Seminars and Colloquia by Series

On the behavior at infinity of solutions to difference equations in Schroedinger form

Series
Math Physics Seminar
Time
Tuesday, December 6, 2011 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Lilian Wong – SoM, Georgia Tech
We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices.Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. This is joint work with Evans Harrell.

Bernstein's theorem, Newton polygons, and tropical intersections

Series
Job Candidate Talk
Time
Tuesday, December 6, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joseph Rabinoff – Harvard University
Bernstein's theorem is a classical result which computes the number of common zeros in (C*)^n of a generic set of n Laurent polynomials in n variables. The theorem of the Newton polygon is a ubiquitous tool in arithmetic geometry which calculates the valuations of the zeros of a polynomial (or convergent power series) over a non-Archimedean field, along with the number of zeros (counted with multiplicity) with each given valuation. We will explain in what sense both theorems are very special cases of a lifting theorem in tropical intersection theory. The proof of this lifting theorem builds on results of Osserman and Payne, and uses Berkovich analytic spaces and extended tropicalizations of toric varieties in a crucial way, as well as Raynaud's theory of formal models of analytic spaces. Most of this talk will be about joint work with Brian Osserman.

The moduli space of points on the projective line and quadratic Groebner bases

Series
Algebra Seminar
Time
Monday, December 5, 2011 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Milena Hering – University of Connecticut
The ring of invariants for the action of the automorphism group of the projective line on the n-fold product of the projective line is a classical object of study. The generators of this ring were determined by Kempe in the 19th century. However, the ideal of relations has been only understood very recently in work of Howard, Millson, Snowden and Vakil. They prove that the ideal of relations is generated byquadratic equations using a degeneration to a toric variety. I will report on joint work with Benjamin Howard where we further study the toric varieties arising in this degeneration. As an application we show that the second Veronese subring of the ring of invariants admits a presentation whose ideal admits a quadratic Groebner basis.

Loose Legendrian Knots in High Dimensional Contact Manifolds

Series
Geometry Topology Seminar
Time
Monday, December 5, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Emmy Murphy – Stanford University
In the talk, I plan to give a definition of loose Legendrian knots inside contact manifolds of dimension 5 or greater. The definition is significantly different from the 3 dimensional case, in particular loose knots exist in local charts. I'll discuss an h-principle for such knots. This implies their classification, a bijective correspondence with their formal (algebraic topology) invariants. I'll also discuss applications of this result, comparisons with 3D contact toplogy, and some open questions.

Discrete Mathematical Biology Working Seminar

Series
Other Talks
Time
Monday, December 5, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Greg Blekherman – Georiga Tech
A discussion of the paper "Evidence for kinetic effects in the folding of large RNA molecules" by Morgan and Higgs (1996).

Tensor Products, Positive Operators, and Delay-Differential Equations

Series
CDSNS Colloquium
Time
Monday, December 5, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Mallet-Paret – Division of Applied Mathematics, Brown University –
We study a class of linear delay-differential equations, with a singledelay, of the form$$\dot x(t) = -a(t) x(t-1).\eqno(*)$$Such equations occur as linearizations of the nonlinear delay equation$\dot x(t) = -f(x(t-1))$ around certain solutions (often around periodicsolutions), and are key for understanding the stability of such solutions.Such nonlinear equations occur in a variety of scientific models, anddespite their simple appearance, can lead to a rather difficultmathematical analysis.We develop an associated linear theory to equation (*) by taking the$m$-fold wedge product (in the infinite dimensional sense of tensorproducts) of the dynamical system generated by (*). Remarkably, in the caseof a ``signed feedback'' where $(-1)^m a(t) > 0$ for some integer $m$, theassociated linear system is given by an operator which is positive withrespect to a certain cone in a Banach space. This leads to very detailedinformation about stability properties of (*), in particular, informationabout characteristic multipliers.

Efficient algorithm for electronic structure calculations

Series
Job Candidate Talk
Time
Thursday, December 1, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Jianfeng Lu – Courant Institute, NYU
Electronic structure theories, in particular Kohn-Sham density functional theory, are widely used in computational chemistry and material sciences nowadays. The computational cost using conventional algorithms is however expensive which limits the application to relative small systems. This calls for development of efficient algorithms to extend the first principle calculations to larger system. In this talk, we will discuss some recent progress in efficient algorithms for Kohn-Sham density functional theory. We will focus on the choice of accurate and efficient discretization for Kohn-Sham density functional theory.

Comments on the BMV Trace Conjecture

Series
Analysis Seminar
Time
Thursday, December 1, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pierre Moussa – CEA/Saclay, Service de Physique Theorique, France
The term "BMV Conjecture" was introduced in 2004 by Lieb and Seiringer for a conjecture introduced in 1975 by Bessis, Moussa and Villani, and they also introduced a new form for it : all coefficients of the polynomial Tr(A+xB)^k are non negative as soon as the hermitian matrices A and B are positive definite. A recent proof of the conjecture has been given recently by Herbert Stahl. The question occurs in various domains: complex analysis, combinatorics, operator algebras and statistical mechanics.

Celebration of Mind: Connecting Mathematics, Magic and Mystery

Series
School of Mathematics Colloquium
Time
Thursday, December 1, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Klaus 1116W
Speaker
Colm Mulcahy – Spelman College

Please Note: Hosts are Ernie Croot and Dan Margalit.

We survey some new and classic recreations in the fields of mathematics, magic and mystery in the style of Martin Gardner, Prince of Recreational Mathematics, whose publishing career recently ended after an astonishing 80 years. From card tricks and counter-intuitive probability results to new optical illusions, there will be plenty of reasons to celebrate the ingenuity of the human mind.

Chip-firing games on graphs

Series
Research Horizons Seminar
Time
Wednesday, November 30, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005.
Speaker
Farbod Shokrieh – Georgia Tech.
I will discuss the theory of chip-firing games, focusing on the interplay between chip-firing games and potential theory on graphs. To motivate the discussion, I will give a new proof of "the pentagon game". I will discuss the concept of reduced divisors and various related algorithmic aspects of the theory. If time permits I will also give some applications, including an "efficient bijective" proof of Kirchhoff's matrix-tree theorem.

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