Trees, Prisms and a Quillen Model Structure on Prismatic Sets
Year of Dissertation:
We define a category which we call the category of prisms, P. This category interpolates between the simplex category, 916; and the box category ☐ . The way in which we interpolate these two categories is considering categories with a tensor functor and a cone functor. It turns out that the objects of the category P are in one to one correspondence with planer rooted trees. Furthermore the morphisms of this category may be defined as certain combinatorial decorations on certain trees. We then show that that the category P is a test category automatically giving a Quillen model structure on the presheaves on P.
Holomorphic Motions and Extremal Annuli
Year of Dissertation:
Holomorphic motions, soon after they were introduced, became an important subject in complex analysis. It is now an important tool in the study of complex dynamical systems and in the study of Teichmuller theory. This thesis serves on two purposes: an expository of the past developments and a discovery of new theories. First, I give an expository account of Slodkowski's theorem based on the proof given by Chirka. Then I present a result about infinitesimal holomorphic motions. I prove the $|\epsilon \log\epsilon|$ modulus of continuity for any infinitesimal holomorphic motion. This proof is a very well application of Schwarz's lemma and the estimate of Agard's formula for the hyperbolic metric on the thrice punctured sphere. One application of this result is that, after the integration of an infinitesimal holomorphic motion, it leads to the Holder continuity property of a quasiconformal homeomorphism. This will be presented in Chapter 3. Second, I compare the proofs given by both Slodkowski and Chirka. Then I construct a different extension of a holomorphic motion in the frame work of Slodkowsk's proof by using the method in Chirka's proof. This gives some opportunity for me to discuss the uniqueness in the extension problem for a holomorphic motion. This will be presented in Chapter 4. Third, I discuss the universal holomorphic motion for a closed subset of the Riemann sphere and the lifting property in the Teichmuller theory. One application of this discussion is the proof of the coincidence of Teichmuller's metric and Kobayashi's metric, a result due to Royden and Gardiner, given by Earle, Kra, and Krushkal by using Slodkowski's theorem. This will be presented in Chapters 5 and 6. Fourth, I study the complex structure of the universal asymptotically conformal Teichmuller space. I give a direct and new proof of the coincidence of Teichmuller's metric and Kobayashi's metric on the universal asymptotically conformal Teichmuller space, a result previously proved by Earle, Gardiner, and Lakic. The main technique that I have used in this proof is Strebel's frame mapping theorem. This will be presented in Chapter 7. Finally, in Chapter 8, I study extremal annuli on a Riemann sphere with four points removed. By using the measurable foliation theory, the Weierstrass P-function, and the variation formula for the modulus of an annulus, I prove that the Mori annulus maximize the modulus for the two army problem in the chordal distance on the Riemann sphere. Gardiner and Masur's minimum axis is also discussed in this chapter. Most of the results in this thesis have been published in several research papers jointly with Fred Gardiner, Jun Hu, Yunping Jiang, and Sudeb Mitra.
The Minimal Resultant And Conductor for Self Maps of the Projective Line
Year of Dissertation:
We develop and study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Szpiro's Theorem (Theorem 5.1), which bounds the degree of the minimal discriminant of an elliptic curve over a function field in terms of the conductor. We also explore a question about Lattes maps: given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes maps for multiplication by n have unstable bad reduction for each n. We then study minimality and semi-stability, considering what conditions imply minimality and whether semi-stable models and presentations are minimal, proving results in the degree two case. We prove the singular reduction of a semi-stable presentation coincides with the bad reduction. Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample to a natural dynamical analog to Szpiro's Theorem. Finally, we consider the notion of "critical bad reduction," and show that a dynamical analog may still be possible using the locus of critical bad reduction to define the conductor.
Drawdowns, Drawups, and Their Applications
Year of Dissertation:
This thesis studies the probability characteristics of drawdown and drawup processes of general linear diffusions. The drawdown process is defined as the current drop from the running maximum, while the drawup process is defined as the current rise over the running minimum. Attention is drawn to the first hitting times of the drawdown and the drawup processes, also known as the drawdown and the drawup respectively, and their applications in managing financial risks and detecting abrupt changes in random processes. The probabilities that the drawdown of a units precedes the drawup of equal size are derived in a biased simple random walk model and a drifted Brownian motion model. It is then shown that there exists an analytical formula for the Laplace transform of the drawdown of a units when it precedes the drawup of b units. The above problem can be related to the arbitrage-free pricing of a digital option related to the drawdowns and the drawups. Several static and semi-static replications are developed to hedge the risk exposure of these options. Finally, we study the properties of the drawups as a means of detecting abrupt changes in random processes with multi-source observations. In particular, we study extensions of the cumulative sum (CUSUM) stopping rule, which is the drawup of the log-likelihood ratio process. It is shown that the N-CUSUM stopping rule is at least second-order asymptotically optimal as the meantime to the first false alarm tends to infinity.