Seminars archive

With the trend of cloud computing, data and computing are moved away from desktop and are instead provided as a service from the cloud. Data-as-a-service enables access to a wealth of data across distributed and heterogeneous data sources in the cloud. It remains a challenge, however, to ensure the privacy, interoperability, and scalability for such services.\\ \\ We designed and developed DObjects, a general-purpose P2P-based query and data operations infrastructure that can be deployed in the cloud and provides access to heterogeneous data sources. The system builds on top of a distributed mediator-wrapper architecture where individual system nodes serve as mediators and/or wrappers and interact with each other in a P2P fashion what guarantees good scalability. As an analogy, the system nodes can be considered as droplets, small elements that provide similar functionality in the cloud. Just as thousands or millions of droplets form a single drop in nature, in cloud computing, groups of droplets that provide similar functionality can form a micro-cloud. Micro-clouds are an integral part of the whole cloud computing system and can provide specific services to users.\\ \\ The dissertation also discusses the novel dynamic query execution engine within the data query infrastructure that dynamically adapts to network and node (or droplet) conditions. The query processing is capable of fully benefiting from all the distributed resources to minimize the query response time and maximize system throughput. In addition to leveraging the traditional distributed query optimization techniques, the (sub)queries are deployed and executed on droplets in a dynamic and iterative manner in order to guarantee the best reaction to network and resource dynamics.\\ \\ Finally, the dissertation presents an extension to the basic DObjects model that enables access to private data that is distributed and needs anonymization. The extension enables droplets to form virtual groups in order to addresses two important privacy issues for the sensitive data: privacy of data subjects and confidentiality of data providers. The dissertation discusses decentralized protocols that enable data sharing for horizontally partitioned databases given these constraints. These protocols can be run by the groups of droplets. Concretely, given a query spanning multiple databases, the query results do not contain individually identifiable information. In addition, institutions do not reveal their databases to each other apart from the query results.

TBA

Let $G$ be a finite connected graph with $E$ edges. The Kirchhoff polynomial of $P(G)$ is a certain homogeneous polynomial in $E$ variables whose degree is the first Betti number of the graph. These polynomials appeared classically in the study of electrical circuits (e.g., Kirchhoff's laws). They also appear in the evaluation of Feynman integrals. Motivated by computer calculations of D.~Kreimer and D.J.~Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of $P(G)$ over a field with $q$ elements is a polynomial function of $q$. P.~Belkale and I disproved this conjecture by relating the schemes $V(P(G))$ to the representation spaces of certain combinatorial objects called matroids. I will discuss this work and subsequent work on the number-theoretical properties of Feynman amplitudes.

Data from real-world graphs often contains incomplete information, so we only observe subgraphs of these graphs. It is therefore desirable to understand how a typical subgraph relates to the underlying host graph. We consider several interrelated problems involving both random trees in host graphs and random subgraphs obtained by taking edges of the host graph independently with probability p. In the second case, we study the emergence of the giant component. We also use the spectral gap to understand discrepancy and expansion properties of a random subgraph. The Erdos-Renyi random graph is the special case of this where the host graph is the complete graph Kn. Additional applications include taking a contact graph as the host graph, and viewing random subgraphs as outbreaks of a disease.

Abstract. We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble.

I will discuss some of the driving questions in the area of Rational points on algebraic varieties (these can be thought of as solutions over fields of number-theoretic interest to systems of homogeneous polynomial equations). I will focus on the case of smooth rational surfaces, and discuss some results concerning the arithmetic of del Pezzo surfaces of degree 1. I will also explain how these results complete a qualitative picture of basic arithmetic phenomena among smooth rational surfaces. Along the way I will go over concepts like weak approximation and the computation of Brauer-Manin obstructions; I will not assume previous knowledge of them.

Correlation inequalities are statements about how events in a probability space positively or negatively reinforce each other. After briefly discussing the better-understood theory of positive correlation, I will talk about some negative correlation inequalities and their relationship to celebrated conjectures of J. Mason about log-concavity properties of certain sequences arising from combinatorial objects. Along the way, I'll mention several interesting open problems.

Lehmer's conjecture (1933) states that the Mahler measure of an algebraic number that is not a root of unity is bounded away from 1. The aim of the seminar is to show the conjecture would imply there is a positive lower bound for closed geodesics in compact arithmetic hyperbolic 3-manifolds of finite volume. In the first lecture, I will introduce the necessary background material on arithmetic hyperbolic 3-manifolds. Then, in the second lecture, I will show how Lehmer's conjecture would imply the existence of the aforementioned positive lower bound. Also, I will prove the existence of a positive lower bound for closed geodesics in non-compact arithmetic hyperbolic 3-manifolds of finite volume to fully address the topic.

In this talk we will present some progress on several extremal problems related to quasiconformal mappings and Beltrami coefficients on the unit disk.

Lehmer's conjecture (1933) states that the Mahler measure of an algebraic number that is not a root of unity is bounded away from 1. The aim of the seminar is to show the conjecture would imply there is a positive lower bound for closed geodesics in compact arithmetic hyperbolic 3-manifolds of finite volume. In the first lecture, I will introduce the necessary background material on arithmetic hyperbolic 3-manifolds. Then, in the second lecture, I will show how Lehmer's conjecture would imply the existence of the aforementioned positive lower bound. Also, I will prove the existence of a positive lower bound for closed geodesics in non-compact arithmetic hyperbolic 3-manifolds of finite volume to fully address the topic.

A new convergence result for an Iteratively Regularized Gauss Newton (IRGN) algorithm with a Tikhonov regularization term using a seminorm generated by a linear operator will be presented [SRK07]. The convergence theorem uses an a posteriori stopping rule and a modified source condition, without any restriction on the nonlinearity of the operator. The theoretical results are illustrated by simulations for a 2D version of the exponentially ill-posed optical tomography inverse problem for the diffusion and absorption coefficient spatial distributions. The modified Tikhonov regularization performs the mapping of the minimization variables, which are the coefficients of the spline expansions for the diffusion and absorption, to physical space. This incorporates the inherently differing scales of these variables in the minimization, and also suggests relative weighting of the regularization terms with respect to each parameter space. The modified IRGN allows greater flexibility for implementations of iteratively regularized solutions of ill-posed inverse problems in which differing scales in physical space hinder standard IRGN inversions.