IITG Mathematics Seminar Series

 Lecture Number: 375 Title: ON THE COLUMN-ROW PROPERTY OF OPERATOR SPACES Speaker: Dr. Samya Kumar Ray Affiliation: Inspire Faculty Fellow, Department of Statistics and Mathematics, ISI Kolkata Date: 18th November, 2022 (Friday) Time: 04:00 PM (Venue: Seminar Hall, Department of Mathematics(by maintaining COVID Protocols))

Abstract: In this talk, we discuss the following question asked by Michael Hartz in a recent paper: Which operator spaces satisfy the column-row property? We provide a complete classification of the column-row property in the cases of $C^*$-algebras (denoted by A) and non-commutative $L^p$-spaces over semifinite von Neumann algebras M. In particular, we prove that both A and $L^p(M)$ for $1 ≤ p \neq 2 ≤ ∞$, have the column-row property if and only if both A and M are subhomogeneous. Moreover, if the column-row constant for A is 1, then it has to be abelian. Also if $L^p(M)$ has column-row property with constant 1 for some $1 ≤ p \neq 2 ≤ ∞$, then M must be abelian. En route we also discuss several other relevant properties of operator spaces that are related to the column-row property. We talk about their existence and non-existence for various natural examples of operator spaces. This is a joint work with Srijan Sarkar.

 Lecture Number: 374 Title: On some questions concerning the ideal class groups of number fields. Speaker: Dr. Jaitra Chattopadhyay Affiliation: PostDoc Fellow, Department of Mathematics, IIT Guwahati Date: 1st November, 2022 (Tuesday) Time: 03:30 PM (Venue: Seminar Hall, Department of Mathematics(by maintaining COVID Protocols))

Abstract: The ideal class group $Cl_K$ of an algebraic number field $K$ is a finite abelian group that measures the extent to which the ring of integers $\mathcal{O}_K$ fails to be a PID. It encodes many important arithmetic information about $K$. The order of $Cl_K$ is the class number of $K$ and is denoted by $h_K$. The distribution of class numbers for quadratic fields is equally important in view of understanding the arithmetic nature of $K$. In this talk, we briefly discuss the simultaneous divisibility and indivisibility of class numbers of pairs of real quadratic fields and the $p$-ranks of $Cl_K$ for imaginary quadratic fields. Towards the end of the talk, we shall study the ring of integer-valued polynomials and its relation with P\'{o}lya group $Po(K)$ of $K$, which is a particular subgroup of $Cl_K$.

 Lecture Number: 373 Title: Adjustment for the uncertainty of predicted expression in transcriptome-wide association study (TWAS): A fusion of measurement error theory and bootstrapping Speaker: Dr. Arunabha Majumdar Affiliation: Assistant Professor, Department of Mathematics, IIT Hyderabad Date: 25th October, 2022 (Tuesday) Time: 04:00 PM (Venue: Seminar Hall, Department of Mathematics(by maintaining COVID Protocols))

Abstract: Transcriptome-wide association study (TWAS) is a promising approach to identifying novel genes associated with complex traits. The method commonly used in a standard TWAS approach combines two subsequent regressions. A significant criticism is that the method ignores the uncertainty of the predicted expression in the second-step regression, which can lead the final inference on the gene trait association subject to an uncontrolled rate of false positives, reducing the reliability of the detected association signals. We propose a novel approach to adjust for the uncertainty of predicted expression in the TWAS approach. We improvise techniques from measurement error theory to derive the adjustment factor that needs to be incorporated in the test statistic obtained from standard TWAS. Next, we apply bootstrapping techniques for penalized regression to estimate the adjustment factor. We use simulations to show that the traditional TWAS approach inflates the type 1 error rate, whereas our adjusted TWAS approach adequately controls it. Due to an inflated false positive rate, the standard TWAS produces a higher power than the adjusted TWAS approach. The adjusted approach has a better-controlled type 1 error rate in a multi-trait setting. Overall, the adjusted TWAS approach provides a valid test for gene-trait association.

 Lecture Number: 372 Title: Supersymmetric cluster algebras and the unity of mathematics Speaker: Ashish K. Srivastava Affiliation: Professor, Department of Mathematics and Computer Science, Saint Louis University, USA Date: 25th July, 2022 (Monday) Time: 03:30 PM (Venue: Seminar Hall, Department of Mathematics(by maintaining COVID Protocols))

Abstract: In this talk I will discuss cluster algebras, their supersymmetric analogues, and demonstrate through various examples how this topic unites several branches of mathematics such as algebra, combinatorics, number theory, geometry, dynamical system, and mathematical physics.

 Lecture Number: 371 Title: Pure contractions on Hilbert spaces Speaker: Srijan Sarkar Affiliation: Inspire Faculty Fellow, Department of Mathematics , IISc Bangalore Date: 20th July, 2022 (Wednesday) Time: 04:00 PM (Venue: Seminar Hall, Department of Mathematics(by maintaining COVID Protocols))

Abstract: Pure contractions serve as an important connection between various topics like operator theory, function theory, and several complex variables. In this talk, we will begin with an elaborate discussion on this connection by focusing on the Sz. Nagy-Foias analytic model and its consequences. Following this, we will concentrate on multiplication operators on Hardy spaces that are pure contractive. These multiplication operators have played an important role in finding models for abstract operators on Hilbert spaces. They have also featured in other contexts. E.g., in a seminal work by Agler and McCarthy, the authors showed that by using this kind of multiplication operator on vector-valued Hardy spaces, one could explicitly describe distinguished varieties in the bidisc |which are important in the context of extremal Pick interpolation problems for the bidisc. Thus, it is evident that these operators play significant roles in several questions in operator theory. However, there hasn't been a detailed study on when such an operator becomes pure contractive. We will aim to establish a complete characterization for these multiplication operators on the Hardy space on both the unit disc and polydisc. If time permits, we will look at other Hilbert spaces of analytic functions and certain applications of our results.

 Lecture Number: 370 Title: On approximation by inner functions Speaker: Prof. Titrthankar Bhattacharyya Affiliation: Dept. of Mathematics, IISC Bangalore Date: 02nd May, 2022 (Monday) Time: 03:00 PM (Venue: Seminar Hall, Department of Mathematics [MS Team Link])

Abstract: A classical theorem of Caratheodory asserts that every holomorphic function from the open unit disc to the closed unit disc can be approximated uniformly on compact subsets of the open unit disc by Blaschke products. A matrix valued generalization is known. We shall explain a recently found proof of the fact that every matrix valued contractive holomorphic function on the open unit disc can be approximated uniformly on compact subsets by rational inner functions. This theorem is then applied to obtain a uniform approximation result for holomorphic functions which extend continuously to the unit circle. The proof uses dilation of a contraction in a novel way. The talk will be accessible to first year Ph.D. students.

 Lecture Number: 369 Title: Nice ideals: applications and some observations Speaker: Prof. Pratulananda Das Affiliation: Dept. of Mathematics, Jadavpur University Date: 08th April, 2022 (Friday) Time: 04:30 PM (Venue: Seminar Hall, Department of Mathematics)

Abstract: In this talk we will primarily talk about the so called "nice ideals" in the realm of set theory. We will in particular describe the very basic notion of ideal convergence to see how these ideals have been found to be very useful in summability theory before moving to more deeper observations.

 Lecture Number: 368 Title: Stark-Heegner/Darmon points on elliptic curves over number fields Speaker: Dr. Amod Agashe Affiliation: Associate Professor at Florida State University, USA Date: 30th March, 2022 (Wednesday) Time: 04:00 PM (Venue: Microsoft Teams, Meeting Link)

Abstract: Darmon points (earlier called Stark-Heegner points) are certain points defined on elliptic curves using analytic constructions, but are conjectured to have algebraic coordinates. They are analogs of Heegner points, which are known to have algebraic coordinates and played a key role in the proof of the Birch and Swinnerton-Dyer conjecture in the analytic rank zero and one case. Darmon points also have a connection with the Birch and Swinnerton-Dyer conjecture. We will describe a geometric construction of Darmon points over arbitrary number fields (assuming some conjectures). Most of this is joint work with Mak Trifkovic.

 Lecture Number: 367 Title: Lane-Emden equations with Hardy potential and measure data Speaker: Dr. Mousomi Bhakta Affiliation: IISER Pune Date: 25th March, 2022 (Friday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: Dr. Mousomi Bhakta is Associate Professor and Rahul Bajaj Chair Professor at IISER Pune. Dr. Bakta has been awarded Swarnajayanti Fellowship (2021), SERB Women Excellence Award Research grant (2020-22) and INSA Young Scientist Award (2018).

Dr. Mousomi Bhakta research work includes the study of existence, multiplicity of positive and sign-changing solutions to local and nonlocal elliptic equations applying tools from nonlinear analysis and to study various qualitative properties of solutions e.g. radial symmetry, regularity, apriori estimate, etc. Dr. Bhakta studied the above questions to various local and nonlocal type semilinear and quasilinear elliptic equations with Hardy potential, equations with Hardy-Sobolev-Maz'ya type nonlinearities either in bounded domain or in the entire Euclidean space.

Another important topic of Dr. Bhakta's research is the study of local and nonlocal type singular elliptic system of equations with measure data. These types of solutions are called very weak solutions. Here too, her interest is to study existence, multiplicity, regularity of positive solutions.

 Lecture Number: 366 Title: Statistical distributions on integer partitions Speaker: Prof. Ken Ono Affiliation: Thomas Jefferson Professor of Mathematics at the University of Virginia, and the Chair of the Mathematics Section of the American Association for the Advancement of Science Date: 02nd February, 2022 (Wednesday) Time: 06:30 PM (Venue: Microsoft Teams)

Abstract: In this talk we examine three types of distributions on integer partitions.

(1)Generalizing a classical theorem of Erdos and Lehner, we determine the distribution of parts in partitions that are multiples of a fixed integer A. For example, the case of A=2 is the case of even parts in partitions of a large integer n. These limiting distributions are of Gumbel type, which are used to predict earthquakes! These results have implications in the algebraic geometry of n point Hilbert schemes.

(2)For a fixed positive integer t, we determine the distribution of the number of hook lengths of size t among the partitions of n. As n tends to infinity, the distributions are asymptotically normal.

(3) For a fixed integer t > 3, we determine the distribution of the number of hook lengths that are multiples of t among the partitions of n. As n tends to infinity, the distributions are asymptotically shifted Gamma distributions.

 Lecture Number: 365 Title: Singular double phase Kirchhoff problems Speaker: Dr. Tuhina Mukherjee Affiliation: Assistant Professor, Department of Mathematics, IIT Jodhpur Date: 18th January, 2022 (Tuesday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: See the PDF file for abstract

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