CMIT | Tangent Tuesdays on 29th August 2024 | 4 pm

Pure mathematics is, in its way, the poetry of logical ideas. - Albert Einstein

Welcome to the fourth episode of our "TANGENT TUESDAYS" talk series-we're thrilled to have you join us! This talk series is going to take you on a journey across the intricate landscape of complex analysis and mathematical physics. From unravelling the secrets of the Riemann Mapping Theorem, a pillar of complex analysis with far-reaching applications, to solving a long-standing mathematical question of holography using the Helmholtz equation, these talks will showcase the beauty and power of mathematics.

Speaker: Visesh Jyothi A (B21 Physics major, Interned at IIT Palakkad)
Title: Conformal Maps and the Riemann Mapping Theorem: Bridging Complex Domains
Date and Time: 29th August 2024 | 4 pm
Venue: Will be informed soon

Abstract: The Riemann Mapping Theorem is a cornerstone of complex analysis, asserting that any non-empty simply connected open subset of the complex plane, which is not the entire plane, can be conformally mapped onto the open unit disk. This profound result, attributed to Bernhard Riemann, provides a powerful tool for understanding complex functions and their behaviour on arbitrary domains.
In this talk, we will explore the Riemann Mapping Theorem, beginning with its statement and historical context. We will then delve into the intricate proof of the theorem, which involves several deep mathematical concepts, including harmonic functions, normal families, and Montel's theorem. The proof also draws on the notion of conformal mappings and the use of the Dirichlet problem, showcasing the beautiful interplay between different areas of mathematics.
We will discuss the theorem's far-reaching implications, such as its role in the classification of simply connected Riemann surfaces and its applications in various branches of mathematics, including differential geometry, dynamical systems, and mathematical physics. Additionally, we will touch upon the limitations of the theorem and the significance of the assumption that the domain is not the entire plane.

Speaker: Arjun V Nair (B20 Math major, Interned at Ecole Polytechnique,France)
Title: Some secrets of the Helmholtz equation
Date and Time: 29th August 2024| 4:30 pm
Venue: Will be informed soon

Abstract: As my summer began, I studied/revised concepts or fundamentals in related areas of Mathematics (Analysis, Inverse Problems, PDEs, etc...). Then I had to decide what open problem I may begin reading further. In the field of Inverse problems, holographic uniqueness problems for various PDEs are important, quite often for Physics. This class of problems was studied well and resolved for the three-dimensional and one-dimensional Helmholtz equation. The two-dimensional setting was seemingly complicated and hence made for an ideal reading project. However, during the course of the internship, I was able to find some mathematical techniques through which we were able to resolve this open problem. We will be publishing two papers in connection to this work and some further implications. In this talk, I intend to discuss the first of these papers and the adventure of this research experience. Refer here for the abstract. The second one may be discussed on some other occasion. "Arxiv themes - Analysis of PDEs, Mathematical Physics".

Hope to see you all there !!!