COOLSTUFF SEMINAR Summer 2007 Abstracts

What is CoolStuff? :: Schedule :: Abstracts Blaschke Products Oleg Ivrii September 7, 2007 Oleg talked about the use of Blaschke products in Hardy spaces and applied them to prove the Muntz-Szasz theorem. Tropical Curves Kristin Shaw August 31, 2008 Kristin continued the theme of tropical geometry. Real Algebraic Curves Yura Burda August 24, 2007 Yura talked about Hilbert's problem about ovals, and explained how tropical geometry can be used to tackle it. Triangulation of Polygons Omar Antolín Camarena August 17, 2007 Omar proved Monsky's theorem (that a square cannot be subdivided in an odd number of equal areas) and some generalizations. Hardy Spaces Oleg Ivrii August 10, 2007 Oleg defined Hardy spaces and proved their basic properties. Subcartesian Spaces Jordan Watts August 3, 2007 Jordan discussed differential forms on subcartesian spaces, like manifolds but admit singularities. The Theta Function Zavosh Amir-Khosravi July 27, 2007 Zavosh defined the theta function, discussed some transformation properties and applied it to find the number of representations of a positive integer as the sum of two or four squares. Virtual Knots Fionntan Roukema July 20, 2007 Fiontann defined the virtual knots and explained why they might be useful. Oleg Ivrii Differential Galois Theory July 13, 2007 Oleg explained why you cannot evaluate the integral of exp(-x2). Zavosh Amir-Khosravi Representation Theory July 6, 2007 Zavosh discussed how representation theory enters contemporary number theory. Oleg Ivrii Waring's Problem June 29, 2007 Oleg proved Gauss lemma (primes of type 4k+1 are the sum of two squares) using Minkowski's geometry of numbers. Then he defined Schnirel'man density and proved the theorem of Erdos that every basis is an essential component. Zavosh Amir-Khosravi Dedekind Domains Jube 15, 2007 Zavosh began his talk by noting the lack of unique factorization for number fields and what people did to overcome this difficulty. János Krámar Probability June 1, 2007 János gave a brief introduction to probability and gave some applications of martingales. Oleg Ivrii p-adic Numbers May 25, 2007 Oleg defined the p-adic numbers, computed their algebraic closure and gave some examples of "p-adic interpolation". Zavosh Amir-Khosravi Modular forms May 18, 2007 Zavosh defined modular forms and gave some examples.