daniel Egli, Arick Shao.

Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. This talk will be focused on one of the famous mean-field spin glasses, the Sherrington-Kirkpatrick model. We will present results on the conjectured properties of the Parisi measure including its uniqueness and quantitative behaviors. This is based on joint works with A. Auffinger.