Seminars archive

In 1946 Erdos asked to determine or estimate the minimum number of distinct distances determined by an n-element planar point set V. He showed that a square integer lattice determines \Theta(n/\sqrt{log n}) distinct distances, and conjectured that any n-element point set determines at least n^{1−o(1)} distinct distances. In 2010-2015, Guth and Katz answered Erdos’s question by proving that any n-element planar point set determines at least \Omega(n/log n) distinct distances. In this talk, we consider a variant of this problem by Erdos and Gyarfas. For integers n, p, q with p \geq q \geq 2, determine the minimum number D(n,p,q) of distinct distances determined by a planar n-element point set V with the property that any p points from V determine at least q distinct distance. In a recent paper, Fox, Pach and Suk prove that when q = {p \choose 2} - p + 6, D(n,p,q) is always at least n^{8/7 - o(1)}. We will discuss a recent improvement of their result and some new bounds for a related (graph theoretic) Ramsey problem of Erdos and Shelah which arise. This is joint work with Adam Sheffer.

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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.

Data privacy is an important issue to address when multiple data owners are required to integrate and share sensitive information for data analysis. We study the privacy threats caused by distributed data sharing and present the first cloud-based data sharing framework to integrate horizontally partitioned data from multiple data owners. The cloud performs the anonymization in a top-down fashion. It proceeds from the most generalized values of attributes (serve as the root of the tree) and specializes them (i.e., generate less generalized values as siblings of the parent node) in every iteration. A candidate value is selected for specialization in each iteration based on its score. The score of each candidate is calculated securely using multiple cryptographic protocols to ensure security. Finally, the cloud adds noise to the integrated data and releases them in a differentially private manner.

It is a classical result going back to Kronecker and Hurwitz that class numbers of binary quadratic forms satisfy interesting recurrence relations, which relate them to arithmetic functions. In this talk, we will see how the theory of mock modular forms may be used to prove infinite families of such identities.

Computational modeling of healthy and diseased electrocardiology (EC) has a great potential for use in improved diagnosis and prognosis of cardiac arrhythmia and also better therapy planning. However, recent computational methods in electrocardiology usually suffer from three major limitations that hinder their use in the clinic: lack of efficient model personalization strategy; high computational demand from the EC solver; lack of good trade-off between the simplification of cellular ionic models and the demand on keeping sufficient biophysical details. The work in this thesis aims at solving these challenging issues. The principal part of the work is on the estimation of cardiac conductivities that parameterize the bidomain/monodomain model—the current standard model for simulating cardiac potential propagation. We consider a variational data assimilation approach by regarding the parameters as control variables to minimize the mismatch between computed and measured potentials. The existence of a minimizer of this misfit function is proved. We significantly improve the numerical approaches in the literature by resorting to a derivative-based optimization method with the settlement of some challenges due to discontinuity. The core of our numerical results is in 3D, on both idealized and real geometries. The reliability and stability of the conductivity estimation approach are demonstrated in presence of noise and with an imperfect knowledge of other model parameters. We then focus on the computational cost reduction for the inverse conductivity problem. The Proper Orthogonal Decomposition (POD) approach was taken for forward model reduction, along with the Discrete Empirical Interpolation Method (DEIM) for tackling nonlinearity. The POD-DEIM combination is finally applied for the inverse problem of conductivity estimation. In this application, we obtain a very small set of samples by sampling the parameter space using the polar coordinates and densifying the boundary layer utilizing Gauss-Lobatto nodes. In usage of the POD-DEIM reduced order model, the computational effort can be reduced to up to 10% in conductivity estimation. The last part of the work is on the development of a data-driven approach to the reduction of state-of-the-art cellular models used for atria simulation. The reduced model predicts cellular action potentials (AP) in a simple form but is effective in capturing the physiological complexity of the original model. We start from an AP manifold learning, and continue with a regression model construction to predict few leading components in the reduced AP manifold. The reduced cellular model drastically improves the performance of atrial tissue-level electrophysiological modeling (up to two order of magnitudes) and enables almost real-time computations. The same modeling technique can be extended to the study of other excitable myocardial tissues.

Coronary stents expand the vessel to alleviate blockage, and they have been used for decades to save lives from coronary heart disease. Technological development has driven the evolution of stenting from bare or drug-coated metallic stents to the newly invented bioresorbable vascular stent (BVS). Rather than leaving a permanent metallic structure in the vessel, a BVS can dissolve in the body after opening the blocked artery, restoring the diseased coronary artery to its natural state. The BVS compensates for its decreased radial strength by having thicker struts that could cause disturbed blood flow, resulting in delayed healing and other devastating complications. Computational fluid dynamics (CFD) simulations had been used to analyze the potential risk of the BVS, but those simulations were conducted on either arbitrary geometry or partially patient-specific geometry with overly smoothed or virtually deployed stents. In this thesis, we present a novel methodology to reconstruct true patient-specific CFD simulations: the geometry of a deployed stent inside a living patient is reconstructed from optical coherence tomography (OCT) images; the shape and the curvature of the stented vessel are obtained from angiography; and a real pulsatile flow rate profile can also be prescribed as inflow condition, when it is available. With patient-specific geometry, the CFD results reflect the true hemodynamics after stent deployment and describe the wall shear stress (WSS) and other quantities. We also aim to make the reconstruction and simulation process automatic, so that a large number of patients can be processed in a short time in order to draw statistically convincing conclusions. In fact, the entire process of the computational patient-specific analysis is expected to become a routine in clinical trials and activities (Computer Aided Clinical Trials, CACT). This has created significant challenges in the methodological approach, ranging from image analysis, image processing, computational geometry and eventually fluid dynamics. These work witnesses the different challenges of this multi-component procedure. We hope the patient-specific reconstruction and simulations study can further the understanding of the BVS, improve its design, and ultimately expedite the tedious validation process so that it can soon become a soldier for us in the battle against coronary heart disease.

The Big Data phenomenon is posing new challenges in our modern society. In addition to requiring information systems to effectively manage high-dimensional and complex data, the privacy and social implications associated with the data collection, data analytics, and service requirements create new important research problems. First, the high volume of personal data generated by users' devices (e.g. credit card transactions, GPS trajectories from mobile devices and medical data) can be used, much like a fingerprint, to identify the person who created it with the risk of disclosing sensitive information such as: political inclination, financial status and medical condition. Second, popular social networks (e.g. Facebook, Foursquare, Yelp) not only enable users to share locations and preference but also create opportunities for them to establish complex interactions (e.g. forming communities, planning trip). This creates the needs for location based services to provide services to groups of users rather than individuals. In this dissertation, we present effective solutions for both these privacy and social challenges. In the privacy domain, we propose new privacy preserving techniques to provide individual users with formal guarantee of privacy while at the same time preserve meaningful information of the data released. We demonstrate the effectiveness of our solutions in different domains such as: sequential pattern mining, record linkage, and computation of statistics over data streams. In the social domain, we propose a new type of group query aiming to find a route that all users can traverse while maximizing the group preference for the locations jointly visited. The ability of solving such query can greatly benefit many existing and emerging tools that allow users to share route information (e.g. Uber, Waze) and plan group outings or trips (e.g. QuickCliqs). Extensive empirical studies demonstrate the effectiveness of our solutions and provide us with important insights for future research directions.

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