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Speaker: Dr. Annapurna Banik (Postdoctoral Fellow, TIFR CAM)
Title: A Comparative Study of Subharmonic and Plurisubharmonic Functions
Abstract: Click HERE to view the abstract.
Date: 26th May, 2025 (Monday) Date: 30th July, 2025 (Wednesday) Date: 2nd August, 2025 (Saturday)
Time: 6:00 PM - 7:00 PM (IST) Time: 6:00 PM - 7:00 PM (IST) Time: 6:00 PM - 7:00 PM (IST)
Speaker: Dr. Ravi Shankar Jaiswal (Postdoctoral Fellow, TIFR CAM Bangalore)
Title: Strongly Pseudoconvex Domains
Date: 1st February, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: This lecture will focus on presenting and proving key properties of strongly pseudoconvex domains.
Title: Levi Pseudoconvexity of Domains
Date: 18th January, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In the last lecture, we defined the analytic convexity of domains and studied some of their important properties. In this talk, our goal is to pass from analytic convexity to a complex analytic analogue of convexity known as the Levi pseudoconvexity. Notably, Levi pseudoconvexity is invariant under biholomorphic maps.
Title: Defining Functions and Convexity of Domains
Date: 11th January, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: The goal of this lecture series is to introduce and formalize the notion of domains in C^n that remain invariant under biholomorphic maps. This notion, broadly speaking, can be described as a form of convexity in complex tangential directions.
In this talk, we will begin by defining the concept of a defining function for a domain and explore its key properties. Subsequently, we will delve into the analytic convexity of domains, demonstrating how it can be understood through the defining function.
Speaker: Agniva Chatterjee (Ph.D. Student, IISc Bangalore)
Title: Hartogs’s Extension Theorem in SCV
Date: 11th December, 2024 (Wednesday)
Time: 5:00 PM - 6:00 PM (IST)
Abstract: As seen in the last talk, the phenomenon of holomorphic extension plays a central role in SCV. In this talk, we discuss a very powerful theorem regarding holomorphic extension, known as Hartogs’s extension theorem.
The proof technique of this theorem allows us to introduce the concept of the d-bar- problem, which plays an instrumental role to various concepts in SCV.
Title: Holomorphic Extensions in SCV
Date: 6th December, 2024 (Friday)
Time: 5:00 PM - 6:00 PM (IST)
Abstract: In the last talk, we saw that there are domains in C^n, where any holomorphic functions can be extended to a larger domain. This particular phenomenon of holomorphic functions is not present in univariate complex analysis. In this talk, we will see that the domain of convergence of a power series has to have some nice convexity property, known as logarithmic convexity.
In the later part of the talk, we will introduce the concept of the domain of holomorphy, which is related to the problem of holomorphic extension and plays one of the most important roles in SCV.
Title: Power Series in SCV
Date: 30th November, 2024 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In univariate complex analysis, power series plays a crucial role in the understanding of holomorphic functions. Also, we have a comprehensive idea regarding the region of convergence of power series in this setting, which solidifies the richness of analysis in complex variables. However, the behavior of power series is vastly different in higher dimensions.
In this talk, we discuss various aspects of power series in SCV, and explore geometry of domains on which holomorphic functions admit power series representations. This talk will also set up a platform to explore holomorphic extension problems in higher dimensions, one of the most important avenues in SCV.
Title: Similarity and Differences between One Complex Variable and SCV
Date: 23rd November, 2024 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: The analysis of univariate complex variables stands out significantly from that of real variables, primarily due to the elegant field structure of complex numbers. However, in higher complex dimensions, such field structures are no longer present. This raises an intriguing question: how does the analysis of several complex variables differ from the analysis of a single complex variable?
In this talk, we begin by proposing definitions of holomorphicity in higher dimensions and drawing parallel comparisons with the properties of holomorphic functions in one complex variable. In the latter half of the talk, we present a version of the Cauchy integral formula for higher dimensions, which plays a fundamental role in the study of several complex variables.
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