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Spring-Summer 2009 Abstracts |
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Oleg Ivrii August 7, 2009 Domino tilings have very nice combinatorial properties, whose proofs are completely elementary, which make them an indispensable tool in modern mathematics. In this talk, I will give a selection of these features which make dominoes so attractive. Some topics I plan to touch on:
Jordan Watts July 31, 2009 A number of decades ago, Smale proved that the space of orientation-preserving diffeomorphisms of the 2-sphere has a deformation retraction onto SO(3), the space of rotations of the sphere. The paper in which the proof resides is somewhat fun: the deformation retraction is broken into smaller retractions of other spaces of maps, and they are all put together in the end like pieces of a puzzle. In this talk, we shall examine these pieces (modulo details, of course), and put them together to get this wonderful result. Oleg Ivrii July 29, 2008, 3:00 - 4:30, BA 6183 Can a square be subdivided into smaller squares, all of different size? Can a square of unit length be subdivided into squares not all of which have rational sides? Can a rectangle with a non-rational aspect ratio be tiled by rectangles of rational aspect ratio? For which x, can a square be tiled by rectangles similar to 1 by x? Come to find out the answers to these and many more exciting questions. Jordan Watts July 24, 2009 A standard theorem in symplectic geometry is that of Darboux, which can be summarized as stating that symplectic manifolds have no local invariants. Thus turning to the global picture, we find a type of "symplectic rigidity" that geometers are still trying to understand to this day. One of the foundational results that inspired much activity in this subject was that of Gromov; in particular, the nonsqueezing Theorem. It says that the radius of a (symplectic) ball that is symplectically embedded into a (symplectic) cylinder must have a radius smaller than or equal to that of the cylinder -- we can't squeeze anything bigger inside. In this talk, we will quickly go through the basics of symplectic geometry, and give a sketch of the proof of this remarkable theorem. Some prerequisite knowledge of differential geometry/topology would be helpful. Omar Antolín Camarena July 17, 2009 While mathematicians are perfectly happy to prove theorems, what every mathematician really wants is to discover a Lemma: like Fatou's in analysis, Gauss's in number theory, Burnside's in combinatorics, Zorn's in set theory, etc. As Paul Taylor says, "Lemmas do the work in mathematics: Theorems, like management, just take the credit. A good lemma also survives a philosophical or technological revolution". We will discuss a marvelous Lemma about determinants discovered by Lindstrom in 1972 and rediscovered and popularized by Gessel and Viennot in 1985 that (1) allows you to "see" all the basic properties of determinants by drawing appropriate pictures and (2) is the source of many results relating determinants to enumerative problems in combinatorics. Omar Antolín Camarena July 10, 2009 In 1956, Claude Shannon investigated the maximum rate of transmission across a communication channel (that might distort some symbols) in such a way that the receiver may recover the original message without errors. Which symbols might be confused is indicated in the confusion graph (which simply has edges between pairs of confusable symbols) and the resulting maximum rate of transmission is (now) called the Shannon capacity of the graph. Shannon himself computed the capacity of many graphs including all graphs of 5 vertices or else except for the 5-cycle. It wasn't until 1979 that Laszlo Lovász gave a beautiful computation of the capacity of the pentagon (and many other interesting graphs). Yura Burda June 19, 2009 Have you ever looked closely at the models of confocal quadrics that stand in the glass box near the big seminar room? What are they? What interesting properties do they have? Why do they stand near a model of ellipsoid ruled by its geodesics? My talk will answer these questions. We will discuss very old theorems about family of confocal quadrics, how they lead to newer results about integrability of some dynamical systems (free particle on ellipsoid, particle on sphere in quadratic potential) and reveal properties of billiards in ellipsoids. Oleg Ivrii June 5, 2009 In this talk, I will discuss Lewy's operator, the first known example of a differential operator which isn't "locally solvable" (this phenomenon came as a shock to the mathematical community when it was discovered some sixty years ago). After I prove the above statement, I will give a short introduction to a branch of complex analysis known as CR geometry and explain the geometric meaning of Lewy's operator. Hrant Hakobyan April 17, 2009 Can a hyperbolic manifold have many hyperbolic structures? Is it always possible to glue two copies of the unit disc along a boundary homeomorphism and obtain the Riemann sphere? Given a metric space homeomorphic to the plane, is it actually bi-Lipschitz to the plane? If a homeomorphism of the sphere is conformal off a set E, is it in fact a Mobius transformation? These seemingly unrelated questions lead to the study of the so-called quasisymmetric or quasiconformal mappings. I will define these classes of maps and explain the answers to some of the questions above. and compactifications of covering spaces Damir Kinzebulatov April 3, 2009 In his celebrated 1962 paper, L. Carleson gave an affirmative answer to a long standing question: is it true that the unit disk is dense in the maximal ideal space of the algebra H^\infty of bounded analytic functions defined on {z : |z| < 1}? In my talk I will state the generalized Corona problem, and present several known results on the topological structure of the maximal ideal space of H^\infty. These results are based on a nice construction, involving "compactifications" of certain covering spaces. Oleg Ivrii February 13, 2009 Loosely speaking, a "circle packing" is a collection of circles in the plane, many of which are tangent. One can ask the converse statement: does there exist a collection of circles with prescribed tangencies? How unique is it? A famous theorem of Koebe, Andreev and Thurston says that the answer is yes, but one must go beyond the plane and pack Riemann surfaces. In this talk, we will see the analogy between circle packings and complex analysis. If one thinks of an analytic function as one which takes "infinitesimal" circles to infinitesimal circles, it is appropriate to define the discrete analogue to be a function between circle packings which preserves tangency relations. Much of classical complex function theory can be recast in these discrete terms, although a full correspondence is not yet known. In future talk(s), we will see that "discrete" analytic functions are not only a fun analogue of analytic functions, but are also an approximation (the Rodin-Sullivan theorem). I will follow the wonderful book of Kenneth Stephenson, Introduction to circle packing, the theory of discrete analytic functions, Cambridge University Press, 2005. Omar Antolín Camarena February 6, 2009 Call two polygons or polyhedra equidecomposable if one can be divided into finitely many pieces that can be rearranged to form the other. The Bolyai-Gerwein theorem says that any two plane polygons of equal area are in fact equidecomposable, but Hilbert believed that the same is not true in three dimensions. He was right, of course, and less than a year after his famous list, Max Dehn found an example of two tetrahedra with the same height and congruent bases which are not equidecomposable. I will prove the Bolyai-Gerwein theorem and give Boltianskii's simplified proof of Dehn's example. Yura Burda January 30, 2009 Have you ever wondered what finite groups can act on a sphere without fixed points? We will start with simple and beautiful variation on the theme of Borsuk-Ulam theorem, found by Milnor, which gives one necessary condition such groups must satisfy. A different route will lead us to consider the notion of group cohomology and formulate another restriction on such groups. Along the way we will encounter a marvellous creature discovered by Poincare: a homology 3-sphere, which is not homotopic to the usual one. The talk should be accessible to students, who understand the word "cohomology". Mikhail Mazin January 23, 2009 The Tarski-Seidenberg theorem basically says that the projection of a real semi-algebraic set in Rn+1 to Rn is again semi-algebraic. The same also holds in the complex setting. I will present a proof which works (up to a small modification) for both cases. The proof also gives an explicit algorithm of how, given a system of equations and inequalities in n+1 variables, to construct a finite set of systems in n variables such that the projection of the set given by the original system is the union of sets given by the systems we get. Oleg Ivrii January 16, 2009 Suppose a holomorphic function converges on the unit disk and extends continuously up to the boundary; must the Taylor series approximate it uniformly on the closed unit disk? Are there natural sequences of polynomials which do approximate the holomorphic function? If so, how fast is the rate of convergence? Suppose that the Taylor series does converge uniformly on the closed unit disk, must it converge absolutely as well? What if the holomorphic function is of a special form, for example, a conformal mapping? What if the Taylor coefficients have restricted growth? What other kinds of conditions are possible? Are the above questions consequences of some general theorem? Oleg Ivrii January 9, 2009 I will explain what the holomorphic functional calculus is about in the context of general Banach algebras and present applications to functions which have Absolutely Convergent Fourier Series (ACFS) and also to approximation theory in several complex variables. The second half of the talk will be specific to ACFS. Namely, I will give present several interesting sufficient conditions due to Bernstein, Zygmund, Beurling, Kamaly and others, some with sharp constants, for functions to have ACFS. |