Seminars and Colloquia by Series

Massey products in Galois cohomology via rational points

Series
Job Candidate Talk
Time
Thursday, December 8, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kirsten Wickelgren – AIM/Harvard University
The cohomology ring of the absolute Galois group Gal(kbar/k) of a field k controls interesting arithmetic properties of k. The Milnor conjecture, proven by Voevodsky, identifies the cohomology ring H^*(Gal(kbar/k), Z/2) with the tensor algebra of k* mod the ideal generated by x otimes 1-x for x in k - {0,1} mod 2, and the Bloch-Kato theorem, also proven by Voevodsky, generalizes the coefficient ring Z/2. In particular, the cohomology ring of Gal(kbar/k) can be expressed in terms of addition and multiplication in the field k, despite the fact that it is difficult even to list specific elements of Gal(kbar/k). The cohomology ring is a coarser invariant than the differential graded algebra of cochains, and one can ask for an analogous description of this finer invariant, controlled by and controlling higher order cohomology operations. We show that order n Massey products of n-1 factors of x and one factor of 1-x vanish, generalizing the relation x otimes 1-x. This is done by embedding P^1 - {0,1,infinity} into its Picard variety and constructing Gal(kbar/k) equivariant maps from pi_1^et applied to this embedding to unipotent matrix groups. This also identifies Massey products of the form <1-x, x, … , x , 1-x> with f cup 1-x, where f is a certain cohomology class which arises in the description of the action of Gal(kbar/k) on pi_1^et(P^1 - {0,1,infinity}). The first part of this talk will not assume knowledge of Galois cohomology or Massey products.

Dynamics of the support of the equilibrium measure in a quartic field

Series
Analysis Seminar
Time
Wednesday, December 7, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrei Martinez Finkelshtein – University of Almeria, Spain –
The asymptotic analysis of orthogonal polynomials with respect to a varying weight has found many interesting applications in approximation theory, random matrix theory and other areas. It has also stimulated a further development of the logarithmic potential theory, since the equilibrium measure in an external field associated with these weights enters the leading term of the asymptotics and its support is typically the place where zeros accumulate and oscillations occur. In a rather broad class of problems the varying weight on the real line is given by powers of a function of the form exp(P(x)), where P is a polynomial. For P of degree 2 the associated orthogonal polynomials can be expressed in terms of (varying) Hermite polynomials. Surprisingly, the next case, when P is of degree 4, is not fully understood. We study the equilibrium measure in the external field generated by such a weight, discussing especially the possible transitions between different configurations of its support. This is a joint work with E.A. Rakhmanov and R. Orive.

Polytope Algebra and Tropical Cycles

Series
Research Horizons Seminar
Time
Wednesday, December 7, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005.
Speaker
Josephine Yu – Georgia Tech
A polytope is a convex hull of a finite set of points in a vector space. The set of polytopes in a fixed vector space generate an algebra where addition is formal and multiplication is the Minkowski sum, modulo some relations. The algebra of polytopes were used to solve some variations of Hilbert's third problem about subdivision of polytopes and to give a combinatorial proof of Stanley's g-Theorem that characterizes face numbers of simplicial polytopes. In this talk, we will introduce McMullen's version of polytope algebra and show that it is isomorphic to the algebra of tropical cycles which are balanced weighted polyhedral fans. The tropical cycles can be used to do explicit computations and examples in polytope algebra.

CANCELED: (Geometric flow for biomolecular solvation)

Series
Mathematical Biology Seminar
Time
Wednesday, December 7, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nathan Baker – Pacific Northwest National Laboratory
Implicit solvent models are important components of modern biomolecular simulation methodology due to their efficiency and dramatic reduction of dimensionality. However, such models are often constructed in an ad hoc manner with an arbitrary decomposition and specification of the polar and nonpolar components. In this talk, I will review current implicit solvent models and suggest a new free energy functional which combines both polar and nonpolar solvation terms in a common self-consistent framework. Upon variation, this new free energy functional yields the traditional Poisson-Boltzmann equation as well as a new geometric flow equation. These equations are being used to calculate the solvation energies of small polar molecules to assess the performance of this new methodology. Optimization of this solvation model has revealed strong correlation between pressure and surface tension contributions to the nonpolar solvation contributions and suggests new ways in which to parameterize these models.

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.

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