Seminar Calendar
for events the day of Thursday, March 15, 2012.

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Thursday, March 15, 2012

Joint number theory / algebraic geometry seminar
11:00 am   in 217 Noyes,  Thursday, March 15, 2012
 Del Edit Copy
Submitted by ford.
 Noam Elkies (Harvard Math)On the areas of rational trianglesAbstract: By a "rational triangle" we mean a plane triangle whose sides are rational numbers. By Heron's formula, there exists such a triangle of area $\sqrt{a}$ if and only if $a > 0$ and $x y z (x + y + z) = a$ for some rationals $x, y, z$. In a 1749 letter to Goldbach, Euler constructed infinitely many such $(x, y, z)$ for any rational $a$ (positive or not), remarking that it cost him much effort, but not explaining his method. We suggest one approach, using only tools available to Euler, that he might have taken, and use this approach to construct several other infinite families of solutions. We then reconsider the problem as a question in arithmetic geometry: $xyz(x+y+z) = a$ gives a K3 surface, and each family of solutions is a singular rational curve on that surface defined over $\mathbb{Q}$. The structure of the Neron-Severi group of that K3 surface explains why the problem is unusually hard. Along the way we also encounter the Niemeier lattices (the even unimodular lattices in $\mathbb{R}^{24}$) and the non-Hamiltonian Petersen graph.

Harmonic Analysis and Differential Equations
1:00 pm   in 345 Altgeld Hall,  Thursday, March 15, 2012
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Submitted by ekirr.
 Irina Nenciu (UIC Math)Essential self-adjointness criteria for Schroedinger operators on bounded domainsAbstract: We consider a Schroedinger operator on a bounded domain in R^n, and search for optimal growth criteria for the potential close to the boundary of the domain insuring essential self-adjointness of the associated operator. We find an abstract integral criterion for the potential, from which we prove that one can add optimal logarithmic type corrections to the classical criteria. As a consequence of our method, we study the question of confinement of spinless and spin 1/2 quantum particles on the unit disk in R^2, and achieve magnetic confinement solely by means of the growth of the magnetic field.