There are two main ways in which modern technology (computers of all kinds) can make the study of mathematics easier:
It is getting to the point where calling yourself a mathematician means you can use a computer to do those things as part of problem solving, when appropriate.
Differential Geometry is full of interesting things that can be understood more easily after routine, tedious computations, or after playing with a 3d visualization.
We do not expect that you have any significant experience with computing or programming before you enroll. This semester, you will learn some of the basics of how to use a mathematical software system. This will help you with linear algebra and, if you stick with it, it can help you with other parts of your study. To get through some of the basics, I have made available some materials for an introductory workshop as a web-page.
This semester, I have chosen to use the Computer Algebra System Sage. Sage is free and open-source, which has some big benefits. The most important one is that you will learn a system you can take with you, and will always be free to use. Another point in favor of Sage is that it is built using Python, a powerful, high-level programming language. Eventually, you will want to write some small program, and Python is easy to learn and easy to use.
To do our work, we will use the SageMathCloud service. SMC allows us to use Sage (and lots of other software) through a web browser. It essentially turns whatever web browser you have on you into a terminal for a powerful linux machine. (You won’t need all that power at first, but you might come to appreciate it later. It is a system designed by mathematicians and programmers for doing what they need.)