Abstract: Intersection theory studies how subvarieties of an algebraic variety X intersect. Algebraically, this information is encoded in the Chow ring A(X), which is very difficult to describe in general. When X is the toric variety of a simplicial fan, there are several combinatorial and polyhedral descriptions of this ring due to Fulton-Sturmfels, Billera, and Danilov-Brion. These descriptions lead to interesting combinatorial problems, and in some cases, they are important ingredients in the proofs of long-standing conjectures. This talk will survey some examples of problems that arise in combinatorial intersection theory and a few approaches to solving them. It will feature joint work with Mont Cordero, Graham Denham, Chris Eur, June Huh, Nayeong Kim, Carly Klivans, and Raúl Penaguião, and will not assume previous familiarity with intersection theory.
BackAbstract: Positroids are certain representable matroids originally studied by Postnikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defining positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson-Lam-Speyer, and Pawlowski studied geometric and cohomological properties of these varieties. In this talk, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids. This allows us to give several formulas for counting the number of smooth positroids according to natural statistics on decorated permutations. Furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at the torus fixed points using an induced subgraph of the Johnson graph. We will conclude with some open problems in this area.
This talk is based on joint work with Jordan Weaver and Christian Krattenthaler.
BackAbstract: Kontsevich and Soibelman defined the cohomological Hall algebra \(\mathcal{H}_Q\) attached to a quiver \(Q\). When \(Q\) is symmetric and has at least one loop at each vertex, Efimov proved that \(\mathcal{H}_Q\) is freely generated as a supercommutative algebra by a certain multigraded vector space \(V\). For any dimension vector \(\gamma\), the dimension and Hilbert series of the \(\gamma^{th}\) piece of \(V\) are the numerical and quantum Donaldson-Thomas invariants. We give the first combinatorial interpretation of the numerical DT invariant as an orbit count of lattice points in a polytope of break divisors attached to \((Q, \gamma)\). We also interpret the quantum DT invariant in terms of graded rings studied by Postnikov-Shapiro and Ardila-Postnikov. Our central technique is the orbit harmonic method dating back to Kostant which has widespread application to combinatorial representation theory.
Joint work with Markus Reineke and Vasu Tewari.
BackAbstract: Young diagrams and standard tableaux on them parameterize irreducible representations of the symmetric group and their bases, respectively. There is a similar story for a nice family of representations of the double affine Hecke algebra (DAHA) or for the rational Cherednik algebra (a.k.a. rational DAHA) with appropriate modifications. This construction of the basis makes use of an alternate presentation of the rational DAHA and the basis diagonalizes the action of its Dunkl-Opdam subalgebra.
We can describe one such representation using the geometry of parabolic Hilbert schemes of points on plane curve singularities. The ``tableau" basis that diagonalizes the Dunkl-Opdam subalgebra is the basis of equivariant homology that comes from torus fixed points.
This is joint work with Eugene Gorsky and José Simental.
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