One half of mirror symmetry for Fano varieties is typically stated as a relation between the symplectic geometry of a Fano variety Y and the complex geometry of a Landau-Ginzburg model, realized as a pair (X,W) with X a quasi-projective variety and W a regular function on X. The pair (X,W) itself is expected to reflect a pair on the Fano side, namely a decomposition of Y into a disjoint union of an affine log Calabi-Yau and an anticanonical divisor D, thought of as mirror to W. We will discuss recent work which shows how the intrinsic mirror construction of Gross and Siebert naturally produce potential LG models assuming milder conditions on the singularities of D than typically required for the intrinsic mirror construction. In particular, we show that classical periods of this LG model recover the quantum periods of Y. In the setting when Y\D is an affine cluster variety, we will describe how these LG models naturally give rise to Laurent polynomial mirrors and encode certain toric degenerations of Y. As an example, we consider Y = Gr(k,n), D a particular choice of anticanonical divisor with affine cluster variety complement and give an explicit description of the intrinsic LG model in terms of Plücker coordinates on Gr(n-k,n), recovering mirrors constructed and investigated by Marsh-Rietsch and Rietsch-Williams. 

The basic operations of free probability - additive free convolution, multiplicative free convolution, and free compression - describe how the eigenvalues of large random matrices interact under the basic matrix operations, such as addition, multiplication, and taking minors.

In this talk we discuss how these free probability operations can be formulated in terms of an “entropic optimal transport” problem – an optimal transport problem but with an entropy penalty for the coupling measure.

Our proof of this formulation uses the quadrature formulas of Marcus, Spielman and Srivastava, which relate the expected characteristic polynomial of matrices under random unitary vs symmetric conjugation. The approach involves an asymptotic analysis of the quadrature formulas using a large deviation principle on the symmetric group.

This is joint work with Octavio Arizmendi (CIMAT).