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A virtual seminar series on complex analysis, function spaces and complex geometry
OUR OBJECTIVE
OUR OBJECTIVE
We plan to facilitate a series of online seminars featuring expository lectures and individual research presentations focused on complex analysis, function spaces, and complex geometry. Visit the Seminar talks page to explore key topics in detail.
iota Seminar: Members of COMPASS meet regularly to discuss a wide range of research topics in areas related to complex analysis, with the goal of expanding knowledge through mutual effort and fostering a collaborative environment. Join our discussions to become a member.
Please fill out the Google form to register for the complex analysis seminar series and receive weekly updates. For any queries, comments, or suggestions, you can write to us at compass.talks.updates@gmail.com.
RESOURCES
The recorded videos of some of the seminar lectures can be found on our YouTube channel. Please subscribe, to stay updated.
You can find the lecture notes for the COMPASS seminar talks here.
iota Seminar
Speaker: Riddhi Mishra (Ph.D. student, Jyväskylä University, Finland)
Title: "Quasiconformal almost parametrizations of metric surfaces" by D. Meier and S. Wenger.
Talk 3: 5th September, 2025 (Friday)
Talk 2: 26th August, 2025 (Tuesday)
Talk 1: 16th August, 2025 (Saturday)
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. Behnam Esmayli (Visiting Assistant Professor, University of Cincinnati)
Title: Removable Sets for Sobolev Functions on Weighted R^n
Date: 11th June, 2025 (Wednesday)
Time: 06:30 PM - 07:30 PM (IST)
Abstract: Let the Euclidean space R^n be equipped with a doubling weight. Define H as the completion of smooth functions in the weighted Sobolov norm. Suppose E is a set of (weighted) measure zero. If $u$ is a Sobolev function on R^n\E then is $u$ a Sobolev function on R^n? If this is true for every $u$ then we say E is removable. The subtlety is that none of the smooth functions that approximate $u$ on R^n\E might be smooth on R^n. Removability is a difficult question even in the unweighted Sobolev theory. I will discuss a recent work of mine with Riddhi Mishra on when a set E is removable for Sobolev functions on R^n equipped with a weight that is doubling and satisfies a Poincare inequality.
Title: Sobolev Functions on Weighted R^n
Date: 4th June, 2025 (Wednesday)
Time: 06:30 PM - 07:30 PM (IST)
Abstract: Problems in PDE (e.g. degenerate elliptic equations) and in geometry motivate and necessitate the study of R^n equipped with a measure other than the Lebesgue measure. What should we define as the class of Sobolev functions on this space? I discuss the intricacies lying behind this question and explain the two main classical approaches: the space W via the notion of weak differentiability and the space H as the closure of smooth functions. I will aim to convince you that H is the more natural choice, but I will review the literature on when H=W.
Speaker: Dr. Abhay Jindal (Postdoctoral Fellow, University of Ljubljana)
Title: Realization Formula of the Functions on the Unit Disc
Date: 15th May, 2025 (Thursday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will explore the realization formula for Schur functions—analytic functions mapping the unit disc to itself. [Notes]
Speaker: Sukanta Das (Ph.D. Student, IIT Bhubaneswar)
Title: Some Properties of the Julia Set and Structure of the Fatou Components
Date: 19th April, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will explore the relationship between periodic points and the Julia set of a rational map of degree at least two. We will also discuss the expansivity property of the Julia set. Following that, we will discuss various types of Fatou components and introduce two new components arising in the context of transcendental entire functions. Additionally, we will present some results concerning the structure of the Fatou set.
Title: Iteration of Rational Functions
Date: 12th April, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will explore the iteration of rational functions, with a particular focus on how the dynamics of a polynomial are influenced by the forward orbit of a critical point. We will then examine key properties of the Julia set associated with rational maps of degree at least two. Following that, we will discuss the classification of periodic points and the various types of Fatou components. Throughout the talk, we will also highlight how these results and ideas extend to the setting of transcendental entire functions.
Title: An Invitation to Complex Dynamics: Iterating Functions in the Complex Plane
Date: 5th April, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will explore the fundamentals of iterated complex functions, focusing on the dynamics that arise from repeated application. We will delve into the properties of Fatou sets (regions of stability), and Julia sets (regions of chaos), highlighting their significance in understanding complex behavior. Additionally, we will analyze some functions to visualize and interpret their iterative behavior in the complex plane.
Speaker: Dr. Abhishek Pandey (Research Associate, IISc Bangalore)
This is a Crash Course in Metric Geometry. This course is also a part of PMRF student lecture series by ISSS.
Title: BHK Uniformization and Characterization of Gromov Hyperbolic Spaces
Date: 22nd March, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will discuss the Bonk-Heinonen-Koskela (BHK) uniformization in detail and explore geometric conditions for Gromov hyperbolicity: the Gehring-Hayman condition and the ball separation condition. This discussion will lead to a characterization of the Gromov hyperbolicity of the quasihyperbolic metric. We will also compare the Gromov hyperbolicity of the hyperbolic and quasihyperbolic metrics in planar hyperbolic domains. If time permits, we will discuss some results on the Gromov hyperbolicity of the Kobayashi metric on convex domains.
Title: Gromov Hyperbolicity
Date: 18th March, 2025 (Tuesday)
Time: 4:00 PM - 5:30 PM (IST)
Abstract: Gromov hyperbolicity is a property of a general metric space to be negatively curved in the sense of coarse geometry. Introduced by Gromov (1987) in geometric group theory, it has since played an increasing role in analysis on general metric spaces with applications to the Martin boundary, invariant metrics in several complex variables. From a complex analysis perspective, Gromov hyperbolicity is significant because:
Many important metrics in complex analysis are frequently Gromov hyperbolic.
The Gromov boundary is a useful concept, both as an alternative way of treating the topological boundary and as a way of defining boundary extensions of maps.
While finding geodesics in invariant metrics is challenging, quasigeodesics are more accessible, and in Gromov hyperbolic spaces, geodesics stay close to quasigeodesics.
This talk will introduce Gromov hyperbolicity for geodesic metric spaces, with examples and non-examples. If time permits, we will discuss geodesic stability and the Gromov boundary.
Title: Notions of Curvature in Metric Spaces
Date: 10th March, 2025 (Monday)
Time: 4:00 PM - 5:30 PM (IST)
Abstract: Some of the important metrics that are used in complex analysis and potential theory, including the Poincaré, Carathéodory, Kobayashi, Hilbert, and quasihyperbolic metrics are quite often negatively curved. The study of nonpositively curved spaces has a long and rich history, dating back to Hadamard’s pioneering work in 1898. The first notion of curvature in metric spaces was introduced by Karl Menger and his student Abraham Wald, laying the foundation for the modern development of this theory. Over time, various formulations of nonpositive and negative curvature have emerged, including Busemann convexity, the CAT(k) condition, and Gromov hyperbolicity. In this talk, we will explore these key concepts, focusing on Busemann convexity and the CAT(k) condition, along with illustrative examples of such spaces.
Title: Hausdorff Measure and Length
Date: 2nd March, 2025 (Sunday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will introduce the concept of Hausdorff measure, starting with intuitive examples that motivate its definition. We will then explore its connection to the length of curves. If time permits, we will examine interesting examples, including the Hausdorff measure of curves and Cantor-type sets.
Title: Length and Geodesic Spaces
Date: 22nd February, 2025 (Saturday)
Time: 11:00 AM - 12:30 PM (IST)
Abstract: In this talk, we will begin by exploring conformal metrics, delving into their properties and applications. Following this, we will examine length spaces and geodesic spaces, providing illustrative examples to demonstrate their structure and significance.
Title: Length of Curves, Rectifiability, Length Formula, and Line Integral
Date: 12th February, 2025 (Wednesday)
Time: 04:00 PM - 05:30 PM (IST)
Abstract: In this talk, we will begin by exploring fundamental concepts in metric geometry, including the length of curves, rectifiability, and arc-length parametrization. We will then examine the conditions under which the length formula holds. Finally, we will conclude with a discussion on the line integral.
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.
Speaker: Abhishek Pandey (Ph.D. Student, IIT Bhubaneswar)
Title: Classification of Riemann Surfaces and the Moduli Problem
Date: 15th November, 2024 (Friday)
Time: 4:30 PM - 5:30 PM (IST)
Abstract: In this talk, we will first discuss the classification of Riemann surfaces whose universal covering is the complex plane or the unit disk. We will then explore the different complex structures on the torus and the Riemann moduli problem.
Title: Gauss Classical Problem
Date: 8th November, 2024 (Friday)
Time: 4:30 PM - 5:30 PM (IST)
Abstract: In this talk, we will explore Gauss's problem of mapping a surface locally conformally into the plane. Gauss demonstrated that this problem is equivalent to finding isothermal coordinates for a given surface S. We will show that the problem of finding isothermal coordinates reduces to solving the Beltrami equation. Gauss established the existence of a solution to the Beltrami equation under certain smoothness assumptions. Thus, he solved the problem of finding a locally conformal map of a smooth, orientable surface that fit into the plane. This result not only has intrinsic significance but also leads to important modern interpretations and far-reaching generalizations:
In other words, Gauss proved that a smooth orientable surface in R^3 can always be made into a Riemann surface.
Gauss's method can be extended to more general contexts, allowing any abstract surface with a Riemannian metric to acquire a complex-analytic structure.
Title: Uniformization, Representation and Classification of Riemann Surfaces
Date: 3rd November, 2024 (Sunday)
Time: 11:30 AM - 1:00 PM (IST)
Abstract: In this talk, we will first discuss the uniformization result of simply connected Riemann surfaces, which indicates that given a Riemann surface, its universal covering must be one of the three Riemann surfaces: the unit disk, the complex plane, or the extended plane. Further, using the uniformization result and tools from the covering space theory, we will prove the basic representation theorem for Riemann surfaces. At last, we will classify Riemann surfaces that have universal covering as the extended plane, the complex plane, the unit disk in terms of the fundamental group of the Riemann surface.
Title: Covering Space Theory
Date: 19th October, 2024 (Saturday)
Time: 11:30 AM - 1:00 PM (IST)
Abstract: Covering surfaces provide the unifying link between the theory of abstract Riemann surfaces and complex analysis in the plane. In elementary function theory, Riemann surfaces are initially encountered in relation to the mapping z^n. This defines the plane as covering of itself, but in such a way that projection mapping has branch points. More generally, non-constant holomorphic maps between Riemann surfaces are covering maps possibly having branch points. In this talk, we will first discuss several concepts of covering spaces and a significant topological fact known as the Monodromy theorem. Further, we will discuss covering groups and cover transformations in detail. This will assist us in demonstrating an important result: every surface is homeomorphic to the quotient of the covering group of its universal covering surface.
Speaker: Athira E V (Ph.D. Student, KSoM)
Title: Holomorphic Maps between Riemann Surfaces
Date: 9th October, 2024 (Wednesday)
Time: 5:00 PM - 6:00 PM (IST)
Abstract: In this talk, I will introduce meromorphic functions on Riemann surfaces. We will also explore holomorphic maps between Riemann surfaces, examining their local behavior and some elementary properties.
Title: Introduction to Riemann Surfaces
Date: 5th October, 2024 (Saturday)
Time: 11:30 AM - 12:30 PM (IST)
Abstract: In the mid-1800s, Bernhard Riemann developed the notion of Riemann surfaces to address the problem of multivalued functions in one variable. On a Riemann surface, which is not simply connected, the analytic continuation of a holomorphic function along different paths may lead to different branches of that function. Riemann’s key idea was to consider a universal covering surface of a given Riemann surface, ensuring that the analytic continuation of the given function becomes single-valued on this cover.
Riemann surfaces are connected, one-dimensional complex manifolds. In this talk, I will introduce manifolds, present some examples of Riemann surfaces, and discuss some theorems on Riemann surfaces that are analogous to theorems in the complex plane.
Speaker: Sheetal Wankhede (Ph.D. Student, IIT Indore)
Title: Fundamental study of univalent function theory in the plane
Date: 28th September, 2024 (Saturday)
Time: 11:30 AM - 12:30 PM (IST)
Abstract: Click HERE to view the abstract. [Lecture Slides]
Speaker: Riddhi Mishra (Ph.D. Student, Jyväskylä University, Finland)
Title: Sobolev Inequalities
Date: 14th September, 2024 (Saturday)
Time: 11:30 AM - 12:30 PM (IST)
Abstract: In this talk, we will discuss Sobolev inequalities. More specifically Gagliardo- Nirenberg-Sobolev inequality, Sobolev-Poincare inequality, Morrey inequality, and the summary of Sobolev embeddings.
Title: Approximation of Sobolev functions
Date: 7th September, 2024 (Saturday)
Time: 11:30 AM - 12:30 PM (IST)
Abstract: In this talk, we will deal with the approximation of a Sobolev function by smooth functions. At first, we will see the local approximation in Sobolev space and then the global approximation.
Title: An introduction to Sobolev spaces
Date: 31st August, 2024 (Saturday)
Time: 11:30 AM - 12:30 PM (IST)
Abstract: This talk will be a short introduction to Sobolev spaces. First, we will discuss the notion of weak derivatives and explore some of the examples. Later, we will see the definition of Sobolev space and its completeness property.
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