Mathematics

This talk will focus on low-energy resolvent expansions for a wide class of scattering settings.  We concentrate on operators acting on \(\mathbb{R}^2\), with particular attention to the Dirichlet or Neumann Laplacian on the exterior of an obstacle and to Aharonov-Bohm operators.  We give applications to long-time behavior of the solutions of the wave equation and low-energy behavior of the scattering phase.

This is based on joint work with Kiril Datchev.  A portion is also with Mengxuan Yang and Pedro Morales.

A weakening of Frobenius splitting, Quasi-F-Splittings have proven to be a vital invariant in the study of varieties in positive characteristic, with numerous applications to arithmetic and birational geometry. This weaker condition extends the application of Frobenius to study singularities of arithmetically supersingular varieties, encompassing a much broader class of examples.  In this talk I'll provide an overview of Quasi-F-Splittings and introduce a local analogue, Quasi-F-Purity. I will also discuss how quasi-F-pure hypersurfaces are "as close to being F-pure as possible" by computing the F-pure Threshold of an arbitrary quasi-F-pure hypersurface. This talk includes joint work with Jack J Garzella. 

We consider the problem of the asymptotic stability of a front type solution to a weakly two dimensional model for the propagation of undular bores. This model includes solutions of the KdV-Burgers equation, so this can be considered a study of the stability of those solutions within a more general two dimensional model. We prove that for domains that are sufficiently narrow in the transverse direction and for a range of the dispersion parameter, the family of fronts is a global asymptotic attractor. 

The weakly étale ring maps -- those which are flat and have flat diagonal -- have gained attention recently for their role in defining the pro-étale site of a scheme. We will propose a new definition of weakly étale ring maps via a lifting property analogous to the one used to define formally étale ring maps. We will use a result of Gabber on the cohomology of Henselian pairs to deduce the equivalence of the two definitions. If time permits, we will discuss an example of a weakly étale morphism which does not lift along a surjective ring map. This is joint work with Johan de Jong. 

Consider a rooted d-regular tree with \ell layers, where each vertex is colored either blue or green. Starting from the root, the color propagates down the tree so that each child inherits its parent’s color but flips with probability 30%. Now, suppose you only observe the colors of the leaves—can you infer the color of the root?

This setting describes a broadcasting process on trees, where in general we have q possible 'colors' and a transition matrix specifying the probability that a child receives color a given that its parent has color b. The associated inference problem is known as the Tree Reconstruction Problem.

A classical result, the Kesten–Stigum bound, characterizes a sharp threshold: above the bound, simply counting the colors at the leaves provides enough information to make a reliable guess of the root color, whereas below it, counting reconstruction is impossible.

In our recent work, we identify the Kesten–Stigum bound as a threshold of computational complexity. Specifically, we show that while it may still be statistically possible to infer the root color below the bound, any algorithm achieving this must overcome a complexity barrier.

I will aim to make this talk accessible to a broad audience, beyond probability.

This is a joint work with Elchanan Mossel.

We define two new versions of integral and Frobenius closures of ideals which incorporate an auxiliary ideal and a real parameter. These additional ingredients are commonly used to adjust old definitions of ideal closures in order to generalize them to pairs. In the case of tight closure, similar generalizations exist due to N. Hara and K. I. Yoshida, as well as A. Vraciu. We study their basic properties and give computationally effective calculations of the adjusted tight, Frobenius, and integral closures in the case of affine semigroup rings in terms of the convex geometry of the associated exponent sets. Finally, we study submodules of the fraction field of a domain defined in terms of our adjusted closures and the application of the new closures to an F-nilpotent property for ideal pairs. This is a joint work with Kyle Maddox and Lance Miller.

This talk is about joint work with I. Swanson.

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A longstanding question in algebraic geometry is the classification of  reduced and 
irreducible local complete one--dimensional domains $R$ over an algebraically closed 
field $k$. It is known that such a ring is completely determined once it is known up 
to a "sufficiently high" power of its maximal ideal, where this sufficiently 
high power depends on the singularity degree $\delta$ of the ring.

In this talk we show that two curve singularities $(R, \mathfrak m)$ and $(R', \mathfrak m')$ 
are already isomorphic if there exists an isomorphishm 
$\varphi: R/ \mathfrak m^{j+1} \longrightarrow R'/ {\mathfrak m'}^{j+1}$ of 
$k$--algebras for some $j \geq 2 \delta +1$, and that the isomorphism may be chosen to agree with $\varphi 
\pmod{\mathfrak m^{j-2 \delta+1}}$. This strengthens a result of Hironaka, who obtained the
bound $3 \delta + 1$. 

We analyze the space of steady rotating solutions to the two-dimensional incompressible Euler equations nearby vortex patch solutions satisfying a nondegeneracy condition. We address the question of desingularization and prove that such vortex patch states are the limit of rotating Euler solutions that are smooth to infinite order, have compact vorticity support, and respect dihedral symmetry. Our nondegeneracy condition is proved to be satisfied by Kirchhoff ellipses and along the local bifurcation curves emanating from the Rankine vortex. The construction, that is based on a local stream function formulation in a tubular neighborhood of the patch boundary, is a synthesis of analysis on thin domains, nonlinear a priori estimates, and Newton's method. Our techniques additionally allow us to construct nearby exotic families of singular rotating vortex patch-like solutions. This is joint work with Razvan-Octavian Radu.

The Bernstein-Sato Polynomial is a classical \(D\)-module invariant that has been used to measure the singularities of a hypersurface \(f\). In this talk we will review some of the classical theory before considering the Bernstein-Sato polynomial over a numerical semigroup ring \(R\). We will see that in this case the Bernstein-Sato polynomial detects not just of an element in \(R\) but of the ambient numerical semigroup ring itself. 

We study minimizers of Cahn-Hilliard energies under strong anchoring (Dirichlet) boundary conditions using a Nehari manifold approach with spectral analysis of the Dirichlet Laplacian.

For the de Gennes energy with quartic potential, we reveal bifurcation phenomena governed by the boundary value and transition layer thickness parameter. When the boundary value equals the phase average, and the parameter exceeds a critical threshold, the minimizer is unique and homogeneous; below this threshold, two symmetric minimizers emerge. Deviating boundary values restore uniqueness with asymmetric minimizers. We derive rigorous bounds for these solutions.

We extend this framework to the Flory-Huggins logarithmic potential, which models polymer blends and presents singular behavior at boundaries. Our analysis, supported by numerical simulations, reveals temperature-mediated bifurcations and demonstrates how the Nehari manifold technique provides a unified treatment of both functionals under strong anchoring conditions.