Algebraic Cycles, Chow Varieties, and Symmetric Polynomials

Jeremy Jankans

November 7th, 2012 - 4:00 - 4:50pm - RH 440R

Abstract

We want to understand spaces that parameterize projective subvarieties. One way to do this is to look at Algebraic Cycles. An Algebraic Cycle is a formal sum of irreducible closed subvarieties. If we take a family of irreducible subvarieties, its limit may have several irreducible components, i.e. the limit may be a general sycle.

We want to study this phenomenon and the Chow Varietes are a way of doing thins. Simply put, the points of a Chow variety are Algebraic Cycles. We will explain at the Chow - Van der Waerden Theorem that imbeds the variety into projective space. Finally we move on to a specific example, 0-cycles. We can use symmetric polynomials to work with 0-cycles. Using this we will look at the tangent space, and derive a formula for the tangent space of a multiple of smooth point.

About the Speaker

Jeremy Jankans is a native of the city of roses Portland, Oregon. Jeremy Jankans studies Algebraic Geometry.

Advisor and Collaborators

Jeremy's advisor is Vladimir Baranovsky.

Hilbert Polynomials of Modules

Roger Dellaca

October 24th, 2012 - 4:00 - 4:50pm - RH 440R

Abstract

The Hilbert polynomial of an homogeneous ideal I gives a lot of information about the ideal. This talk will describe the Hilbert function and the Hilbert polynomial, describe their significance, and give some classical theorems by Macaulay and Gotzmann. We will show how some of these results have been extended to the case of finitely presented graded modules, and how some of these results fail as given for ideals. Lastly, we will describe work in progress to generalize some results.

About the Speaker

Roger has an M.S. in Information Systems from Cal State Fullerton and an M.S. in Math from Cal State Long Beach. He is an avid chess player.

Advisor and Collaborators

Roger's advisor is Vladimir Baranovsky.

Modeling Quasicrystals: An Application of Hyperbolic Dynamics

May Mei

October 10th, 2012 - 4:00 - 4:50pm - RH 306

Abstract

The Nobel Prize-winning discovery of quasicrystals has spurred much work in aperiodic sequences and tilings. One such example is the family of one-dimensional discrete Schrödinger operators with potentials given by primitive invertible substitutions on two letters, which are a one-dimensional model of quasicrystals. We prove results about spectral properties of these operators using tools from hyperbolic dynamics.

About the Speaker

May Mei received her B.A. in mathematics from the University of California, Berkeley in 2007 and has been a graduate student in mathematics at the University of California, Irvine ever since. When she isn't working on research, she can be found at the Dollar Spot at Target or scouting new restaurants on Yelp.

Advisor and Collaborators

May's advisor is Anton Gorodetski.

Analytical Study of Multi-Layer and Continuously Stratified Barotropic Models of Ocean Dynamics

Aseel Farhat

May 4th, 2012 - 4:00 - 4:50pm - RH 440R

Abstract

In geophysics, multilayer models are derived under the assumption that the fluid consists of a finite number of homogeneous layers of distinct densities. We introduce a two-layer model that was derived to study the perturbation about a vertical shear flow. We show that the model is linearly unstable, however the solutions of the nonlinear model are bounded in time. We prove the existence of finite dimensional compact attractor and derive upper bounds on its dimension.

In plasma physics, the 3D Hasegawa-Mima equation is one of the most fundamental models that describe the electrostatic drift waves. In the context of geophysical fluid dynamics, the 3D Hasegawa- Mima equation appears as a simplified model of a reduced Rayleigh-Bénard convection model that describes the motion of a fluid heated from below. Investigating the 3D Hasegawa-Mima model is challenging even though the equations look simpler than the 3D Euler equations. Inspired by these models, we introduce and study a simplified mathematical model that has a nicer mathematical structure. We prove the global existence and uniqueness of solutions of the 3D simplified model as well as a continuous dependence on the initial data result. These results are one of the first results related to the 3D Hasegawa-Mima equation.

About the Speaker

Aseel will be a Zorn Postdoctoral Fellow at Indiana University this fall. She enjoys flashy earrings and has an impressive collection of shoes.

Advisor and Collaborators

Aseel's advisor is Edriss Titi.

Almost Commuting Matrices

Mustafa Said

April 25th, 2012 - 4:00 - 4:50pm - RH 440R

Abstract

We investigate a variant of an old problem in linear algebra and operator theory that was popularized by Paul Halmos: Must almost commuting matrices be nearly commuting?

About the Speaker

Mustafa Said is a 5th year graduate student working on problems in functional analysis.

Advisor and Collaborators

Mustafa's advisor is Timur Oikhberg.

Problems for Quasiperiodic Schrodinger Operators

Rajinder Mavi

March 8th, 2012 - 4:00 - 4:50pm - RH 440R

Abstract

The almost Matthieu operator arises as a model for Bloch electrons in a magnetic field. Aubry and Andre famously made a conjecture about the spectral properties of this operator more than thirty years ago. In the process of its study, variations of the conjecture arose naturally. Although the original conjecture was recently settled, new problems remain unsettled. We discuss some of these open problems and possible methods (and their shortcomings) to their solution.

About the Speaker

TBA

Advisor and Collaborators

Rajinder's advisor is Svetlana Jitomirskaya.