Macdonald polynomials and related structures in geometry

This project belongs to an exciting and lively area of mathematical research at the intersection of algebra, geometry, and combinatorics. From the combinatorial point of view, consider the classical binomial ! coefficients ( ) = . They appear all over the place! We use them to compute the number of ways to !(-)! pick k elements out of n. Replacing the factorials by the q-factorials, [] = 1 (1 + ) (1 + + 2 ) (1 + +. . . +-1 ), we arrive at the definition of quantum or simply q-binomial coefficients. They are not numbers anymore, but polynomials in the variable q. It is surprising that these q-binomials sometimes behave very much like the classical ones. In fact, many theories in algebra and combinatorics admit q-analogies. It was discovered in the 90`s that sometimes it is possible and indeed a good idea to introduce two parameters q,t and form q,t-analogies. So we have a pattern classical mathematics q-mathematics q,t-mathematics. Macdonald polynomials serve as building blocks of this new theory. Since their discovery, they have been surrounded by a cloud of conjectures which all witness that the passage from q to q,t is much more subtle than the passage from classical mathematics to q. In recent joint work with Erik Carlsson, we have solved one of these questions, the shuffle conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov. There is no reason to stop at algebra and combinatorics. What would be q,t-geometry? To a space , we associate the sequence of Betti numbers () which, roughly speaking, count the number of k- dimensional holes in . The classical invariant associated to is the Euler characteristic, which is the alternating sum of the Betti numbers (). Its q-version is the Poincaré polynomial, the polynomial with coefficients (). In this project, we study more refined invariants of spaces, which produce q,t- polynomials, and compare them to the q,t-polynomials found by people in combinatorics. In another direction, we study q,t-polynomials associated to knots: take a piece of thread and wrap it around the torus (think: doughnut) in the natural way, keeping count of how many times we pass through the hole, and how many times we complete the full turn. Suppose the counts are m and n at the end. Now remove the torus. The thread will form a knot called the m,n-torus knot (, ). We also compute certain q,t-invariants for these knots and again obtain the same q,t-polynomials as before. In this project, we will attempt to understand the deep reasons behind the appearance of q,t-polynomials in so different mathematical settings.

This project belongs to an exciting and lively area of mathematical research at the intersection of algebra, geometry, and combinatorics. One motivation comes from q,t-analogies. Take for instance binomial numbers, which count the number of ways to pick k elements out of n. It is known that the binomial numbers and many other numbers have generalizations which are not numbers anymore, but functions of one variable "q". Not long ago it was discovered that sometimes we can go further and have functions in two variables q,t which are symmetric in q and t. Functions called "Macdonald polynomials" serve as building blocks of this new theory. Since their discovery, they have been surrounded by a cloud of conjectures which all witness that the passage from "q" to "q,t" is much more subtle than the passage from numbers to "q". In recent joint work with Erik Carlsson, we have solved one of these questions, the shuffle conjecture. But as soon as the solution was published, a more general conjecture appeared. One of the project's outcomes is a solution of this more general conjecture called "delta conjecture". What it basically says is that certain sum of q,t-monomials corresponding to combinatorial objects called Dyck paths can be related to a certain mysterious operator delta which acts on functions, and the way it acts is described in terms of Macdonald polynomials. Since Dyck paths are counted by the Catalan numbers, we have constructed, in a way, generalized q,t-Catalan numbers. This explains a connection between combinatorics (Dyck paths) and algebra (operators). What about geometry? For instance, often we see that some geometric spaces can be obtained by patching together simpler pieces. Then we expect that counting these pieces produces some interesting numbers, and maybe even the pieces themselves correspond to interesting combinatorial objects. What if the operators can also be explained geometrically, for instance by some transformations of the space? This is the kind of questions we are after. Another outcome of the project is a construction of a space whose basic pieces precisely match the Dyck paths we already mentioned. The main conjecture which would relate all this to some other spaces coming from gauge theory remains open. In yet another direction, we study q,t-polynomials associated to knots. Since long ago mathematicians wanted to be able to classify knots, and for that purpose constructed sophisticated invariants, numbers which would be the same for identical knots and different for different knots. Another outcome of the project is a "knot-like" interpretation of the operators we have mentioned. And so Dyck paths, operators, spaces and knots all become part of one fascinating story, where new and new connections keep being discovered.