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

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 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. 

In algebraic geometry, the existence and geometry of quotient schemes is a delicate issue.  Even when quotients exist, they may not reflect enough properties of the original group action to be useful.  The machinery of geometric invariant theory is one prescription for identifying open subsets of the original scheme that admit useful quotients, but it can be shown that there are, in general, other open sets that also admit well-behaved quotients.  In this talk, we examine particular actions of diagonalizable groups on affine space and illustrate the wide variety of properties that quotients arising from this action can have.

In algebraic geometry, the existence and geometry of quotient schemes is a delicate issue.  Even when quotients exist, they may not reflect enough properties of the original group action to be useful.  The machinery of geometric invariant theory is one prescription for identifying open subsets of the original scheme that admit useful quotients, but it can be shown that there are, in general, other open sets that also admit well-behaved quotients.  In this talk, we examine particular actions of diagonalizable groups on affine space and illustrate the wide variety of properties that quotients arising from this action can have.

In this talk, we will consider the NLS system of the third-harmonic generation. Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor, discussed global well-posedness, finite time blow-up, as well as other aspects of the dynamics. These authors have also constructed solitary wave solutions, via the method of the Nehari manifold, in an appropriate range of parameters. Specifically, the waves exist only in spatial dimensions \(n=1,2,3\). They have also established some stability/instability results for these waves.

In this work, we systematically build and study solitary waves for this important model. We construct the waves in the largest possible parameter space, and we provide a complete classification of their spectral stability.

Finally, we showed instability by a blow-up, for dimension 3, and for a more restrictive set of parameters, we use virial identities methods to derive the strong instability, in the spirit of Ohta's approach. This is joint work with Atanas Stefanov.

Friday 4-5 PM, MSB 110