Randomness extractors are combinatorial objects used to generate perfectly random bits from imperfect (correlated) sources of randomness. Extractors are fundamental objects in the theory of pseudorandomness which have found many surprising applications in algorithm design, coding theory, and cryptography beyond their original motivation. Standard applications of the probabilistic method show that most functions are extractors with amazing parameters. However, constructing *efficient* extractors (and other types of pseudorandom objects, such as error-correcting codes and expander graphs) with good parameters has proved to be extremely challenging. Designing efficient extractors and related pseudorandom objects with improved parameters is a major line of research in theoretical computer science. For much more about this, see Vadhan's excellent monograph at https://people.seas.harvard.edu/~salil/pseudorandomness/.

(Seeded) Condensers are weaker objects than (seeded) extractors, closely related to expansion properties of graphs. We can see a seeded condenser as an "unbalanced" regular bipartite expander graph. This is a graph G = (L,R,E) whose right vertex set R is significantly smaller than the left vertex set L, each vertex in L has the same degree D, and G satisfies an "expansion" property -- every large enough subset S of left vertices has a large neighborhood (ideally with size close to D*|S|, the maximal size of such a neighborhood).

The goal of this project is to explore improvements to an interesting geometric construction of an efficient condenser due to Zuckerman that is based on line-point incidence theorems in finite fields (https://theoryofcomputing.org/articles/v003a006/v003a006.pdf, Section 8). In particular, we would like to understand the improvements that can be achieved by considering "higher-dimensional incidences" (say, between planes and lines instead of between lines and points).