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Abstract: Let W = C[t, t^{-1}]\del_t be the Witt algebra of algebraic vector fields on C* and let Vir be the Virasoro algebra, the unique nontrivial central extension of W . In [Petukhov, Sierra 2022], it was shown that Poisson primitive ideals of S(W ) and S(Vir) can be constructed from elements of W* and Vir* of a particular form, called local functions. In this paper, we show how to use a local function on W or Vir to construct a representation of the Lie algebra. We further show that the annihilators of these representations are new completely prime primitive ideals of U(W ) and U(Vir). We use this to define a Dixmier map from the Poisson primitive spectrum of S(Vir), respectively S(W ), to the primitive spectrum of U(Vir), respectively U(W ), successfully extending the orbit method from finite-dimensional solvable Lie algebras to the countable-dimensional setting.

Our method involves new ring homomorphisms from U(W ) to the tensor product of a localized Weyl algebra and the enveloping algebra of a finite-dimensional solvable subquotient of W. We further show that the kernels of these homomorphisms are intersections of the primitive ideals constructed from natural subsets of W*.

Many people working in graph theory today are interested in thinking of graphs as analogous to spaces, and developing homotopy theories for graphs. One particular categorical point of view on this idea can be found in the work of Lawvere, dating back to the 1980s. Lawvere observed that some elementary toposes seem to deserve to be thought of as…

Algebraic theories, such as the theories of groups, rings, and
modules over a given ring, are often studied category theoretically
as finitary monads on Set. Which such theories have the
property that all their algebras are free? It turns out that we can
write them all down relatively easily! Proving this result, however, is
…

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…

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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