The Hilbert-Samuel multiplicity of an ideal is a fundamental invariant of singularities of the ideal and is known to satisfy various convexity properties. In this talk, I will discuss more general convexity properties for ideal sequences (rather than single ideals) and provide an application to Chi Li’s normalized volume of a singularity. 

A framework was developed in a joint work with F. J. Gallego and M. Gonzalez to systematically deal with the deformation of finite morphisms, multiple scheme structures on algebraic varieties and their smoothing. There are several applications of this framework. In this talk I will talk about some of them. First is the description of some components of the moduli space of varieties of general type in all dimensions. In particular, we show the existence of components of the moduli space of general type in all dimensions that are analogue of the moduli space of curves of genus g≥2�≥2. Secondly, we give a new method to construct smooth varieties in projective space embedded by complete sub canonical linear series within the range of the Hartshorne conjecture and beyond. Are all of them complete intersections? We also construct systematically, smooth non complete intersection subvarieties embedded by complete linear series outside the range of the Hartshorne conjecture. As a byproduct, we construct simple canonical varieties of any dimension, expanding the original question posed by Enriques for algebraic surfaces. This is joint work with F. J. Gallego, J. Mukherjee and D. Raychaudhury. 

The log minimal model program has recently been completed for klt threefolds over excellent base schemes of residue characteristic p>5�>5.  In this talk I will survey the known results, together with some motivations and applications for working in this more general setup.

The talk will be on Zoom. Here is the link: https://umsystem.zoom.us/j/95107302505?pwd=Y1BMTWw1TDl0My9Haks0ZTFadk5hdz09

The passcode is: 967345

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.

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.