We will discuss some number theoretic motivation for studying minimal exponent of a hypersurface, which is the nonarchimedean counterpart of the Archimedean zeta function. Then I will explain a toy model, i.e. Milnor fiber of an isolated singularity, for the proof of the main theorem. We will see why Hodge theory naturally enters into the picture.
Minimal exponent of hypersurfaces, p-adic integration and Milnor fibers Sponsored by the Meyer Fund
Mar. 20, 2025 2:30pm (MATH 3…
Functional Analysis
Laura Scull (Fort Lewis College)
X
One common way of representing spaces with singularities is via topological groupoids. However, such a representation is not unique, and a space may be represented by many different groupoids, leading to the idea of Morita equivalence of groupoids. To work with these spaces, we define a localised bicategory of groupoids in which Morita equivalences become isomorphisms. This localisation can be done in several equivalent ways, as developed by Pronk and Roberts.
Our interest lies in a particularly nice and ubiquitous class of examples, the action groupoids which represent the quotient of a space by an action of a group. Such groupoids come with their own notion of morphisms given by equivariant functors. Our aim is to connect these two viewpoints. We create a localised bicategory built on equivariant functors of groupoids, and show that for action groupoids, this is equivalent to the bicategories of Pronk and Roberts. Using this viewpoint, Morita equivalences may be defined using a certain class of equivariant projection functors, and many common conditions on the singularities can be integrated into this construction. This also gives us a way of examining invariants from equivariant homotopy to see if they can be made Morita invariant and extended to our setting.
This talk is based on joint work with J. Watts and C. Farsi. The talk will include many pictures and examples illustrating the underlying geometric intuition behind the abstraction of the bicategories.
Localising Action Groupoids
Mar. 20, 2025 3:35pm (MATH 3…
Probability
Giorgio Cipolloni (University of Arizona)
X
We study the Brownian evolution of large non-Hermitian matrices and show that their log-determinant converges to a 2+1 dimensional Gaussian field in the Edwards-Wilkinson regularity class, i.e. logarithmically correlated for the parabolic distance. This gives a dynamical extension of the celebrated result by Rider and Virag (2006) proving that the fluctuations of the eigenvalues of Gaussian non-Hermitian matrices converge to a 2 dimensional log-correlated field. Our result, previously not known even in the Gaussian case, holds out of equilibrium for general matrices with i.i.d. entries. We also study the extremal values of these fields and demonstrate their logarithmic dependence on the matrix dimension.
Logarithmically correlated fields from non-Hermitian random matrices
Minimal exponent of hypersurfaces, p-adic integration and Milnor fibers