University of Georgia
Department of Mathematics

Seminar Schedule
March 5 – March 9, 2007

All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.

MONDAY, March 5, 2007

Algebra
2:30pm, Room 410
Speaker: Martin Montgomery, Piedmont College
Title of talk: A Naive Approach to Calculating Global Dimension

Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea and Cookies

Colloquium
3:30pm, Room 302
Speaker: Endre Szemeredi, Rutgers University
Title of talk: Finite and infinite arithmetic progressions in sumsets
Abstract: We prove that if A is a subset of at least cn^{1/2} elements of {1,2,...,n}, (where c is a sufficiently large constant), then the collection of sums formed from the subsets of A contains an arithmetic progression of length n. As an application, we confirm a long standing conjecture of Erdos and Folkman on complete sequences. Joint work with Van Vu.

Joint UGA - Georgia Tech Topology Seminar
Room: Boyd 304
3.15 pm:
Speaker: Bill Goldman, University of Maryland
Title of talk: Cubic surfaces and SL(2)-character varieties of surfaces
Abstract: This talk will survey the relationship between elementary invariant theory of SL(2) and moduli spaces of geometric structures over surfaces. In particular I will describe a relationship (first noted by Fricke and Klein) between affine cubic surfaces, the one- holed torus and the four-holed sphere.

4.20 pm:
Speaker: Anar Akhmedov, Georgia Tech
Title of talk: Construction of New Symplectic Cohomology S2 x S2 and Small Exotic Manifolds
Abstract: In this talk we present new examples of symplectic manifolds with same
integral cohomology as S2 x S2. We also discuss the generalization of
these examples as well as its application in the construction of simply
connected exotic 4-manifolds with small Euler characteristic.

TUESDAY, March 6, 2007

VIGRE-Graduate Student Seminar
2:00pm, Room 304
Speaker: Adrian Jenkins, Purdue University
Title of talk: An introduction to discrete dynamical systems
Abstract: I will talk about (discrete) dynamical systems in the local and global cases, of one and several complex variables. Complex dynamics is an interesting field of study, and very popular. All relevant definitions will be given, and proofs will be kept to a minimum. Basically, I would like to give an idea of some of the beauty of this field.

Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea and Cookies

Colloquium
3:30pm, Room 304
Speaker: Bill Goldman, University of Maryland
Title of talk: Dynamics of surface group representations
Abstract: The space of representations of the fundamental group of a surface
in a Lie group is a rich geometric object, with an algebraic structure enjoying much symmetry. The simplest examples include symplectic vector spaces, Jacobi varieties, and moduli spaces of holomorphic vector bundles.

Fricke-Teichmueller spaces also arise as representation spaces. They are a special case of deformation spaces of locally homogeneous geometric structures in the sense of Ehresmann and Thurston. The underlying algebraic structure of deformation spaces closely relates to the geometric structures they parametrize. Understanding the geometric structures is often a key for understanding the topology and dynamics of these spaces.

The mapping class group of the surface acts on this space preserving a natural Poisson geometry. Natural Hamiltonian flows on the deformation space generalize the classical Fenchel-Nielsen twist flows on Teichmueller space. For compact Lie groups, the mapping class group action is chaotic. The proof of ergodicity can be regarded as an analog of the Fenchel-Nielsen coordinates for Teichmuller space. For representations corresponding to uniformizations by geometric structures, the action is proper.

In general the dynamics falls between these two extremes. In the case of a one-holed torus, the dynamics reduces to an action of the modular group on cubic surfaces related to the Markoff equation, where both chaotic and proper dynamics coexist.

WEDNESDAY, March 7, 2007

Algebraic Geometry
2:30pm, Room 410
Speaker: TBA
Title of talk: TBA

Faculty and Graduate Student Social
3:00pm, Room 409
Coffee, Cookies, Tea

Number Theory/Arithmetic Geometry
3:30pm, Room 304
Speaker: Pete Clark, University of Georgia
Title of talk: Ramanujan Graphs, Part II: Spectral theory and zeta-functions.

Special Analysis Seminar
3:45pm, Room 323
Speaker: Adrian Jenkins, Purdue University
Title of talk: Structural stability of hyperbolic polynomial automorphisms of {\mathbf{C}}^{2}
Abstract: I plan to discuss recent results developed with Greg Buzzard regarding the structural stability of hyperbolic polynomial automorphisms of {\mathbf{C}}^{2}. The results have roots in the theory of diffeomorphisms on compact manifolds, but also utilize holomorphic motions in two dimensions. Generally, such higher-dimensional motions have been of limited use in the theory of dynamical systems (unlike their one dimensional counterparts).

VIGRE – Quantum Mechanics
4:00pm, Room 302

THURSDAY, March 8, 2007

VIGRE – ODE
2:00pm, Room 326

VIGRE – Moduli spaces
2:00pm, Room 304

VIGRE – Geometry
2:00pm, Room 410

Faculty and Graduate Social
3:00pm, Room 409
Coffee, Cookies, Tea

Colloquium
3:30pm, Room 304
Speaker: Herbert Lange, Erlangen, Germany
Title of talk: Schur and Kanev correspondences.
Abstract: Correspondences on curves are used to construct Prym-Tyurin varieties which represent a generalization of Prym varieties: special types of abelian varieties. In order to construct Prym-Tyurin varieties, several people associated to every finite Galois covering of smooth projective curves a type of correspondences, which are equivalent to Schur's character relations and which we therefore call Schur's correspondences. Another type of correpondences was introduced by Kanev using the monodromy of a spectral covering. In the talk the relation between both correspondences will be explained and several examples will be given. This is joint work with Anita Rojas.

FRIDAY, March 9, 2007

Applied Math Seminar
12:20pm-1:10pm, Room 326
No meeting this week

Geometry
2:30pm, Room 410
Speaker: Joe Fu, University of Georgia
Title of talk: The principal kinematic formula for the unitary group
Abstract: Although we have a complete understanding of the structure of the algebra of convex valuations invariant under the unitary group, and although this structure determines completely the integral geometry of this group, it is not so easy to write down the resulting formulas explicitly. The most important formula (cf. the title of the talk) is equivalent to an expression for the expected number of intersections of two real submanifolds M,N in general position in CP^n in terms of certain canonical integrals over M and N separately; this was worked out in low dimensions (n <= 4 or so) by Kang/Tasaki and by Park. Making use of the structure of the algebra, including its structure as a representation of sl_2, we give a complete, albeit complicated, answer for all n. This is joint work with A. Bernig.

VIGRE–Algebra
3:30pm, Room 304

VIGRE - Hodge Theoretic questions in Algebraic Geometry
3:30pm, Room 303