The Department of Mathematics hosted its 9th Distinguished Lecture, “PDE through Fourier” on September 18, 2024. The speaker, Prof. Mythily Ramaswamy, a renowned scholar in the fields of partial differential equations, control theory, and fluid dynamics addressed a crowd of faculty members, PhD students, MSc students, and attendees from various other Departments.

Prof. Mythily Ramaswamy’s lecture delved into the intricate development of partial differential equations (PDE) as influenced by the groundbreaking work on heat conduction by Fourier in the early 19th century. Prof. Ramaswamy, skillfully traced the history of Fourier’s discovery, leading to the formulation of the Fourier Series—a fundamental aspect of solving PDEs. She also explored modern developments in Fourier analysis and its vital applications in the realm of PDEs.

Prof. Mythily is currently serving as a NASI Senior Scientist at the International Centre for Theoretical Sciences (ICTS-TIFR) in Bengaluru, she has held significant positions such as Dean at the TIFR Centre for Applicable Mathematics. A recipient of the prestigious P C Mahalanobis Medal from the Indian National Science Academy, Prof. Ramaswamy is recognised for her substantial contributions to nonlinear functional analysis and optimal control, and she takes pride in mentoring the next generation of mathematicians.

Following the lecture, a lively 15-minute Q&A session provided attendees the opportunity to engage with Prof. Ramaswamy. Participants raised thoughtful questions, which sparked invigorating discussions on the applications of PDEs and the pivotal role of Fourier analysis in modern mathematics. The speaker’s valuable insights enriched the audience’s understanding and prompted further interest in the subject.

The event was deemed a resounding success, significantly enriching the academic experience of all participants. Both PhD students and faculty members gleaned crucial knowledge from Prof. Ramaswamy’s expertise, enhancing the intellectual atmosphere of the university. This lecture is poised to positively impact the institution’s academic reputation and foster future research collaborations. The Department of Mathematics looks forward to hosting more events that contribute to the vibrant academic community.

On October 4, 2024, Department of Mathematics at SRM University-AP hosted its 10th Distinguished Lecture, featuring renowned mathematician and educator Prof. R Ramanujam. The event attracted a diverse audience, including BSc and BTech students, PhD candidates, and faculty members, all eager to engage with the critical topic of mathematics and science education for students from socially and economically marginalised backgrounds.

Prof. Ramanujam’s lecture posed a thought-provoking question: “What do mathematics and science education mean to a student from socially and economically marginalised sections?” Drawing from his extensive experience in various educational contexts, he provided valuable insights into the intersection of education and social equity. His work with the Tamil Nadu Science Forum, government curriculum bodies, and teacher education programs at Azim Premji University in Bengaluru informed his perspective on the necessity of aligning educational practices with the realities faced by marginalised communities.

During his talk, Prof. Ramanujam emphasised the importance of conducting educational research that is deeply rooted in social contexts. He argued that curriculum-making decisions should be informed by such research to ensure that education is socially inclusive and responsive to the unique challenges faced by disadvantaged students. His advocacy for an education system that addresses the needs of all learners resonated strongly with the audience, highlighting the potential for education to serve as a transformative force in society.

Prof. R Ramanujam is a distinguished figure in the fields of mathematics and education. He completed his PhD at the Tata Institute of Fundamental Research (TIFR) and pursued postdoctoral work at the City University of New York (CUNY), USA. His long-standing association with the Institute of Mathematical Sciences (IMSc) in Chennai and his current role as a visiting professor at Azim Premji University further underscore his commitment to advancing educational practices.

The lecture provided a stimulating exploration of how mathematics and science education can help address social inequalities. Prof. Ramanujam’s reflections on education, grounded in his experiences with marginalised communities, left a lasting impact on attendees. The event concluded with an engaging Q&A session, where participants raised questions about the challenges of implementing socially rooted educational reforms. This discussion reinforced the lecture’s key themes of inclusivity and the transformative potential of education when designed to meet the diverse needs of learners.

The 10th Distinguished Lecture at SRM University-AP not only highlighted the importance of educational equity but also inspired dialogue on how institutions can better serve all students, particularly those from marginalized backgrounds.

The Department of Mathematics, SRM University-AP, is pleased to announce that Assistant Professor Dr Subha Sandeep Repaka has published a significant research paper titled “On Reducibility of Induced Representations of Odd Unitary Groups: The Depth Zero Case.” This accomplishment reflects Dr Repaka’s expertise and dedication to advancing mathematical research, further enriching the academic contributions of the department and the university.

Abstract:

We study a problem concerning parabolic induction in certain $p$-adic unitary groups. More precisely, for $E/F$ a quadratic extension of $p$-adic fields the associated unitary group $G=\mathrm{U}(n,n+1)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E) \times \mathrm{U}_1(E)$. Let $\pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $\iota_P^G \pi$ is reducible.

Future Research Plans:

We would like to solve the same problem which I had solved in this paper for the groups U(n,n) and U(n,n+1) over p-adic fields using L-Functions which is a very novel approach.

The link to the article:

http://nyjm.albany.edu/j/2024/30-50.html

Dr Shekhar Singh Negi from the Department of Mathematics has published a research paper titled “A note on Sugeno exponential function with respect to distortion.” Dr Negi’s research investigates the Sugeno exponential function. This research develops new mathematical tools and rules to work with a different way of measuring things, which can be useful in various fields like economics, biology, or any area where traditional measurements don’t quite fit the problem at hand.

Abstract:

This study explores the Sugeno exponential function, which is the solution to a first order differential equation with respect to nonadditive measures, specifically distorted Lebesgue measures. We define k-distorted semigroup property of the Sugeno exponential function, introduce a new addition operation on a set of distortion functions, and discuss some related results. Furthermore, m-Bernoulli inequality, a more general inequality than the well-known Bernoulli inequality on the real line, is established for the Sugeno exponential function. Additionally, the above concept is extended to a system of differential equations with respect to the distorted Lebesgue measure which gives rise to the study of a matrix m-exponential function.

Finally, we present an appropriate m-distorted logarithm function and describe its behaviour when applied to various functions, such as the sum, product, quotient, etc., while maintaining basic algebraic structures. The results are illustrated throughout the paper with a variety of examples.

Collaborations:

Prof. Vicenc Torra, Professor at the Department of Computing Science at Umea University. His area of research include artificial intelligence, data privacy, approximate reasoning, and decision making.

Future Research Plans:

To explore the aforementioned derivative and investigate results with applications in real life.

The link to the article

Dr Koyel Chakravarty, Assistant Professor in the Department of Mathematics, has made a significant contribution to the field of cancer research with her paper “Analysis and Regulation of Chaos Dynamics in a Cancer Model through Chemotherapeutic Intervention and Immune System Augmentation,” which was recently published in the International Journal of Dynamics and Control. In her paper, Dr Chakravarty delves into the intricate world of chaos dynamics within a cancer model and explores the potential for regulating these dynamics through the combined approach of chemotherapeutic intervention and immune system augmentation.

Her research offers insights into understanding the complex behaviour of cancer cells and how such insights can be leveraged to develop more effective treatment strategies. Dr Chakravarty’s work marks a crucial step forward in the ongoing efforts to combat cancer, shedding light on the dynamic interplay between therapeutic interventions and the body’s immune response.

The publication of this paper not only underscores Dr Koyel’s expertise in the field of mathematical analysis in cancer research but also signifies a promising advancement in the collective pursuit of understanding and addressing the challenges posed by cancer.

Abstract

The focus of the current investigation lies in the formulation and analysis of a dynamic model depicting cancer growth, incorporating the joint influences of chemotherapy and immune system augmentation. The primary emphasis of this study revolves around the analysis of the dynamic behaviour within a living-cell closed carcinoma system, specifically one devoid of external vitamin support, with a particular exploration of chaos dynamics. Subsequently, the authors aim to scrutinise the pivotal impact of infused vitamins in attaining stable system dynamics through the application of chaos control techniques.

The formulated model exhibits fundamental mathematical properties, revealing a spectrum of co-dimension one and co-dimension two bifurcations. The identification of specific bifurcation types relies on algebraic criteria techniques, where conditions necessary and sufficient for bifurcation types are developed. Notably, these criteria are distinct from traditional approaches based on the characteristics of the eigenvalues of the Jacobian matrix, instead relying on coefficients derived from characteristic equations. The accuracy of the analytical conclusions is validated through numerical findings, elucidating diverse bifurcation structures. The article enriches its contribution by delving into the control of chaos through the reinforcement of the internal immune system and the maintenance of the biological system’s stability. This work culminates in proposing future directions aimed at advancing a more realistic approach to eradicating cancer.

Research in Layperson’s Terms

This study focuses on developing and analysing a model that simulates how cancer grows, considering both chemotherapy and the immune system’s response. The main goal is to understand how cancer behaves over time in a system that doesn’t have external vitamin support, especially looking at how chaotic or unpredictable the growth can become. The researchers also investigate how adding vitamins might help stabilise this chaotic system using specific control techniques. The model they created has certain mathematical features that show different types of changes, called bifurcations, which can occur under specific conditions.
Additionally, the study explores how strengthening the immune system might help control this chaos and stabilise the biological system. The paper concludes by suggesting future research directions that could lead to more effective cancer treatment strategies.

Practical implementation

The practical implementation and social implications of analysing and regulating chaos dynamics in a cancer model through chemotherapeutic intervention and immune system augmentation can be profound. Insights gained from this research could be applied to optimize cancer treatment protocols, potentially leading to more effective therapies with reduced side effects. By understanding and controlling the chaotic behaviour in cancer systems, patient outcomes could be improved through personalized treatment strategies.
Socially, the adoption of these findings may lead to increased public confidence in advanced cancer treatments, as well as a broader acceptance of integrating immune system support with traditional therapies. The potential for more stable and predictable treatment outcomes may also reduce the emotional and financial burden on patients and healthcare systems. Additionally, this approach may encourage further interdisciplinary research, bridging gaps between Mathematics, Biology, and Medicine, thus fostering innovation in cancer therapy development.

Collaborations
Dr Lakshmi Narayan Guin, Associate Professor, Department of Mathematics, Siksha Bhavana, Visva-Bharati

Future research plans
Potential areas for further exploration include:

• Personalised Medicine: Developing patient-specific models that consider individual biological variations could lead to more tailored and effective cancer treatments, minimising side effects and improving outcomes.
• Integration with Advanced Therapies: Combining the insights from chaos dynamics with emerging therapies such as immunotherapy, targeted therapy, and gene editing could enhance the precision and efficacy of cancer treatments.