Past Seminars

In Applied Mathematics and Theoretical Computer Science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. Graph Partitioning is a fundamental technique in graph theory. Cutting a graph into smaller pieces is one of the basic algorithmic operations. Partitioning large graphs is often important for complexity reduction or parallelization. In practice, one often seeks to find a partition that minimizes (or maximizes) an objective. Graph partitioning technique is used widely to solve several combinatorial problems. The problem of simulating one network by another is modeled as a graph embedding problem. If a network A can be embedded in a network B, then all the algorithms developed for parallel processing with network A can be easily transported onto another processor network B. However graph embedding includes some combinatorial problems such as bandwidth problem, cutwidth problem, treewidth problem, wirelength problem, forwarding index problem, vertex ordering and topology mismatch. The quality of an embedding can be measured by certain cost criteria, namely dilation, congestion and wirelength. While the general problem of graph embedding is difficult, by exploiting the special structure of the interconnection schemes, a number of results relating to optimal embedding of one class of networks into another have been developed. In this talk, new theoretical results and techniques are explored and applied to find congestion and wirelength of different networks.

Combinatorial expressions are expressions made by composing two kinds of functions --- those with bounded arity and unbounded domain (example sum:\N\times \N \rightarrow \N), and those with unbounded arity and finite domain (example: Boolean disjunction of arbitrary many inputs). We establish lower bound results for such expressions. This is a joint work with Thomas Colcombet.

This paper formulates a structural dynamic programming model of preschool investment choices of altruistic parents and then empirically estimates the structural parameters of the model using the NLSY79 data. The paper finds that preschool investment significantly boosts cognitive and non-cognitive skills, which enhance earnings and school outcomes. It also finds that a standard Mincer earnings function, by omitting measures of non-cognitive skills on the right-hand side, overestimates the rate of return to schooling. From the estimated equilibrium Markov process, the paper studies the nature of within generation earnings distribution, intergenerational earnings mobility, and schooling mobility. The paper finds that a tax-financed free preschool program for the children of poor socioeconomic status generates positive net gains to the society in terms of average earnings, higher intergenerational earnings mobility, and schooling mobility.

Most of the talk will be a gentle introduction to the dilation theory, in the sense of Sz.-Nagy and C. Foias, and resolutions of free modules over the ring of polynomials, in the sense of R. Douglas. We will discuss several (old, new, strange and unknown) topics related to this approach including submodules and quotient modules of (reproducing kernel) Hilbert modules.

In this talk, we prove two density theorems for quadrature domains in $\mathbb{C}^n, n \geq 2$. It is shown that quadrature domains are dense in the class of all product domains of the form $D\times \Omega$ where $D \subset\mathbb{C}^{n-1}$ is a smoothly bounded pseudoconvex domain satisfying Bell's Condition R and $\Omega \subset \mathbb{C}$ is a smoothly bounded domain. It is also shown that quadrature domains are dense in the class of all smoothly bounded complete Hartogs domains in $\mathbb{C}^2$.

Bootstrap is a widely used statistical technique that is able to deal with a wide range of statistical inference problems without stringent structural assumptions on the underlying random process generating the data. Efron, in 1979, developed it for data with identically and independent observations. The literature on Bootstrap method for independent data is quite extensive. In this talk we are going to discuss some of the established Bootstrap methods for non-iid data, for example time series data.

For an electron system in a superconducting state the Meissner effect characterizes the response to a magnetic field. In type-II superconductors there is flux penetration beyond a threshold hc1 in the form of an Abrikosov lattice before superconductivity (SC) is finally lost at the orbital critical field hc2 orb. The magnetic field also couples to the spin of the electrons and tends to break the Cooper pair. This effect is detrimental to SC, and, if orbital effects were irrelevant, SC order would be lost at some “Pauli-limiting” field, hc2 P. The ratio of these critical fields, ?=hc2 orb/hc2 P, defines the Maki parameter. In most superconductors ??1, so the Pauli suppression effects never show up. There are, however, scenarios where it becomes relevant.
In this talk I would discuss about the situation where Pauli limiting behavior indeed becomes relevant. It would be shown that superfluidity in such a scenario no longer remains restricted to an homogeneous BCS like state. Population imbalance gives rise to coexisting magnetic and superfluid order at low polarizations leading to the 'breached pair' state, while beyond a critical polarization the system transits to a modulated superfluid (FFLO) state with finite momentum pairing between the fermions. With tools which can access fluctuations beyond the mean field theory I would map out the resulting phase diagram in comparison with the experimental observations. Further, the novel spectral features expected to be observed in the FFLO superfluid state would also be discussed in this talk.

In this talk, we present Turan type inequalities for various special functions, namely, Struve functions, general Bessel functions and modified Dini functions. Moreover, we study monotonicity patterns and functional inequalities for modified Dini functions, cross-product of Bessel functions and bounds for product of modified Bessel functions. In addition, we obtain interlacing properties of positive real zeros of Dini functions and cross-product of Bessel functions.

In this talk, we present Turan type inequalities for various special functions, namely, Struve functions, general Bessel functions and modified Dini functions. Moreover, we study monotonicity patterns and functional inequalities for modified Dini functions, cross-product of Bessel functions and bounds for product of modified Bessel functions. In addition, we obtain interlacing properties of positive real zeros of Dini functions and cross-product of Bessel functions.

In this talk, we discuss certain geometric properties of Gauss hypergeometric and Heine's hypergeometric functions defined on the unit disk. Using the continued frac- tion technique, we obtain integral representations of some ratios of basic hypergeometric functions. This extends to investigate the geometric properties of the ratios which say that the ratios are analytic in a cut-plane and maps the unit disk or half-planes univalently onto domains that are convex in the direction of the imaginary axis. Also, we focus on sufficient conditions for which the basic hypergeometric functions belonging to a special class of analytic functions and discuss some related open problems.

In joint work with Ji\v{r}\'{\i} Lebl and Alan Noell, we study the extension of CR functions near the CR singularity of real codimension two CR singular submanifolds. We prove regularity of the extension at the CR singularity under natural conditions on the submanifold: holomorphically-flat with a nondegenerate CR singularity. We will use several examples to highlight the key techniques and ideas involved.

We study the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries.

The structure of a complex network comprised of dissimilar units of a system interacting with each other, affects the dynamics of the system and provides deeper insight about the functionality of the system, its evolution, and the role of individual or modular units. Thus, network analysis has become a primary tool in the fields as diverse as biology, ecology, sociology, economics and finance. We discuss the role of networks and simple clustering algorithms in studying two socio-economic phenomena:
?Language dynamics: We construct the Levenshtein distance matrix between the Mazatec dialects in Mexico, and use complex networks and clustering algorithms to study the spatio-temporal dynamics of the dialects. Our analyses reinforce the hypotheses of Gudschinsky regarding the ethno-history and diversification of the Mazatec dialects, and give a more fine-structured evolution of the dialects.
?Financial markets: We construct the nominal return correlation networks from daily data to encapsulate sector-level dynamics and calculate the relative importance of the sectors in the nominal network through centrality measures and clustering algorithms. We show that the sectors that are relatively large constitute the core of the return networks, whereas the periphery is mostly populated by relatively smaller sectors. Further, sector-level nominal return dynamics is anchored to the real size effect, which ultimately shapes the optimal portfolios for risk management.