In this talk, we discuss anti-plane shear deformations on a semi-infinite slab with a non-linear mixed traction displacement boundary condition. We apply global bifurcation theoretic methods and deduce extreme behavior at the terminal end solution curves. It is shown that arbitrarily large strains are encountered for a class of idealized materials. We also consider degenerate materials, prove that ellipticity breaks down, and show that a concurrent blow-up in the second derivative occurs.

The analytic spread of a module M is the minimal number of generators of a submodule that has the same integral closure as M. In this talk, we will present a result that expresses the analytic spread of a decomposable module in terms of the analytic spread of its component ideals. In the second part of the talk, we will show an upper bound for the symbolic analytic spread of ideals of small dimension. The latter notion is the analogue of analytic spread for symbolic powers. These results are joint work with Carles Bivià-Ausina and Hailong Dao, respectively.

The G-transform to a data series is the extension of the Fourier transform from the unit circle to the entire complex plane.I shall introduce the Padé approximant to the G-transform and discuss some of its properties as regard its poles, zeros, and the residues. In particular, I’ll show examples of superresolution with respect to the Nyquist limit, numerical evidence of universality for the behavior of poles and zeros associated with noise and how the presence of signals alters that behavior. I’ll conclude showing a couple of applications. In particular, work in progress on brain waves.

Let \(R\) be a Noetherian ring, \(P\) be a prime ideal of \(R\) such that \(R_P\) is Cohen-Macaulay of dimension \(h\), \(\omega\) be a finitely generated \(R\)-module such that \(\omega_P\) is a canonical module for \(R_P\), and \(W\) be a subset of \(R\) that naturally maps onto the set of nonzero elements of \(R/P\). We show that for every finitely generated $R$-module $M$, there exists \(g \in W\) such that \(H_P^j(M)_g \cong Hom(Ext_R^{h-j}(M, \omega), H_P^h(\omega))_g\), which gives the well-known local duality when we localize at \(P\). Moreover, each \(H_P^j(M)_g\) has an ascending filtration such that all the factors are free over \(R/P\). We use this result to study the purity exponents in Noetherian rings of prime characteristic \(p\). In the case where \(R\) is excellent Cohen-Macaulay (this assumption can be weakened), we establish an upper semicontinuity result for the purity exponent considered as a function on the spectrum of \(R\). This result enables us to prove that excellent strongly F-regular rings are very strongly F-regular (also called F-pure regular). Another consequence is that the F-pure locus is open in an excellent Cohen-Macaulay ring. This is joint work with Mel Hochster.

Let \(R\) be a Noetherian ring, \(P\) be a prime ideal of \(R\) such that \(R_P\) is Cohen-Macaulay of dimension \(h\), \(\omega\) be a finitely generated \(R\)-module such that \(\omega_P\) is a canonical module for \(R_P\), and \(W\) be a subset of \(R\) that naturally maps onto the set of nonzero elements of \(R/P\). We show that for every finitely generated $R$-module $M$, there exists \(g \in W\) such that \(H_P^j(M)_g \cong Hom(Ext_R^{h-j}(M, \omega), H_P^h(\omega))_g\), which gives the well-known local duality when we localize at \(P\). Moreover, each \(H_P^j(M)_g\) has an ascending filtration such that all the factors are free over \(R/P\). We use this result to study the purity exponents in Noetherian rings of prime characteristic \(p\). In the case where \(R\) is excellent Cohen-Macaulay (this assumption can be weakened), we establish an upper semicontinuity result for the purity exponent considered as a function on the spectrum of \(R\). This result enables us to prove that excellent strongly F-regular rings are very strongly F-regular (also called F-pure regular). Another consequence is that the F-pure locus is open in an excellent Cohen-Macaulay ring. This is joint work with Mel Hochster.