Course of differential geometry

Principles of Differential Geometry

The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on tensor calculus as they are based on the materials and conventions given in those documents. They can be used as a reference for a first course on the subject or as part of a course on tensor calculus.

微分几何是什么：曲线与曲面

What is differential geometry: curves and surfaces

本书通过曲线和曲面介绍微分几何，提供基础技术与可视化方法，帮助读者理解该领域的核心概念与研究方向。

微分几何是什么：曲线与曲面

A translation of "What is differential geometry: curves and surfaces"

本书通过曲线与曲面介绍微分几何，强调基础技术与直观理解，为深入学习提供入门指导。

On the development of differential geometry in Estonia

We give a brief review of a research made in the field of differential geometry in Estonia in the period from the beginning of the 19th century to the present time. The biographic data of mathematicians who made a valuable contribution to the development of differential geometry in Estonia in mentioned period are presented. The material presented in the introduction covers the period from the beginning of the 19th century to the middle of the 20th century and it can be considered as a brief historical sketch of the development of differential geometry in Estonia in this period. The next sectio…

Real Algebraic Geometry, Positivity and Convexity

Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and 10 are the lecture notes of its further continuation "Geometry of Linear Matrix Inequalities" from the winter term 2021/2022. These courses have been delivered at the University of Konstanz in Southern Germany. The entirety of these lecture notes is accompanied by a list of 47 long videos which is available from the following YouTube playlist: https://youtub…

Guido Castelnuovo's Last Higher Geometry Lectures (1922-23)

Here we present the transcription of the handwritten notes, carefully taken by an anonymous student, of the last lecture course taught by Guido Castelnuovo in 1922-23. The title of the course is "Curve algebriche piane e sghembe" ("Plane and space algebraic curves"): indeed, the language of both the original text and our brief introduction is Italian. The manuscript belongs to the legacy of Alfredo Franchetta (1916-2011) and the typesetting is due to Natascia Zangani. This edition is our modest contribution to celebrate the 150th birth anniversary of Guido Castelnuovo (1865-1952).

Foundations of geometry for university students and high-school students

This book is a textbook for the course of foundations of geometry. It is addressed to mathematics students in Universities and to High School students for deeper learning the elementary geometry. It can also be used in mathematics coteries and self-education groups.

A Primer on Stochastic Differential Geometry for Signal Processing

This primer explains how continuous-time stochastic processes (precisely, Brownian motion and other Ito diffusions) can be defined and studied on manifolds. No knowledge is assumed of either differential geometry or continuous-time processes. The arguably dry approach is avoided of first introducing differential geometry and only then introducing stochastic processes; both areas are motivated and developed jointly.

Engaging students in conjecturing through homework in Real Analysis and Differential Equations

In this note we report on an implementation of discovery-oriented problems in courses on Real Analysis and Differential Equations. We explain a type of task-design that gives students the opportunity to conjecture, refute and prove. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework assignments. In addition to several concrete examples we also discuss feedback and assessment outcomes of our students.