Abstract: I will present my recent result on homological mirror symmetry for the universal centralizer (a.k.a Toda space) associated to a complex semisimple Lie group..

The A-side is a partially wrapped Fukaya category on the universal centralizer, and the B-side is the category of coherent sheaves on the categorical quotient of the dual maximal torus by the Weyl group (with some modifications if the group has nontrivial center). I will illustrate many of the geometry and ideas of the proof using the example of SL_2 or PGL_2.

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Abstract: We discuss the structure of the Kähler moduli space of Picard rank two Calabi-Yau threefolds, which are given in terms of complete intersections in projective ambient spaces, or as hypersurfaces in toric ambient spaces. As it turns out, flop transitions are ubiquitous in such setups. The triple intersection form of the Kähler cone generators can be brought into four different normal forms, and we use this to solve the geodesic equations in the moduli space for each one of them. Moreover, we will discuss that flops can lead to isomorphic or non-isomorphic Calabi-Yau manifolds. We find that there exist infinite flop chains of isomorphic geometries, but only a finite number of flops to inequivalent manifolds. Physically, the latter result is expected based on the swampland distance conjecture, and mathematically fits to a conjecture due to Kawamata and Morrison for Calabi-Yau threefolds.

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Abstract: I will talk about the problem of determining the birational complexity of moduli spaces of curves and K3 surfaces. I will recall some recently introduced invariants that measure irrationality and talk about what is known for these moduli spaces. In the second half I will report on joint work with D. Agostini and K.-W. Lai, where we study how the degrees of irrationality of the moduli spaces of polarized K3 surfaces grow with respect to the genus g. We provide polynomial bounds. The proof relies on Kudla's modularity conjecture for Shimura varieties of orthogonal type. For special genera we exploit the deep Hodge theoretic relation between K3 surfaces and special hyperkähler fourfolds to obtain much sharper bounds.

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Abstract: Consider a tensor product of representations of a semisimple Lie algebra g. The Gaudin algebra is a commutative algebra which acts on this tensor product, commuting with the action of g. This algebra depends on a parameter which lives in the moduli space of marked genus 0 curves. In previous work, we studied the monodromy of eigenvectors for this algebra as the parameter varies in the real locus of this space. In new work in-progress, we consider trigonometric Gaudin algebras, which act on the same vector space (but do not commute with the g-action). We see that this leads to the action of the affine cactus group, and we describe the action of this group combinatorially using crystals. I will also describe the (conjectural) relation between trigonometric Gaudin algebras and the quantum cohomology of affine Grassmannian slices.

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Abstract: 3d mirror symmetry is a proposed physical duality relating a pair of 3d N=4 field theories. Various mathematical shadows of this result have been studied, but ultimately (after a topological twisting), 3dMS should entail an equivalence between a pair of 2-categories associated to the algebraic (respectively, symplectic) geometry of a pair of holomorphic symplectic stacks. In general, the definitions of these 2-categories are not known, but in this talk we explain how one can define the relevant 2-categories and construct an equivalence between them in the case where the spaces involved are linear quotients by a torus. Potential applications include a Betti geometric version of Tate's thesis and a recovery of earlier results on Koszul duality for hypertoric categories O. This is joint work with Justin Hilburn and Aaron Mazel-Gee.

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Abstract: The Gross-Hacking-Keel mirror is constructed in terms of scattering diagrams and theta functions. The ground of the construction is that scattering diagrams inherit the algebro-geometric analogue of the holomorphic disks counting. With Yu-shen Lin, we made use this idea and gave first non-trivial examples of family Floer mirror. Then with Sam Bardwell-Evans, Hansol Hong, and Yu-shen LIn, we construct a special Lagrangian fibration on the non-toric blowups of toric surfaces that contains n odal fibres, and prove that the fibres bounding Maslov 0 discs reproduce the scattering diagrams. As a consequence, we can then illustrate the mirror duality between the A and X cluster varieties.

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Abstract: The Mumford-Tate group G(M) of a mixed Hodge structure M is a subgroup of GL(M) which satisfies the following property: any rational subspace of any tensor power of M underlies a mixed Hodge substructure if and only if it is invariant under the natural action of G(M). Assuming M is graded-polarizable, the unipotent radical U(M) of G(M) is a subgroup of the unipotent radical U0(M) of the parabolic subgroup of GL(M) associated to the weight filtration on M. Let us say U(M) is large if it is equal to U0(M).

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This talk is a report on a recent joint work with Kumar Murty, where we consider the set of all mixed Hodge structures on a given rational vector space, with a fixed weight filtration and a fixed polarizable associated graded Hodge structure. It is easy to see that this set is in a canonical bijection with the set of complex points of an affine complex variety S. The main result is that assuming some conditions on the (fixed) associated graded hold, outside a union of countably many proper Zariski closed subsets of S the unipotent radical of the Mumford-Tate group is large.

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Abstract: We describe two different bases for irreducible symmetric group representations: the tableaux basis from combinatorics (and from the geometry of a class of varieties called Springer fibers); and the web basis from knot theory (and from the quantum representations of Lie algebras). We then describe new results comparing the bases, e.g. showing that the change-of-basis matrix is upper-triangular, and sketch some applications to geometry and representation theory. This work is joint with H. Russell, as well as with T. Goldwasser and G. Sun.

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Abstract: Kazhdan-Lusztig polynomials lie at the intersection of representation theory, geometry and algebraic combinatorics. Despite their relevance and elementary definition (through a recursive algorithm involving only elementary operations), the explicit computation of these polynomials is still one of the hardest open problems in algebraic combinatorics. In this talk we will present the explicit formulas of the Kazhdan-Lusztig polynomials for a Coxeter system of type affine B_2.

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Abstract: We give an efficient presentation of the Chow ring with integral coefficients of the open part of the moduli space of rational maps of odd degree to projective space. A less fancy description of this space has its closed points correspond to equivalence classes of (r+1)-tuples of degree d polynomials in one variable with no common positive degree factor. We identify this space as a GL(2) quotient of an open set in a projective space, and then obtain a (highly redundant) presentation by considering an envelope of the complement. A combinatorial analysis then leads us to eliminating a large number of relations, and to express the remaining ones in generating function form for all dimensions. The upshot of this work is to observe the rich combinatorial structure contained in the Chow rings of these moduli spaces as the degree and the target dimension vary. This is joint work with Damiano Fulghesu.

Abstract: I will first consider crepant resolutions of Weierstrass models corresponding to elliptically-fibered fourfolds with simple Lie algebras. I will further discuss the fibrations with multisections or nontrivial Mordell-Weil groups. In contrast to the case of fivefolds, Chern and Pontryagin numbers of fourfolds are invariant under crepant birational maps. This fact enables us to be able to compute Chern and Pontryagin numbers, independently from a choice of a crepant resolution, along with various other characteristic numbers such as the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory. For the case of Calabi-Yau fourfolds, F-theory compactification provides the resulting 4d N=1 gauge theories.