Seminar Schedule
September 13 – September 17, 2004

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

MONDAY, September 13, 2004

Algebra
2:30-3:30p.m., Room 410
Speaker: Jon Kujawa, University of Georgia
Title of talk: Representation Type of Schur Superalgebras
Abstract: A fundamental problem in the representation theory of finite dimensional
algebras is to determine how many indecomposible modules a particular algebra has;
that is, its representation type. In very recent work with Hemmer and Nakano we
solve this problem for the Schur superalgebras. This talk should be accessible to all.
Only the basics of module theory will be assumed.

Probability Theory
2:45-3:45pm Room 302
Speaker: M. Pemy, University of Georgia
Title of talk: The valulation of European option in the Black-Scholes model.

Topology
2:30-3:30pm, Room 326
Speaker: Paolo Lisca, University of Pisa
Title of talk: Ozsvath-Szabo invariants and tight contact structures, I
Abstract: This is the first of two talks in which I will describe recent joint work with Andras Stipsicz on the existence of tight contact structures on closed oriented 3-manifolds. I will state our results and then describe the two main tools needed to prove them, namely contact surgery and the Ozsvath-Szabo invariants for contact structures.

3:45-4:45, Room 326
Speaker: Paolo Lisca, University of Pisa
Title of talk: Ozsvath-Szabo invariants and tight contact structures, II
Abstract: This is the second of two talks on joint work with Andras Stipsicz. I will recall the results stated in the first talk and then outline some of their proofs.

Lie Theory
3:30p.m., Room 303
No Meeting this week

TUESDAY, September 14, 2004

VIGRE Graduate Student Seminar
2:00p.m., Room 304
Speaker: Wu Li, NASA Langley Research Center
Title of talk: Multidisciplinary Optimization Branch
Summary: An important NASA mission is to develop new technologies for space exploration and for improvement of life on the earth. To carry out this mission with a limited budget, NASA decision makers need to invest in technologies that have the highest probable impact on future aerospace vehicles. To assist in making an informed decision on technology investment, NASA systems analysts use low-fidelity designs (called conceptual designs) to assess and demonstrate the benefits of new technologies. The analysts either evaluate a baseline conceptual design and assess the impacts of various technologies on that design, or use new technologies to create revolutionary systems concepts. Consequently, methods for credible and efficient analysis and design of complex aerospace systems are a critical part of the NASA technology development program.

For the purpose of this talk, fidelity means the accuracy of a model in representing the physical reality. There are two types of fidelity for systems simulation: (i) accuracy of computed systems performance metrics and (ii) resolution of systems description. For example, analysts can use measured data or a simulation model to estimate the lift and drag coefficients of an aerial vehicle. Of course, measured data has a higher fidelity than any simulation data but appropriate data may not be available. In the absence of measured data, a computational fluid dynamics (CFD) solver can estimate the performance metrics (lift and drag) of the vehicle by using a low-fidelity panel code or a high-fidelity Navier-Stokes code. Moreover, for the same CFD code, the fidelity of generated simulation results depends on the grid resolution; and finer grid resolution usually provides increased accuracy of the estimated lift and drag coefficients. Regardless of the CFD fidelity, the structural finite element model (FEM) can be described by using a few parameters that specify a generic shape and simple load bearing structure, or by using hundreds or thousands parameters that capture many critical features of the vehicle structure. These two types of fidelity for systems simulation are usually related to each other, and high-fidelity analysis is employed only when low-fidelity analysis cannot provide credible assessment of the response of the underlying design concept.

Use of high-fidelity analysis tools creates many problems for systems analysts. For example, an analysis code may take hours or even days to generate one response, high-fidelity analysis codes may be difficult to interface with existing conceptual design codes, the needed CFD and FEM models may not exist, and the existing models may not be parameterized appropriately. These problems in obtaining high-fidelity simulation results could deter analysts from meaningful exploration of the design space and disrupt the interactive design process whereby each analysis result provides insight needed to improve the baseline design. Some of these problems may be solved by replacing high-fidelity analysis codes by approximation models, which is the motivation for a comprehensive survey of existing methods for approximation of system responses in conceptual design.

All of the requirements related to the use of approximation methods in conceptual design can be categorized into the following two use cases: (i) to predict a performance metric that can only be estimated by using historical data or expensive computer simulations, and (ii) to improve the accuracy of an existing low-fidelity analysis tool by using a few high-fidelity data points. The second use case arises from the need for tuning an existing low-fidelity model to reflect more accurate systems behaviors.

Due to the need for solving two use cases in conceptual design mentioned above, it is necessary to look beyond least-squares polynomial approximation or Kriging interpolation, and identify useful approximation methods that are promising for conceptual design applications. The main goal of the talk is to give an overview of advanced approximation methods for multivariate approximation problems that cannot be solved by using quadratic polynomial approximation or Kriging interpolation. We will highlight the pros and cons of advanced approximation methods, and identify some challenging problems in approximation method development from conceptual design perspective.

Dynamics on Berkovich Space
3:30-5:30p.m., Room 326
Speaker: Robert Rumely, University of Georgia
One topic of the seminar will be Rivera-Letelier's work on the dynamics of rational functions, including the action of a rational function on Berkovich space, his description of the Berkovich Julia and Fatou sets, and his classification of fixed points. Another topic will be Favre and Rivera-Letelier's proof of the equidistribution theorem for small points, relative to the canonical height attached to a dynamical system. Normally the seminar will meet every other week, for the rest of the semester.

I will give the first presentation, reviewing the basic facts about the Berkovich Line for those who did not take part in last year's seminar, and describing the action of a rational function on Berkovich space a la Rivera-Letelier.

WEDNESDAY, September 15, 2004

VIGRE Algebraic Geometry Group
12:20-1:10p.m., Room 326

Algebraic Geometry
2:30-3:45 p.m., Room 410
Speaker: Viacheslav Nikulin (University of Liverpool)
Title of talk: On correspondences of a K3 surface with itself
Abstract: Let $X$ be a K3 surface with a polarization $H$ of degree $H^2=2rs$, $r,\,s\ge 1$,
and with a primitive Mukai vector $(r,H,s)$. The moduli space of sheaves over $X$ with the
isotropic Mukai vector $(r,H,s)$ is again a K3 surface $Y$.

We prove that $Y\cong X$, if the Picard lattice $N(X)$ has an element $h_1$ with
$(h_1)^2=f(r,s)$, and the pair $(H,h_1)$ satisfies a finite number of congruence conditions
modulo $N_i(r,s)$. All these conditions are exactly written, and they are necessary, if $X$ is
general with $\rkN(X)=2$.

Existence of such kind a criterion is very surprising, and it also gives some geometric
interpretation of elements in $N(X)$ with negativesquare. We also describe all irreducible
divisorial conditions on moduli of $(X,H)$ which imply $Y\cong X$, and we prove that their number is
infinite if non-empty.

See math.AG/0309348, 0307355, 0304415, 0206158 for details.

VIGRE – Cardiac Physiology
2:30p.m., Room 323

VIGRE – Clifford Algebras
2:30p.m., Room 322

Faculty and Graduate Social
3:00 p.m., Room 409
Coffee, Cookies, Tea

Number Theory
3:45 p.m., Room 304
Speaker: TBA
Title of talk: TBA

THURSDAY, September 16, 2004

VIGRE - Rational points on curves
2:00p.m., Room 304

FRIDAY, September 17, 2004

Student Arithmetic/Algebraic Geometry Seminar
12:15p.m., Room 326
Speaker: Charles Pooh, University of Georgia
Title: TBA

VIGRE-Algebra
2:30p.m., Room 410
No Meeting this week

Spline Analysis
2:30-3:30p.m., Room 303
Speaker: Taytana Sorokina, University of Georgia
Title of talk: Construction of 3D splines, cont.

Geometry
2:30 p.m., Room 323
Speaker: Jason Parsley, University of Georgia
Title of talk: The Biot-Savart operator and linking integrals on S^3, part III

Wavelet Analysis
3:30-4:30 p.m., Room 303
Speaker: O. Cho, University of Georgia
Title of talk: A class of orthonormal refinable functions, cont.