Recent Progress on Sparse Hard Sets:
          The Resolution of a 1978 Conjecture of J. Hartmanis
 
                            D. Sivakumar
                        University of Houston
 
The following question is of fundamental interest in Computer Science:
 
    Is it true that every problem solvable in polynomial time
    can also be solved very efficiently in parallel,
    or using very little computer memory?
 
This question appears extremely hard, given the state of the art of
developments in Theoretical Computer Science.  A closely related
question is the following:
 
    Does there exist a small (polynomial size) database D (one that
    could potentially be hopelessly hard to generate) such that
        every problem solvable in polynomial time
        can also be solved very efficiently in parallel,
        or using very little computer memory,
    IF we are given one-time access to D?
 
Couched in formal terms, the question is the same as asking whether
there is a P-hard sparse set.  This simpler question has many important
ramifications and connections to various questions of central interest
in complexity theory, some of which we will briefly outline in the talk.
 
The main result to be presented is that the two questions are equivalent,
i.e., the answer to the second question is YES if and only if the answer
to the first question is YES.  In other words, if such a database D does
exist at all, then we don't need it!  This result affirms a conjecture
made by Juris Hartmanis in 1978.
 
The core of the talk will be a simple, self-contained exposition of the
main ideas of the proof of the equivalence of the two questions.
The proof makes extensive use of computations in finite fields, in particular
solving a Vandermonde system of equations using the Discrete Fourier Transform.
 
            (Joint work with Professor Jin-Yi Cai, SUNY/Buffalo)