Mathematics

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.

In this talk, I will describe a strong Briançon-Skoda type result (the integral closure of \(J^{n+k-1}\) is contained in \(J^k\), or a slight enlargement of \(J^k\)) which utilizes the Eagon-Northcott
or Buchsbaum-Eisenbud complex.

This result gives a new proof of Lipman-Sathaye's Briançon-Skoda result for regular rings, it gives the
precise version of Briançon-Skoda for generalizations of pseudo-rational rings (improving Lipman-Teissier's result and implying Aberbach-Huneke in equal characteristic), it also implies the tight closure, plus closure, and epf closure versions of the Briançon-Skoda theorem. It also implies effective Briancon-Skoda results for characteristic free versions of Du Bois singularities, generalizing work of Huneke-Watanabe and Wheeler-Zhang in the F-pure case.  Finally, we show how this result, plus Gabber's weak local unformization, can be used as the missing piece to solve Huneke's conjecture of uniform Briançon-Skoda
and uniform Artin-Rees for (reduced) quasi-excellent rings of finite Krull dimension.

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.

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