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Series: Algebra Seminar

We will outline some open questions about rational points on varieties,

and present the results of some computations on explicit genus-2 K3

surfaces. For example, we'll show that there are no rational numbers

w,x,y,z (not all 0) satisfying the equation w^2 + 4x^6 = 2(y^6 +

343z^6).

and present the results of some computations on explicit genus-2 K3

surfaces. For example, we'll show that there are no rational numbers

w,x,y,z (not all 0) satisfying the equation w^2 + 4x^6 = 2(y^6 +

343z^6).

Monday, March 29, 2010 - 13:00 ,
Location: Skiles 255 ,
Luca Gerardo Giorda ,
Dep. of Mathematics and Computer Science, Emory University ,
Organizer: Sung Ha Kang

Schwarz algorithms have experienced a second youth over the lastdecades, when distributed computers became more and more powerful andavailable. In the classical Schwarz algorithm the computational domain is divided into subdomains and Dirichlet continuity is enforced on the interfaces between subdomains. Fundamental convergence results for theclassical Schwarzmethods have been derived for many partial differential equations. Withinthis frameworkthe overlap between subdomains is essential for convergence. More recently, Optimized Schwarz Methods have been developed: based on moreeffective transmission conditions than the classical Dirichlet conditions at theinterfaces between subdomains, such algorithms can be used both with and without overlap. On the other hand, such algorithms show greatly enhanced performance compared to the classical Schwarz method. I will present a survey of Optimized Schwarz Methods for the numerical approximation of partial differential equation, focusing mainly on heterogeneous convection-diffusion and electromagnetic problems.

Series: Probability Working Seminar

In this talk I will present an elementary short proof of the existence of global in time H¨older continuous solutions for the Stochastic Navier-Stokes equation with small initial data ( in both, 3 and 2 dimensions). The proof is based on a Stochastic Lagrangian formulation of the Navier-Strokes equations. This talk summarizes several papers by Iyer, Mattingly and Constantin.

Friday, March 19, 2010 - 14:00 ,
Location: Skiles 269 ,
Alan Diaz ,
School of Mathematics, Georgia Tech ,
Organizer:

Last week we motivated and defined Khovanov homology, an invariant of

oriented links whose graded Euler characteristic is the Jones

polynomial. We'll discuss the proof of Reidemeister invariance, then

survey some important applications and extensions, including Lee

theory and Rasmussen's s-invariant, the connection to knot Floer

homology, and how the latter was used by Hedden and Watson to show

unknot detection for a large class of knots.

oriented links whose graded Euler characteristic is the Jones

polynomial. We'll discuss the proof of Reidemeister invariance, then

survey some important applications and extensions, including Lee

theory and Rasmussen's s-invariant, the connection to knot Floer

homology, and how the latter was used by Hedden and Watson to show

unknot detection for a large class of knots.

Series: Dissertation Defense

In 2001, Fishburn, Tanenbaum, and Trenk published a series of two papers that introduced the notions of linear and weak discrepancy of a partially ordered set or poset. Linear discrepancy for a poset is the least k such that for any ordering of the points in the poset there is a pair of incomparable points at least distance k away in the ordering. Weak discrepancy is similar to linear discrepancy except that the distance is observed over weak labelings (i.e. two points can have the same label if they are incomparable, but order is still preserved). My thesis gives a variety of results pertaining to these properties and other forms of discrepancy in posets.

Series: ACO Colloquium

***Refreshments at 4PM in Skiles 236.***

Forbidding (undirected or directed) paths in graphs, what can be easier? Yet we show that in the context of coloring problems (CSP) and structural graph theory, this is related to the notions tree depth, (restricted) dualities, bounded expansion and nowhere dense classes with applications both in and out of combinatorics.

Series: School of Mathematics Colloquium

Light refreshments will be available in Room 236 at 10:30 am.

A single round soap bubble provides the least-area way to enclose a given volume. How does the solution change if space is given some density like r^2 or e^{-r^2} that weights both area and volume? There has been much recent progress by undergraduates. Such densities appear prominently in Perelman's paper proving the Poincare Conjecture. No prerequisites, undergraduates welcome.

Series: Analysis Seminar

The Drury-Arveson space of functions on the unit ball in C^n has recently been intensively studied from the point of view function theory and operator theory. While much is known about this space of functions, a characterization of the interpolating sequences for the space has still remained elusive. In this talk, we will discuss the relevant background of the problem, and then I will discuss some work in progress and discuss a variant of the question for which we know the answer completely.

Series: Other Talks

Santosh Vempala and I have been exploring an intriguing new

approach to convex optimization. Intuition about first-order interior

point methods tells us that a main impediment to quickly finding an

inside track to optimal is that a convex body's boundary can get in

one's way in so many directions from so many places. If the surface of

a convex body is made to be perfectly reflecting then from every

interior vantage point it essentially disappears. Wondering about what

this might mean for designing a new type of first-order interior point

method, a preliminary analysis offers a surprising and suggestive

result. Scale a convex body a sufficient amount in the direction of

optimization. Mirror its surface and look directly upwards from

anywhere. Then, in the distance, you will see a point that is as close

as desired to optimal. We wouldn't recommend a direct implementation,

since it doesn't work in practice. However, by trial and error we have

developed a new algorithm for convex optimization, which we are

calling Reflex. Reflex alternates greedy random reflecting steps, that

can get stuck in narrow reflecting corridors, with simply-biased

random reflecting steps that escape. We have early experimental

experience using a first implementation of Reflex, implemented in

Matlab, solving LP's (can be faster than Matlab's linprog), SDP's

(dense with several thousand variables), quadratic cone problems, and

some standard NETLIB problems.

approach to convex optimization. Intuition about first-order interior

point methods tells us that a main impediment to quickly finding an

inside track to optimal is that a convex body's boundary can get in

one's way in so many directions from so many places. If the surface of

a convex body is made to be perfectly reflecting then from every

interior vantage point it essentially disappears. Wondering about what

this might mean for designing a new type of first-order interior point

method, a preliminary analysis offers a surprising and suggestive

result. Scale a convex body a sufficient amount in the direction of

optimization. Mirror its surface and look directly upwards from

anywhere. Then, in the distance, you will see a point that is as close

as desired to optimal. We wouldn't recommend a direct implementation,

since it doesn't work in practice. However, by trial and error we have

developed a new algorithm for convex optimization, which we are

calling Reflex. Reflex alternates greedy random reflecting steps, that

can get stuck in narrow reflecting corridors, with simply-biased

random reflecting steps that escape. We have early experimental

experience using a first implementation of Reflex, implemented in

Matlab, solving LP's (can be faster than Matlab's linprog), SDP's

(dense with several thousand variables), quadratic cone problems, and

some standard NETLIB problems.

Series: ACO Student Seminar

Santosh Vempala and I have been exploring an intriguing newapproach to convex optimization. Intuition about first-order interiorpoint methods tells us that a main impediment to quickly finding aninside track to optimal is that a convex body's boundary can get inone's way in so many directions from so many places. If the surface ofa convex body is made to be perfectly reflecting then from everyinterior vantage point it essentially disappears. Wondering about whatthis might mean for designing a new type of first-order interior pointmethod, a preliminary analysis offers a surprising and suggestiveresult. Scale a convex body a sufficient amount in the direction ofoptimization. Mirror its surface and look directly upwards fromanywhere. Then, in the distance, you will see a point that is as closeas desired to optimal. We wouldn't recommend a direct implementation,since it doesn't work in practice. However, by trial and error we havedeveloped a new algorithm for convex optimization, which we arecalling Reflex. Reflex alternates greedy random reflecting steps, thatcan get stuck in narrow reflecting corridors, with simply-biasedrandom reflecting steps that escape. We have early experimentalexperience using a first implementation of Reflex, implemented inMatlab, solving LP's (can be faster than Matlab's linprog), SDP's(dense with several thousand variables), quadratic cone problems, andsome standard NETLIB problems.