### Testing independence of regression errors with residuals as data

- Series
- Job Candidate Talk
- Time
- Tuesday, December 8, 2009 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Xia Hua – Massachusetts Institute of Technology

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- Series
- Job Candidate Talk
- Time
- Tuesday, December 8, 2009 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Xia Hua – Massachusetts Institute of Technology

In a regression model, say Y_i=f(X_i)+\epsilon_i, where (X_i,Y_i) are observed and f is an unknown regression function, the errors \epsilon_i may satisfy what we call the "weak'' assumption that they are orthogonal with mean 0 and the same variance, and often the further ``strong'' assumption that they are i.i.d. N(0,\sigma^2) for some \sigma\geq 0. In this talk, I will focus on the polynomial regression model, namely f(x) = \sum_{i=0}^n a_i x^i for unknown parameters a_i, under the strong assumption on the errors. When a_i's are estimated via least squares (equivalent to maximum likelihood) by \hat a_i, we then get the {\it residuals} \hat epsilon_j := Y_j-\sum_{i=0}^n\hat a_iX_j^i. We would like to test the hypothesis that the nth order polynomial regression holds with \epsilon_j i.i.d. N(0,\sigma^2) while the alternative can simply be the negation or be more specific, e.g., polynomial regression with order higher than n. I will talk about two possible tests, first the rather well known turning point test, and second a possibly new "convexity point test.'' Here the errors \epsilon_j are unobserved but for large enough n, if the model holds, \hat a_i will be close enough to the true a_i so that \hat epsilon_j will have approximately the properties of \epsilon_j. The turning point test would be applicable either by this approximation or in case one can evaluate the distribution of the turning point statistic for residuals. The "convexity point test'' for which the test statistic is actually the same whether applied to the errors \epsilon_j or the residuals \hat epsilon_j avoids the approximation required in applying the turning point test to residuals. On the other hand the null distribution of the convexity point statistic depends on the assumption of i.i.d. normal (not only continuously distributed) errors.

- Series
- Job Candidate Talk
- Time
- Monday, December 7, 2009 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Robert Young – IHES/Courant

The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an openquestion; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n>=4. This differs from the behavior for n=2 (when the Dehn function is linear) and for n=3 (when it is exponential). I have proved Thurston's conjecture when n>=5, and in this talk, I will give an introduction to the Dehn function, discuss some of the background of the problem and, time permitting, give a sketch of the proof.

- Series
- Other Talks
- Time
- Monday, December 7, 2009 - 08:00 for 8 hours (full day)
- Location
- University of Alabama, Birmingham
- Speaker
- Southeast Geometry Seminar – University of Alabama, Birmingham

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions:

- The University of Alabama at Birmingham
- The Georgia Institute of Technology
- Emory University
- The University of Tennessee Knoxville

The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology.

- Natasa Sesum (U Penn)
- Alexandru Ionescu (U Wisconsin)
- Sergiu Klainerman (Princeton U)
- Alex Freire (U Tennessee Knoxville)
- Christian Hainzl (UAB)

A poster session will be hosted. There will also be an evening public lecture by plenary speaker Sergiu Klainerman entitled The Mathematical Magic of Black Holes.

- Series
- Other Talks
- Time
- Friday, December 4, 2009 - 15:00 for 1.5 hours (actually 80 minutes)
- Location
- Skiles 168
- Speaker
- Michelle Delcourt and Leo Chen – School of Mathematics, Georgia Tech

Leo Chen: The Shape and Stability of a Flexible Sheet in a von Karman Vortex Street

Michelle Delcourt: Dessin and Manturov bracket shuffles

In this talk we will explore the connections between knot theory and combinatorics. Links are related to Grothendieck's dessins d'enfants. Cartographic one-vertex dessins can be represented by chord diagrams. The diagrams can be recorded as "words" using a finite alphabet (k-bracket parenthesis system). Many combinatorial objects are related to these Manturov bracket structures.

- Series
- Geometry Topology Working Seminar
- Time
- Friday, December 4, 2009 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Jim Krysiak – School of Mathematics, Georgia Tech

This talk will mostly be exposition on a result of M. Ghomi that
any C^2 knot in R^n can be C^1 perturbed into a knot of constant curvature
while preserving any smoothness properties.

- Series
- Graph Theory Seminar
- Time
- Thursday, December 3, 2009 - 12:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Carl Yerger – Math, GT

A fundamental question in topological graph theory is as follows: Given a surface S and an integer t > 0, which graphs drawn in S are t-colorable? We say that a graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In 1993, Carsten Thomassen showed that there are only finitely many six-critical graphs on a fixed surface with Euler genus g. In this talk, I will describe a new short proof of this fact. In addition, I will describe some structural lemmas that were useful to the proof and describe a list-coloring extension that is helpful to ongoing work that there are finitely many six-list-critical graphs on a fixed surface. This is a joint project with Ken-ichi Kawarabayashi of the National Institute of Informatics, Tokyo.

- Series
- Analysis Seminar
- Time
- Wednesday, December 2, 2009 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Alexander Stokolos – Georgia Southern University

I will speak about an extension of Cordoba-Feﬀerman Theorem on the equivalence between boundedness properties of certain classes of maximal and multiplier operators. This extension utilizes the recent work of Mike Bateman on directional maximal operators as well as my work with Paul Hagelstein on geometric maximal operators associated to homothecy invariant bases of convex sets satisfying Tauberian conditions.

- Series
- Other Talks
- Time
- Wednesday, December 2, 2009 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Kangkang Wang – School of Mathematics, Georgia Tech

We will present a sheaf-theoretic proof of the Riemann-Roch theorem
for projective nonsingular curves.

- Series
- Research Horizons Seminar
- Time
- Wednesday, December 2, 2009 - 12:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 171
- Speaker
- John McCuan – School of Mathematics, Georgia Tech – mccuan@math.gatech.edu

I will describe several geometrical problems that arise from
the minimization of some sort of integral functional and the basic
relation between such minimization and partial differential equations.
Then I will make some further comments on my favorite kind of such
problems, namely those that have something to do with minimizing area of
surfaces under various side conditions.

- Series
- Mathematical Biology Seminar
- Time
- Wednesday, December 2, 2009 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Laura Miller – University of North Carolina at Chapel Hill

The Reynolds number (Re) is often used to describe scaling effects in ﬂuid dynamics and may be thought of as roughly describing the ratio of inertial to viscous forces in the ﬂuid. It can be shown that ’reciprocal’ methods of macroscopic propulsion (e.g. ﬂapping, undulating, and jetting) do not work in the limit as Re approaches zero. However, such macroscopic forms of locomotion do not appear in nature below Re on the order of 1 − 10. Similarly, macroscopic forms of feeding do not occur below a similar range of Reynolds numbers. The focus of this presentation is to describe the scaling effects in feeding and swimming of the upside down jellyﬁsh (Cassiopeia sp.) using computational fluid dynamics and experiments with live animals. The immersed boundary method is used to solve the Navier-Stokes equations with an immersed, flexible boundary. Particle image velocimetry is used to quantify the flow field around the live jellyfish and compare it to the simulations.