Seminars and Colloquia by Series

Color-Critical Graphs on Surfaces

Series
Dissertation Defense
Time
Thursday, August 19, 2010 - 10:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 114
Speaker
Carl YergerSchool of Mathematics, Georgia Tech
A graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In this thesis, we study the structure of critical graphs on higher surfaces. One major result in this area is Carsten Thomassen's proof that there are finitely many 6-critical graphs on a fixed surface. This proof involves a structural theorem about a precolored cycle C of length q. In general terms, he proves that a coloring \phi of C can be extended inside the cycle, or there exists a subgraph H with at most 5^{q^3} vertices such that \phi cannot be extended to a 5-coloring of H. In Chapter 2, we provide an alternative proof that reduces the number of vertices in H to be cubic in q. In Chapter 3, we find the nine 6-critical graphs among all graphs embeddable on the Klein bottle. Finally, in Chapter 4, we prove a result concerning critical graphs related to an analogue of Steinberg's conjecture for higher surfaces. We show that if G is a 4-critical graph embedded on surface \Sigma, with Euler genus g and has no cycles of length four through ten, then |V(G)| \leq 2442g + 37.

Noncommutative Geometry and Compact Metric Spaces

Series
Dissertation Defense
Time
Monday, May 3, 2010 - 11:00 for 2 hours
Location
Skiles 255
Speaker
Ian PalmerGeorgia Tech
A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ—in particular, it does not depend on any self-similarity or regularity conditions on the space or an embedding in an ambient space. The only restriction on the space is that it have positive s-dimensional Hausdorff measure, where s is the Hausdorff dimension of the space, assumed to be finite.

Asymptotic Properties of Muntz Orthogonal Polynomials

Series
Dissertation Defense
Time
Tuesday, April 27, 2010 - 13:00 for 2 hours
Location
Skiles 269
Speaker
Ulfar StefanssonSchool of Mathematics, Georgia Tech
Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics and endpoint limit asymptotics on the interval. The zero spacing behavior follows, as well as estimates for the smallest and largest zeros. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval and the asymptotic properties of the associated Christoffel functions.

Estimates for Discrepancy and Calderon-Zygmund Operations

Series
Dissertation Defense
Time
Monday, April 26, 2010 - 15:00 for 2 hours
Location
Skiles 255
Speaker
Armen VagharshakyanSchool of Mathematics, Georgia Tech
We improve the lower bound for the L_\infty norm of the discrepancy function. This result makes a partial step towards resolving the Discrepancy Conjecture. Being a theorem in the theory of irregularities of distributions, it also relates to corresponding results in approximation theory (namely, the Kolmogorov entropy of spaces of functions with bounded mixed derivatives) and in probability theory (namely, Small Ball Inequality - small deviation inequality for the Brownian sheet). We also provide sharp bounds for the exponential Orlicz norm and the BMO norm of the discrepancy function in two dimensions. In the second part of the thesis we prove that any sufficiently smooth one-dimensional Calderon-Zygmund convolution operator can be recovered through averaging of Haar shift operators. This allows to generalize the estimates, which had been previously known for Haar shift operators, to Calderon-Zygmund operators. As a result, the A_2 conjecture is settled for this particular type of Calederon-Zygmund operators.

A Study of Discrepancy Results in Partially Ordered Sets

Series
Dissertation Defense
Time
Friday, March 19, 2010 - 09:00 for 3 hours
Location
Skiles 269
Speaker
David HowardSchool of Math, Georgia Tech
In 2001, Fishburn, Tanenbaum, and Trenk published a series of two papers that introduced the notions of linear and weak discrepancy of a partially ordered set or poset. Linear discrepancy for a poset is the least k such that for any ordering of the points in the poset there is a pair of incomparable points at least distance k away in the ordering. Weak discrepancy is similar to linear discrepancy except that the distance is observed over weak labelings (i.e. two points can have the same label if they are incomparable, but order is still preserved). My thesis gives a variety of results pertaining to these properties and other forms of discrepancy in posets.

Gabor and Wavelet Analysis with Applications to Schatten Class Integral Operators

Series
Dissertation Defense
Time
Thursday, March 11, 2010 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Van Leer Building Room W225
Speaker
Shannon BishopSchool of Mathematics, Georgia Tech
This thesis addresses four topics in the area of applied harmonic analysis. First, we show that the affine densities of separable wavelet frames affect the frame properties. In particular, we describe a new relationship between the affine densities, frame bounds and weighted admissibility constants of the mother wavelets of pairs of separable wavelet frames. This result is also extended to wavelet frame sequences. Second, we consider affine pseudodifferential operators, generalizations of pseudodifferential operators that model wideband wireless communication channels. We find two classes of Banach spaces, characterized by wavelet and ridgelet transforms, so that inclusion of the kernel and symbol in appropriate spaces ensures the operator if Schatten p-class. Third, we examine the Schatten class properties of pseudodifferential operators. Using Gabor frame techniques, we show that if the kernel of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. This result improves existing theorems and is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class. The implications of this result for the Kohn-Nirenberg symbol of a pseudodifferential operator are also described. Lastly, Fourier integral operators are analyzed with Gabor frame techniques. We show that, given a certain smoothness in the phase function of a Fourier integral operator, the inclusion of the symbol in appropriate mixed modulation spaces is sufficient to guarantee that the operator is Schatten p-class.

Eigenvalue Inequalities for Relativistic Hamiltonians and Fractional Laplacian

Series
Dissertation Defense
Time
Tuesday, October 27, 2009 - 13:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 269
Speaker
Selma Yildirim-YolcuSchool of Mathematics, Georgia Tech
In this thesis, some eigenvalue inequalities for Klein-Gordon operators and restricted to a bounded domain in Rd are proved. Such operators become very popular recently as they arise in many problems ranges from mathematical finance to crystal dislocations, especially the relativistic quantum mechanics and \alpha-stable stochastic processes. Many of the results obtained here concern finding bounds for some spectral functions of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogues results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the operator Hm, are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved.

Some Results on Linear Discrepancy for Partially Ordered Sets

Series
Dissertation Defense
Time
Thursday, October 1, 2009 - 14:00 for 2 hours
Location
Skiles 255
Speaker
Mitch KellerSchool of Mathematics, Georgia Tech
Tanenbaum, Trenk, and Fishburn introduced the concept of linear discrepancy in 2001, proposing it as a way to measure a partially ordered set's distance from being a linear order. In addition to proving a number of results about linear discrepancy, they posed eight challenges and questions for future work. This dissertation completely resolves one of those challenges and makes contributions on two others. This dissertation has three principal components: 3-discrepancy irreducible posets of width 3, degree bounds, and online algorithms for linear discrepancy. The first principal component of this dissertation provides a forbidden subposet characterization of the posets with linear discrepancy equal to 2 by completing the determination of the posets that are 3-irreducible with respect to linear discrepancy. The second principal component concerns degree bounds for linear discrepancy and weak discrepancy, a parameter similar to linear discrepancy. Specifically, if every point of a poset is incomparable to at most \Delta other points of the poset, we prove three bounds: the linear discrepancy of an interval order is at most \Delta, with equality if and only if it contains an antichain of size \Delta+1; the linear discrepancy of a disconnected poset is at most \lfloor(3\Delta-1)/2\rfloor; and the weak discrepancy of a poset is at most \Delta-1. The third principal component of this dissertation incorporates another large area of research, that of online algorithms. We show that no online algorithm for linear discrepancy can be better than 3-competitive, even for the class of interval orders. We also give a 2-competitive online algorithm for linear discrepancy on semiorders and show that this algorithm is optimal.

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