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Open Richardson varieties play an important role in Kazhdan-Lusztig theory as they geometrize the R-polynomials. In the '80s, Deodhar introduced a decomposition of open Richardson varieties of any flag varieties with the remarkable property that each component is isomorphic to a product of an affine space and a torus. In a recent work on standard extension algebras, Eberhardt and Stroppel introduced a further decomposition of the open Richardson varieties of Grassmannians to which they referred to as Bruhat-type stratification. The components of this decomposition are again of the same shape as the components of the Deodhar decomposition. In this talk, we will see that the Deodhar and Bruhat-type decomposition actually coincide.

A 2-group is a groupoid with a structure analogous to that of a group. By a representation of a 2-group G I mean a (weak) 2-functor from G[1], the one-object 2-groupoid with G as 2-group of self-equivalences of the unique object, to some target 2-category C. Taking the 2-category of Baez-Crans 2-vector spaces as C leads to a bad…

Diffusion geometry is a new framework for geometric and topological data analysis that defines Riemannian geometry for probability spaces. This lets us apply the huge wealth of theory and methods from classical differential geometry as tools for data analysis. In this talk, I will outline the basic theory of diffusion geometry, like the construction of vector fields and differential forms. I will also survey a range of new data analysis tools, including vector calculus, solving spatial PDEs on data, finding integral curves and geodesics, and finding circular coordinates for de Rham cohomology classes. In the very special case of data from manifolds, we can compute the curvature tensors and dimension. These methods are highly robust to noise and fast to compute when compared with comparable methods like persistent homology.

The study of monodromies of differential equations has been a rich area of mathematical physics, interconnected with various fields in mathematics and physics. Recent discoveries reveal that monodromy varieties naturally possess the structure of cluster varieties, significantly enhancing our understanding of their connections to string theory and Donaldson–Thomas invariants. A key technique in these developments is the (exact) WKB approximation.

In string theory, q-difference equations (qDEs) naturally appear as an "M-theory completion" of differential equations, though defining monodromy in this context remains an active research area. In this seminar, I will discuss how the WKB approximation, traditionally formulated for second-order ODEs, can be effectively generalized to second-order q-difference equations, providing a natural characterization of their monodromies. Central to this approach is the WKB Stokes diagram, known in the physics literature as the exponential network, which offers a framework for defining cluster coordinates for monodromies of qDEs.

I will illustrate this formalism through explicit examples, including the q-difference Mathieu equation. Remarkably, its monodromy around the origin—known in topological string theory as the quantum mirror map— takes the form of the Hamiltonian of a cluster integrable system within these cluster coordinates.

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