You are wandering through the British Museum. Along the way, you end up in Room 4, which is dedicated to Egyptian sculpture. Why would we send you here to understand mathematical concepts? Not because of some sarcophagus or statue, but rather a large black slab inscribed with three versions of the same decree from ancient Egypt: the Rosetta Stone. These three texts are in Greek, Demotic and Hieroglyphic.
Even though the inscriptions themselves do not deal with mathematics, the complex history of its decipherment illustrates the concept, and difficulty, of "correspondences". Before 1799, when the stone was found, occidental Egyptologists were faced with the major difficulty that they could not read Hieroglyphic (or Demotic, to some extent). But, hoping that the three texts contain only minor differences, the Rosetta Stone allowed these texts to be used to understand each other. Unfortunately, there was the added complication that some parts of each text are missing, though the three versions glued together give the full decree.
The Rosetta Stone played a key and singular role in the struggle to decipher hieroglyphs, as it allowed one to build bridges or partial dictionaries - a mathematician would say "establish correspondences" - between the various languages (with missing parts!) at stake.
In the setting of this project, the role of languages is taken by the "irreducible smooth representations of G and of H, where (G,H) is a dual pair in a symplectic group". The so-called "theta correspondence" associates to certain irreducible smooth representations of G, an irreducible smooth representation of H: so, if we think of G as Greek and H as Hieroglyphic, then the theta correspondence is a Rosetta Stone, giving a translation from certain words in Greek to words in Hieroglyphic. To complete the picture, there is also an analogue of Demotic ("galois representations into the Dual group of G") and a correspondence which translates Greek to Demotic: the "Langlands correspondence", which has been a focus of effort for a wide range of mathematicians over the last 50 years. Moreover, unlike the Rosetta Stone, there are infinitely many pairs (G,H) that one can consider (so infinitely many "stones"), which means a lot of information and cases involved.
In this project, we will study the theta correspondence in a more refined way. If we take an imprint of the Rosetta Stone to a certain small depth, then we see only a partial contour of each word - and different words may give the same partial contour. Nonetheless, we may still be able to find a correspondence between the Greek partial contours and those in Hieroglyphic - and one which matches the original theta correspondence so that, if a Greek and Hieroglyphic word match, then their partial contours also match. We can even allow the depth to vary and look for a correspondence which matches all of these together.
In our project, these partial contours are "l-modular representations", where l is a prime number representing the depth. While there are reasons that a simple correspondence cannot happen for certain primes l, we expect to find both a correspondence for the remaining l and partial results (for example, a weaker correspondence) for the difficult primes l, as well as a pathway towards a correspondence "in families" which would help explain all of these simultaneously.
The theta correspondence, and the Langlands correspondence, are of interest because they encode a lot of arithmetic meaning - ultimately, this means information about properties encoded in the integers ... -1, 0, 1, 2, ... Indeed, even knowing which "words of Greek" we are able to translate tells us a lot, and the correspondence has numerous applications across Mathematics, developed over a century or so, ranging from representation theory to analytic number theory.
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