TBA

TBA

Given a finitely generated module \(M\) over a noetherian local ring \(R\), one may assign to it a conical affine variety, called the cohomological support variety of \(M\) over \(R\). This theory was first developed by Luchezar Avramov for local complete intersection rings in 1989, and by the work of many has recently been extended to encompass all commutative noetherian local rings. Geometric properties of this variety encode important homological information about \(M\) as well as \(R\). In this talk I will discuss what cohomological support varieties are, why they are useful, and some recent work on how they behave when restricting along a local homomorphism.

Note the non-standard day

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.

Abstract: Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever and however far from the body we want, to illuminate its entire surface. What is the minimum number of such light sources that we would have to use? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases.

The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). In this talk I will briefly discuss some of its history, and then focus on joint works with Wen Rui Sun that settle the conjecture for all 1-symmetric convex bodies (by complementing a method previously developed by K. Tikhomirov for this class of bodies), and that also deal with certain cases of 1-unconditional bodies, and possible extensions of those.

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 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.