Abstract. Delta Conjecture of Haglund, Remmel and Wilson is the identity describing the action of Macdonald operators on elementary symmetric functions. The conjecture was proved independently by Blasiak-Haiman-Morse-Pun-Seelinger, and D'Adderio-Mellit. In this talk, I will give a geometric model for Delta conjecture using affine Springer fibers. This is a joint work with Sean Griffin and Maria Gillespie.

Abstract. The amoeba and coamoeba of a subvariety $Z \subset (\bC^\times)^n$ are the images under its projections to $\bR^n$ and $T^n$, respectively. In this talk we discuss joint work with Chris Kuo studying the coamoebae of Lagrangian submanifolds of $(\bC^\times)^n$, specifically how the combinatorics of their degenerations encodes the homological algebra of mirror coherent sheaves. Concretely, we associate to a free resolution $F^\bullet$ of a coherent sheaf on $(\bC^\times)^n$ a tropical Lagrangian coamoeba $T(F^\bullet)$, a certain simplicial complex in $T^n$. We show that the discrete information in $F^\bullet$ can be recovered from $T(F^\bullet)$ in common situations, and that in general there is a constructible sheaf supported on $T(F^\bullet)$ which is mirror to the coherent sheaf in the relevant sense. This sheaf can be interpreted as a singular Lagrangian brane supported on the stratified conormal bundle of $T(F^\bullet)$, and in some cases can be expressed as a degeneration of smooth Lagrangian branes. The resulting interplay between coherent sheaves on~$(\bC^\times)^n$ and simplicial complexes in $T^n$ provides a higher-dimensional generalization of the spectral theory of dimer models in~$T^2$, as well as a symplectic counterpart to the theory of brane brick models.

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Abstract. A natural class of dynamical systems is obtained by iterating polynomial maps, which can be viewed as maps from projective space to itself. One can ask which other projective varieties admit endomorphisms of degree greater than 1. This seems to be an extremely restrictive property, with all known examples coming from toric varieties (such as projective space) or abelian varieties. We describe what is known in this direction, with the new ingredient being the "Bott vanishing" property. Joint work with Tatsuro Kawakami.

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