### Sharp Weighted Estimates for Singular Integrals

- Series
- Dissertation Defense
- Time
- Tuesday, March 1, 2011 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Maria del Carmen Reguera Rodriquez – School of Mathematics, Georgia Tech

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- Series
- Dissertation Defense
- Time
- Tuesday, March 1, 2011 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Maria del Carmen Reguera Rodriquez – School of Mathematics, Georgia Tech

- Series
- Dissertation Defense
- Time
- Thursday, February 3, 2011 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Sergio Angel Almada – School of Mathematics, Georgia Tech

A stochastic differential equation with vanishing martingale term is
studied.
Specifically, given a domain D, the asymptotic scaling properties of both
the exit time from the domain and the exit distribution are considered under
the additional (nonstandard) hypothesis that the initial condition also has
a scaling limit. Methods from dynamical systems are applied to get more
complete estimates than the ones obtained by the probabilistic large
deviation theory.
Two situations are completely analyzed. When there is a unique critical
saddle
point of the deterministic system (the system without random effects), and
when
the unperturbed system escapes the domain D in finite time. Applications to
these results are in order. In particular, the study of 2-dimensional
heteroclinic networks is closed with these results and shows the existence
of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare
events are further studied using these results as building blocks.

- Series
- Dissertation Defense
- Time
- Tuesday, October 5, 2010 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 114
- Speaker
- Tobias Hurth – School of Mathematics, Georgia Tech

We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute the unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero or infinity and derive analogues of classical probability theory results such as central limit theorem and large deviation principle.

- Series
- Dissertation Defense
- Time
- Thursday, August 19, 2010 - 10:00 for 1.5 hours (actually 80 minutes)
- Location
- Skiles 114
- Speaker
- Carl Yerger – School 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.

- Series
- Dissertation Defense
- Time
- Thursday, July 1, 2010 - 10:30 for 1.5 hours (actually 80 minutes)
- Location
- Skiles 154
- Speaker
- Ramazan Tinaztepe – School of Mathematics, Georgia Tech

- Series
- Dissertation Defense
- Time
- Monday, May 3, 2010 - 11:00 for 2 hours
- Location
- Skiles 255
- Speaker
- Ian Palmer – Georgia Tech – icpalm@math.gatech.edu

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.

- Series
- Dissertation Defense
- Time
- Tuesday, April 27, 2010 - 13:00 for 2 hours
- Location
- Skiles 269
- Speaker
- Ulfar Stefansson – School 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.

- Series
- Dissertation Defense
- Time
- Monday, April 26, 2010 - 15:00 for 2 hours
- Location
- Skiles 255
- Speaker
- Armen Vagharshakyan – School 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.

- Series
- Dissertation Defense
- Time
- Friday, March 19, 2010 - 09:00 for 3 hours
- Location
- Skiles 269
- Speaker
- David Howard – School 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.

- 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 Bishop – School 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.