## Mathematics Seminar Series 2018

Year   2018    2019

Title:     Enumeration formulae for self-dual, self-orthogonal and complementary-dual cyclic additive codes

Speaker:    Dr. Anuradha Sharma, Associate Professor, IIIT-Delhi

Date:    November 15, 2018 (Thursday)

Abstract:    Cyclic additive codes over finite fields form an important family of error-correcting codes and are natural generalizations of cyclic codes. These codes have rich algebraic structures and have nice connections with quantum stabilizer codes. In this talk, we will study their dual codes with respect to three different trace bilinear forms. We will also present enumeration formulae for self-dual, self-orthogonal and complementary-dual cyclic additive codes, which are useful in classifying these three classes of cyclic additive codes up to equivalence.

Title:     Perfect Powers in Binary Recurrence Sequences

Speaker:    Dr. Shanta Laishram, Associate Professor, Indian Statistical Institute, Delhi

Date:    November 9, 2018 (Friday)

Title:     The algorithm which took us to the moon

Speaker:    Dr. Sanat K Biswas, Assistant Professor, IIIT-Delhi

Date:    November 2, 2018 (Friday)

Abstract:    This talk is about an estimation algorithm which was developed for Apollo missions around 50 years ago. Over the years, this algorithm has found its way to almost every disciplines of engineering and science. This ubiquitous estimator- the Kalman Filter is used in a wide range of practical applications - from navigation to the stock market price estimation. The mathematics behind the Kalman Filter is the topic of this discussion. I will talk about the Linear and Extended Kalman Filter. Then I will present the modifications that have been done to handle non-linearity in the estimation problems. Finally, I will discuss about few estimators which are termed as "beyond Kalman Filter

Title:     Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise Control and State Constraints

Speaker:    Dr. Kamana Porwal, Assistant Professor, Indian Institute of Technology, Delhi

Date:    October 23, 2018 (Tuesday)

Abstract:    In this talk, we study conforming and nonconforming finite element methods for elliptic distributed optimal control problems with pointwise state and control constraints. The state control constrained minimization problem is solved for the state variable by reducing it into a fourth order variational inequality and convergence of the state error is established in the H2-like energy norm. The key ingredients are constraint preserving properties of the interpolation operator and the enriching map. We also discuss post-processing methods to obtain the approximation of the control from the discrete state. Finally, we present numerical results to illustrate theoretical findings.

Title:     What is Logic?

Speaker:    Prof. Mihir Chakraborty, Retired Professor, Department of Pure Mathematics, University of Calcutta

Date:    October 22, 2018 (Monday)

Abstract:    The title should have been what are logics? It is an interesting discovery to me that Google is underlining with red whenever I write logics but does not do so when I write logic. This would be exactly my topic of discussion. I will give a very broad characterization of the term following basically Tarski. Then I will develop a few mathematical consequences of this definition, show some typical examples from both the Western as well as Eastern tradition. Finally, I would like to argue with a few examples in favour of the claim that it makes real sense to attach the mark of plurality to the word 'Logic'.

Title:     Models and Algorithms for space efficient algorithms

Speaker:    Venkatesh Raman, Institute of Mathematical Sciences

Date:    October 11, 2018 (Thursday)

Abstract:    Read-only memory is a classical model used to understand space-time tradeoffs.We first look at algorithms for sorting and selection in this model using small amount of extra space. Then we consider fundamental graph algorithms like BFS and DFS in this model using space considerably less than those used in classical algorithms. For problems like sorting (and BFS, DFS) where some output is desired, we assume that there is a write-only output tape where the output is written.

We conclude with a discussion on algorithms in recent models like "in-place" and "restore" models.

Title:     Symmetrically-Normed Ideals and Characterizations of Absolutely Norming Operators

Speaker:    Dr. Satish K. Pandey from the University of Waterloo, Canada

Date:    October 9, 2018 (Tuesday)

Abstract:    We begin by presenting a spectral characterization theorem that settles Chevreau's problem of characterizing the class of absolutely norming operators --- operators that attain their norm on every closed subspace. We next extend the concept of absolutely norming operators to several particular (symmetric) norms and characterize these sets. In particular, we single out three (families of) norms on $\mathcal B(\mathcal H, \mathcal K)$: the Ky Fan $k$-norm(s)", the weighted Ky Fan $\pi, k$-norm(s)", and the $(p,k)$-singular norm(s)", and thereafter define and characterize the set of absolutely norming operators with respect to each of these three norms. We then restrict our attention to the algebra $\mathcal B(\mathcal H)$ of operators on a separable infinite-dimensional Hilbert space $\mathcal H$ and use the theory of symmetrically-normed ideals to extend the concept of norming and absolutely norming from the usual operator norm to arbitrary symmetric norms on $\mathcal B(\mathcal H)$. In addition, we exhibit the analysis of these concepts and present a constructive method to produce symmetric norm(s) on $\mathcal B(\mathcal H)$ with respect to each of which the identity operator does not attain its norm. Finally, we introduce the notion of universally symmetric norming operators" and universally absolutely symmetric norming operators" and characterize these classes. These refer to the operators that are, respectively, norming and absolutely norming, with respect to every symmetric norm on $\mathcal B(\mathcal H)$. In effect, we show that an operator in $\mathcal B(\mathcal H)$ is universally symmetric norming if and only if it is universally absolutely symmetric norming, which in turn is possible if and only if it is compact. In particular, this result provides an alternative characterization theorem for compact operators on a separable Hilbert space.

Title:     Rectilinear Crossing Number of Uniform Hypergraphs

Speaker:    Rahul Gangopadhyay (Ph.D. student, CSE department, IIIT-D)

Date:    October 5, 2018 (Friday)

Abstract:    A graph is a collection of vertices and edges spanned by these vertices. Embedding of a graph G=(V, E) in a plane is a mapping of its vertices to the points in general position $\mathbb{R}^2$ and joining the edges by simple curves. Given a graph $G=(V, E)$, we define the rectilinear drawing of $G$ as an embedding of $G$ in $\mathbb{R}^2$ such that the vertices are mapped to the points in general position in $\mathbb{R}^2$ and the edges are drawn as straight line segments joining corresponding vertices. Euler's formula gave us a condition to check the planarity of a graph. Indeed a simple connected planar graph with $n$ vertices can contain at most 3n-6 edges. In an embedding of a graph, two edges are said to be crossing if they are vertex disjoint and they intersect. Fary proved that any planar graph has a rectilinear drawing such that no two of its edges cross. The crossing number of a graph is defined as the minimum number of crossing pair of edges among all embedding of it. Crossing number inequality states that for a simple undirected graph G with |E|>4|V|, the crossing number is greater than c|E|^3/|V|^2. A lot of research has been done on the special graphs (e.g., complete graph, complete bipartite graph, complete tripartite graphs, product graphs of cycles) where the structure of the graph is well known. A uniform hypergraph is a natural generalization of the graph. A $k$-uniform hypergraph is a collection (V, E) where $E \subseteq V^k$. A $d$-dimensional rectilinear drawing of a $d$-uniform hypergraph (also termed as geometric hypergraph) is an embedding of the hypergraph in $\mathbb{R}^d$ where vertices are paced as points in general position in $\mathbb{R}^d$ and hyperedges are drawn as $(d-1)$-simplices. In such an embedding two hyperedges are said to have a non-trivial intersection if they contain a common vertex in their relative interiors. Two vertex disjoint non-trivially intersecting hyperedges are said to be crossing. Dey and Edelsbrunner proved that a $3$-uniform geometric hypergraph always contains a non-trivial intersection if has more than $n^2$ hyperedges. They also proved that a $3$-uniform geometric hypergraph always contains a crossing pair of hyperedges if it has more than $1.5n^2$ hyperedges. Later Dey and Pch generalized this result for $d$-uniform hypergraph. They proved that a$d$-uniform geometric hypergraph with $n$ vertices can have at most $O(n^{d-1})$ hyperedges if it does not contain a crossing pair of hyperedges. A $d$-dimensional convex drawing of a $d$-uniform hypergraph is a $d$-dimensional rectilinear drawing of the hypergraph with vertices in convex position in $\mathbb{R}^d$. In this talk, we focus on the complete $d$-partite $d$-uniform hypergraph and complete $d$-uniform hypergraph. In particular, we discuss the lower bounds and upper bounds on the $d$-dimensional rectilinear crossing number of these hypergraphs. We will also discuss some special embeddings of these hypergraphs (e.g. whre vertices are in non-convex position, or form a neighborly polytope or lie on a $d$-dimensional moment curve). We will also focus on embedding the complete $3$-uniform hypergraphs in $\mathbb{R}^3$. We will also establish that the convex crossing number of $K_{n,n}$ is $n^4/6+\Theta(n^5)$. To obtain these results, we use techniques like Gale transformation, $k$-sets and $k$-edge, Ham-sandwich theorem etc.

Title:    Tsirelson's problems and non-closure of the set of quantum correlations

Speaker:    Jitendra Prakash

Date:    September 24, 2018 (Monday)

Abstract:    We consider a bipartite system with two observers, Alice and Bob, who are performing measurements in their labs. There are two models of quantum mechanics which describe the joint lab of Alice and Bob --- the quantum model and the commuting quantum model. Tsirelson's original question asked whether these two models were essentially the same. We show that these two models are different for bipartite systems with five quantum experiments and binary outcomes for each experiment, by using the notion of correlation functions of graphs. (This is a joint work with K. Dykema and V. I. Paulsen.)

Title:    A visual-search model observer for detection-localization tasks in nuclear medicine

Speaker:    Dr. Anando Sen Research Investigator, MD Anderson Cancer Center

Date:    September 04, 2018 (Tuesday)

Abstract:    Model observers are mathematical models intended for performing diagnostic tasks. Scanning observers (based on point-by-point evaluation) have been proposed for detection-localization tasks in medical imaging, but handling anatomical noise with these observers can be challenging. We have introduced visual-search (VS) observers as an alternative. The VS observer is a two-step process which mimics human perception through an initial search before a more detailed candidate analysis. Both the scanning and VS observers often outperform humans. We propose three additional means for bridging this human-model gap - (1) Task equivalence between model and human observer studies. In particular using both functional and anatomical images for the tumor localization decision; (2) Introducing inefficiencies into the model that a human might be affected by (e.g. internal noise, background approximation, perceptual thresholds and search noise); (3) Moving from a dual-feature to a multi-feature adaptive VS observer that relies less on prior information, particularly the background. Applications studied were SPECT-CT and planar nuclear imaging. Detailed localization receiver operator characteristic (LROC) studies with human and model observers were performed. Area under the LROC curve was used for observer performance evaluation. Results indicate that the VS observer applied with a combination of factors mentioned above can quantitatively match human performance.

Title:    Infection spread and stability in random graphs

Speaker:    Dr. Ghurumuruhan Ganesan from NYU Abu Dhabi

Date:    August 21, 2018 (Tuesday)

Abstract:    In the first part of the talk, we discuss infection spread in random geometric graphs where n nodes are distributed uniformly in the unit square centred at the origin and two nodes are joined by an edge if the Euclidean distance between them is less than r_n , the connectivity distance. Assuming that the edge passage times are exponentially distributed with unit mean, we obtain upper and lower bounds for speed of infection spread in the sub-connectivity regime. In the second part of the talk, we discuss convergence rate of sums of locally determinable functionals of Poisson processes and establish bounds for the rates of convergence of spatial averages of such functions, in terms of the radius of determinability.

Title:    Planar support for non-piercing regions and applications

Speaker:    Rajiv Raman, Assistant Professor (CSE, CB, Applied Mathematics), IIIT-D

Date:    August 10, 2018 (Friday)

Abstract:    Given a hypergraph H=(X,E), a planar support is a planar graph G on the vertices X, such that for each hyperedge e in E, the induced subgraph of G on the vertices of e is connected. A set S of compact, connected regions in the plane is said to be non-piercing if for any pair of regions A,B in S, the sets A\B, and B\A are both connected. Examples of non-piercing regions include disks, unit-height rectangles, homothets of convex sets, etc. Given two families of non-piercing regions R and B, the intersection hypergraph is a hypergraph whose vertex set is the family B of non-piercing regions, and each region r in R defines a hyperedge consisting of all regions in B intersecting the region r.In this talk, I will prove that intersection hypergraphs of non-piercing regions have a planar support, that further can be computed in polynomial time. This result also has several applications, including unified PTASs for several packing and covering problems on non-piercing regions, as well as coloring hypergraphs of non-piercing regions.

Title:    Multi-twisted codes over finite fields and their dual codes

Speaker:    Varsha Chauhan, Ph.D. student in Mathematics

Date:    July 26, 2018 (Thursday)

Abstract:    Aydin and Halilovic (2017) introduced and studied multi-twisted (MT) codes over finite fields, which are generalizations of well-known classes of linear codes, viz. constacyclic codes and quasi-cyclic codes, having rich algebraic structures and containing record-breaker codes. They also obtained multi-twisted codes with best-known parameters over GF (3), over GF (5), over GF (7) and optimal parameters over GF(7). Apart from this, they proved that the parameters over GF(5) and over GF(3) can not be attained by constacyclic or quasi-cyclic codes, which suggests that this larger class of multi-twisted codes is promising to find codes with better parameters than the current best known linear codes. This motivated us to further study multi-twisted codes over finite fields.

Title:    On the Structure and distances of repeated-root constacyclic codes

Speaker:    Tania Sidana, Ph.D. student in Mathematics

Date:    July 24, 2018 (Tuesday)

Abstract:    The main aim of coding theory is to construct codes that are easier to encode and decode, can detect and correct many errors, and contain a sufficiently large number of codewords. To study error-detecting and error-correcting properties of a code with respect to various communication channels, several metrics (e.g. Hamming metric, Lee metric, Rosenbloom-Tsfasman (RT) metric, symbol-pair metric, etc.) have been introduced and studied in coding theory.