What can a real algebraic curve, defined by an equation f(x, y) = 0 where f is a real polynomial of degree n, look like? For n = 2 the answer -- an ellipse, a hyperbola, or a parabola -- is classical. The cases n = 3 and n = 4 have been solved by Newton and by Descartes. The degree 4 curve can have at most 4 ovals and if their number is exactly 4 then none of them lies inside a disc bounded by another closed oval (prove it!).
At the Paris International Congress of Mathematicians in 1900 David Hilbert announced his (wrong) answer for the degree 6 case. Today the shapes of the degree 6 curves are known: there exist 3 arrangements of their 10 ovals. This was discovered in 1970 by D. Gudkov, who was continuing the bifurcation theory studies of his teacher, physicist Andronov.
For the degree 8 curves the problem is still open in spite of its astonishing relation to the topological quantum field theory of 4-dimensional smooth manifolds.
The talk will describe old and recent results in this active part of mathematical studies of the real world.
Note: Prof. Arnold has also offered a few problems for high school students. Please look at them here.
Vladimir I. Arnold is one of the best world mathematicians. The number of disciplines to which he has contributed significantly is astounding. During his 50-year career following his first main result for his PhD thesis -- the solution of Hilbert's 13th problem -- Prof. Arnold published over 300 articles and 20 books. He is best known for the Kolmogorov-Arnold-Moser theorem regarding the stability of integrable Hamiltonian systems. However, he has also made important contributions in dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics, magnetohydrodynamics, and singularity theory. Arnold is a rare teacher with a special gift for finding new beautiful problems for young researchers, and he has had a profound influence on an entire generation of mathematicians. The face of modern mathematics would be unrecognizable without his work.
Prof. Arnold is a member of at least 10 world renowned scientific societies -- the Russian Academy of Sciences, the US National Academy of Sciences, the Royal Society of London, the Academy of Sciences of Paris among them. He has been awarded honorary degrees from many top-notch universities, and has received numerous prizes. In 1981, the minor planet 10031 Vladarnolda was named after him.
How to get to Santa Clara University:
|101||From US Highway 101, take the De La Cruz Boulevard/Santa Clara exit and follow the signs to El Camino and the main campus entrance.|
|280||From I-280, take I-880 north toward Oakland to the Alameda exit. Turn left onto The Alameda (which becomes El Camino Real) to the main campus entrance.|
|880||From I-880, take the Alameda exit, travel north (The Alameda becomes El Camino Real) to the main campus entrance.|