Annual Progress Seminar
Speaker: Aamir Yousuf (IIT Bombay)
Host: Neela Nataraj
Title: Numerical analysis of a thermoelastic diffusion plate model
Time, day and date: 11:00:00 AM, Monday, February 24
Venue: Online (Link in attached file)
Abstract: We investigate the well-posedness of a coupled 
hyperbolic-parabolic system modeling diffusion in thermoelastic plates, 
consisting of a fourth-order hyperbolic PDE for plate deflection and 
second-order parabolic PDEs for the first moments of temperature and 
chemical potential. The unique solvability is established via the 
Galerkin approach, and additional regularity of the solution is obtained 
under appropriately strengthened given data. For numerical 
approximation, we employ the Newmark method for time discretization and 
a C 0 -interior penalty (C0-IP) scheme for the spatial discretization of 
displacement. For the first moments of temperature and chemical 
potential, we use the Crank-Nicolson method for time discretization and 
conforming finite elements for spatial discretization. The convergence 
of the fully discrete scheme is established, achieving quasi-optimal 
rates in space and a quadratic rate in time. Several numerical examples 
will be presented to validate the theoretical findings.
 

Annual Progress Seminar
Speaker: Aratrika Pandey (IIT Bombay)
Host: Ravi Raghunathan
Title: Exact Approximation in the field of formal series
Time, day and date: 4:00:00 PM, Monday, February 24
Venue: Ramanujan Hall
Abstract: We prove a lower bound for the Hausdorff dimension of the set 
of exactly $\psi$-approximable matrices with values in a local field of 
positive characteristic. This is the analogue of the corresponding 
theorem of Bandi and de Saxc\'e over the reals, and is a higher 
dimensional version of a theorem.

Seminar
Speaker: Dr. Rajeev Das
Host: Radhendushka Srivastava
Title: Design and Applications of Robust Metaheuristics for Constrained 
Optimization Problems
Time, day and date: 4:00:00 PM, Monday, February 24
Venue: Online (https://meet.google.com/dtw-ztpe-jnc)
Abstract: The presentation includes a detailed discussion of my Doctoral 
research based on the development and applications of efficient 
metaheuristic algorithms for solving constrained optimization problems 
(COPs). The primary focus is on the Quadratic Approximation guided Jaya 
Algorithm (JaQA) and its enhanced variant, the Self-Adaptive 
Multi-Population Quadratic Approximation guided Jaya (SMP-JaQA). These 
two algorithms address challenges in solution diversity and balancing 
exploration-exploitation, making them highly effective in solving 
complex real-world optimization problems. The presentation begins with 
an introduction to COPs and the significance of metaheuristics in 
overcoming traditional optimization limitations. A comprehensive 
discussion on the research motivation, problem statement, and the 
identified gaps in existing methodologies follows. The design and 
implementation of these two proposed methodologies is elaborated upon, 
along with their computational complexity analysis. Experimental 
validation is carried out using constrained benchmarks from CEC 2006, 
CEC 2010, and CEC 2017. The outcomes demonstrate the superior 
performance of the proposed algorithms when compared to state of-the-art 
techniques, validated through statistical analysis (say, Friedman Rank 
Test and Wilcoxon Signed Rank Test), exploration-exploitation analysis, 
convergence analysis, etc. Real-life applications of the research are 
also highlighted in areas such as single-stage and two-echelon uncertain 
Fixed Charge Transportation Problems (FCTP), dairy feed ration 
optimization, mechanical design and electronics design optimization. The 
discussion concludes with an overview of research contributions, key 
findings, and potential future research directions. Moreover, insights 
into the publication status of my research so far is also provided 
showcasing peer-reviewed journal and conference publications along with 
communicated works.

Analysis seminar
Speaker: Kapil Jaglan (IIT Ropar)
Host: Prachi Mahajan
Title: A study on Planar Harmonic Mappings and Minimal surfaces
Time, day and date: 5:00:00 PM, Monday, February 24
Venue: Online (https://meet.google.com/woo-psdw-wdj)
Abstract: This talk explores the properties of univalent harmonic 
functions from a geometric function theoretic perspective and 
establishes connections between these functions and minimal surfaces. 
The talk consists of four sections followed by future direction, with 
the initial section serving as an introduction and containing basic 
definitions and results from the existing literature.

In the second section, our main aim is to determine a geometric 
condition under which a locally univalent harmonic mapping f defined on 
the unit disk D is univalent, and maps D onto a linearly accessible 
domain of order β for some β ∈ (0, 1). A linearly accessible domain (a 
non-convex domain) is important because, under certain sufficient 
conditions stated by Dorff et al., minimal graphs over these domains are 
area-minimizing. As a consequence, we derive sufficient conditions for f 
to map D onto a linearly accessible domain of order β, in the form of a 
convolution result, and a coefficient inequality. By extending the ideas 
of Dorff et al., we construct one-parameter families of globally 
area-minimizing minimal surfaces over a linearly accessible domain of 
order β.

In the third section, we explore the properties of odd univalent 
harmonic functions. Our starting point of investigation is to obtain the 
sharp coefficient estimates for odd univalent functions exhibiting 
convexity in one direction. We then advance our investigation to more 
generalized classes, including major geometric subclasses of 
sense-preserving univalent harmonic mappings. We examine the growth 
pattern of odd univalent harmonic functions and extend the range of 'p' 
for which these functions belong to the Hardy space h^p. Our results, in 
particular, add to the understanding of the growth pattern between odd 
univalent harmonic functions and the harmonic Bieberbach conjecture.

In the final section, we deal with the fundamental problem of 
determining the location of the zeros of complex-valued harmonic 
polynomials. The best-known results available in this direction are up 
to harmonic trinomials only. The exploration of the zeros of a general 
harmonic polynomial has been limited due to various challenges. Our 
research takes a leap further in identifying the regions encompassing 
the zeros of a general harmonic polynomial of arbitrary degree using 
various techniques, such as the matrix method and certain other matrix 
inequalities. Additionally, we employ the harmonic analog of the 
argument principle to examine the distribution of zeros, which we 
demonstrate through illustrative examples.