Seminars archive

The query processing of relational data has been studied extensively throughout the past decade. A number of theoretical and practical solutions to query processing have been proposed under various scenarios. With the recent popularity of cloud computing, data owners now have the opportunity to outsource not only their data, but also the data management tasks to the cloud. Because of data security and personal privacy concerns, sensitive data (e.g. medical records) should be encrypted before being outsourced to a cloud, and the cloud should perform query processing tasks on the encrypted data only. In this talk, I will present our current research on the development of secure distributed protocols to facilitate query processing over encrypted data hosted in the cloud. I will also explore possible future research directions.

Image registration is a key task in image analysis. Its applications range from fusing multimodal data over object tracking to motion modeling, e.g. for the respiratory system. In the latter example large motions occur and inside the lungs no foldings of tissue are to be expected. Hence it is appropriate to model the movement as a diffeomorphic nonlinear trans- formation. As requirements like diffeomorphic transformations and the capability of capturing large motions are often necessary in image registration the ”Large Deformations Diffeomorphic Metric Mapping (LDDMM)” approach is very useful. The theoretical foundations for LDDMM were laid in the late 1990s and the beginning of the 2000s by Grenander, Christensen, Miller, Trouve, Younes and others. In 2005 Beg et al. [1] provided a practical algorithm to solve the LDDMM image registration problem. LDDMM is related to optical flow. In [2] optical flow problems were solved using an optimal control approach. Following a similar approach in 2009 the LDDMM model was used for image registration from an optimal control perspective in [3]. In the talk I will give an introduction to LDDMM following loosely [3] and start with the matching of scalars, which results in a tool for linear regression. In this example, the two different concepts of relaxation and shooting are illustrated. In the relaxation approach the optimization is performed regarding the complete temporal velocity field. However, in the shooting approach the optimization is only over the initial condition, i.e., slope and possibly y-intercept of the line. I will then discuss how to extend these ideas to the problem of image matching and how to discretize the optimization problem using consistent Runge-Kutta methods for the transport equation and its adjoint. References [1] Mirza Faisal Beg, Michael I. Miller, Alain Trouve, and Laurent Younes. Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61(2):139–157, 2005. [2] Alfio Borzi, Kazufumi Ito, and Karl Kunisch. Optimal Control Formulation for Determining Optical Flow. SIAM Journal on Scientific Computing, 24(3):818–847, 2003. [3] Gabriel L. Hart, Christopher Zach, and Marc Niethammer. An optimal control approach for deformable registration. 2009 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2(1), 2009.

The behavior of norms of roots of univariate trinomials with respect to the choice of coefficients is a classical late 19th and early 20th century problem. In 1908, P. Bohl characterized the parameter space, but only in an algebraic way. By using amoeba theory we uncover a beautiful geometric and topological structure in the corresponding parameter space. More precisely, we show that norms of roots of trinomials are geometrically characterized by hypo-epitrochoids and its parameter space is topologically characterized by torus knots.

A graph is called diameter-k-critical if its diameter is k, and the removal of any edge strictly increases the diameter. In this talk, we will present several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-k-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and Haggkvist (that in every diameter-2-critical graph, the average edge-degree is at most the number of vertices), which promised to completely solve the extremal problem for diameter-2-critical graphs. On the other hand, we prove that the same claim holds for all higher diameters, and is asymptotically tight, resolving the average edge-degree question in all cases except diameter-2. We also apply our techniques to prove several bounds for the original extremal question, including the correct asymptotic bound for diameter-k-critical graphs, and an upper bound of (\frac{1}{6} + o(1))n^2 for the number of edges in a diameter-3-critical graph. This is a joint work with Po-Shen Loh.

Glycomics is the study of the structure and function of carbohydrates in biological systems. In comparison to the expansion of the more established fields of genomics and proteomics, the integration of glycans and glycomics in biomedical research has lagged far behind. Glycomics has the potential to be included as another foundational science in the study of human disease, since glycans play have major roles in certain hereditary diseases, infectious diseases, and cancer. The structural and functional complexity of glycans coupled with the lack of robust bioinformatics impedes the integration of glycoscience into the scientific mainstream. The central objective of this thesis is to develop novel computational methods and bioinformatics tools to advance the understanding of structure and function relationships of glycans and their recognition and binding by Glycan Binding Proteins (GBPs). We have developed a method to automate the interpretation of glycan microarray data to identify the glycan determinants that are necessary for binding. We evaluate this method against GBPs of known specificities to validate the results. We demonstrate this approach revealed new recognition motifs that had not been previously reported. We also present a novel computational approach to automate the sequencing of glycans based on a method known as “Metadata-Assisted Glycan Sequencing” (MAGS), which combines analyses of glycan structures by mass spectrometry (MS) and glycan microarray technology to fully characterize glycan sequences. We target the soluble glycans in the human milk glycome as the first meta-glycome to be defined using this method. To facilitate access by scientists to glycomics information, we developed an open-source, web-based bioinformatics platform for glycan microarray analysis. The platform provides interactive visualization features to view, search, and compare experimental data and also includes glycan motif mining and analysis. In addressing these research areas, we have developed novel methods, algorithms, and software tools applied to the field of glycomics. These contributions will aid in the elucidation of the human glycome and a greater understanding of the diverse and important biological functions of glycans.

We will introduce how the geometry of the upper half plane model of the hyperbolic plane can be use to calculate Dedekind sums. Using this geometric perspective, "generalize'' Dedekind sums can emerge from the existence of very particular (non-arithmetic) Fuchsian groups sitting inside of mathrm{PSL}_2 (mathbb{Q}). We will conclude by discussing the difficulties in forming an analogue of Dedekind reciprocity and other avenues for further investigation.

See downloadable flyer.

In recent work, we found a new proof of a conjecture due to Erdös and Sós stating that large weakly quasirandom 3-uniform hypergraphs with density greater than 1/4 contain four vertices spanning at least three hyperedges. This was proved earlier by Glebov, Kral, and Volec using flag algebras and computers. The new proof is based on the hypergraph regularity method and gave rise to further developments some of which are surveyed in this talk. This is joint work with Vojtech Rodl and Mathias Schacht.

A tile is a finite subset T of mathbb{Z}^n. It may or may not tile mathbb{Z}^n, in the sense of mathbb{Z}^n having a partition into copies of T. However, Chalcraft conjectured that every TA does tile mathbb{Z}^d for some d. In this talk, we will discuss some examples, and also a proof of the conjecture, recently obtained in joint work with Vytautas Gruslys and Ta Sheng Tan.

The Palmetto Number Theory Series (PANTS) is a series of number theory meetings held at colleges and universities in South Carolina, the Palmetto State, as well as neighboring states. The conference will last until Sunday afternoon. The goal of the PANTS meetings is to provide an opportunity for number theorists in South Carolina, and more generally, in the Southeast, to hear about recent research in all areas of number theory, pure and applied. See http://www.mathcs.emory.edu/pantsxxiv/ for more details.