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Algorithmic phenomena in Biology:

We are in the early stages of a new scientific revolution. This time the revolution is driven by the flood of discoveries and associated data in Biology. One of the underlying themes in these discoveries is that the great variety of observed phenomena about life are outcomes of well-defined processes (algorithms) following well-defined execution rules (programs).

One algorithmic phenomenon I have been examining is related to a process occurring in ciliates: Ciliates are single celled eukaryotes with two types of nuclei. The one type of nucleus is an encrypted version of the other type. Some events in the cell's life cycle trigger decryption of the encrypted version, while the previously unencrypted nuclei present during the decryption process are degraded. Currently very limited data is available about this whole process, and the process is not yet understood. Models for parts of the process suggest that the ciliate micro nuclear decryption apparatus has highly nontrivial computational capabilities. 

Research opportunities for students

Set Theory and its relatives:

Set Theory is a foundation for mathematics. Several topics have been developed in Set Theory but ought to be part of the general knowledge of all mathematicians working on infinitary objects. These include ordinal arithmetic,  parts of cardinal arithmetic, hypotheses such as the Continuum Hypothesis or Martin's Axiom and the notions of a measurable cardinal or a real-valued measurable cardinal. The most specialized among these may be Martin's Axiom. These few items are in widespread use in several areas of mathematics, but are not treated as a necessary part of graduate level education in mathematics. These few topics constitute a set of mathematical tools that are in several subdisciplines of mathematics already as indispensable as mathematical induction.

Modern Set Theory draws heavily on mathematical logic. Set Theory is developing techniques for establishing consistency of, or independence of, mathematical statements relative to a given axiom system that includes the Zermelo-Fraenkel axioms. Set Theory also is developing techniques for measuring the relative strength of statements whose consistency or independence cannot be established  by merely postulating that the Zermelo-Fraenkel axioms are consistent. An outgrowth of this foundational activity is the formulation of powerful mathematical hypotheses whose relative consistency status is known, and which have rich mathematical consequences. Martin's Axiom is by now a classical example of such an outgrowth from the method of forcing. There are numerous hypotheses of this kind now in use in the usual laboratory for refining the products of Set Theory - General Topology. Analysis and Algebra are fast becoming standard testing grounds for the products of Set Theory.

Game Theory:

Game theory is the mathematics of competitive behavior, whether this behavior is deliberate or not. The players need not be what we normally think of as conscious beings, but might for example be two enzymes competing for a binding site on a strand of DNA. Examples where game theory is a useful tool include Biology, Economics, Anthropology and the analysis of security notions used in cryptology and information security.

Cryptology:

Cryptology is the mathematics of information assurance in a hostile environment. The intuitive notion of ``security" is one of the central concepts to be made precise in developing and evaluating cryptosystems.

Elementary number theory:

With the rise of distance communication and e-commerce over public channels (phone lines, wireless, etc.) number theory has become one of the most applied subjects in pure mathematics. Number theory is one of the foundations for practical crypto systems.