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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.
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