I need advice and good reference in geometric modeling? I am a student of computer science and this semester is my first semester of MSc. Next semester I must choose an advisor and the subject of my dissertation. I somewhat know what geometric modeling is and I want to research in this area, but I don't know where should I start. Can anyone who is a professional researcher in geometric modeling help me with where I should start?

Does this field have problems to work on?
As a graphics professional, not a researcher, it's a bit hard to know what exactly is or is not part of geometric modeling, but looking at this page (http://graphics.cs.ucdavis.edu/~joy/GeometricModelingLectures/) it looks like it's just all about different ways to model surfaces, whether that's with triangles, mathematical descriptions, or other things.

If that is the case, yeah, there are still open problems.

Here is a journal with some papers on the topic:
http://jcgt.org/

I would also look at the SIGGRAPH conference topics over the years, there is a lot of that sort of thing there.

IMO you are going to find that the problems are broken up into two main groups:


Real Time Graphics
Non Real Time Graphics


Video games and real time simulation software has the primary need of needing representations that can be rendered in real time, and are also fairly efficient as far as size, so that they don't take up too much memory.

Modern day movies, and also "offline processes" (such as baking out lighting into textures to use in real time applications) use non real time rendering techniques to get high quality results.  Speed and memory still matter to them of course, but the restrictions are much more relaxed as it isn't uncommon for a single frame of a movie to take a day to render, using a cluster of machines in a rendering farm.

Both real time and non real time graphics also have needs beyond rigid body geometry though, and also are interested in finding good ways to model liquids, gasses like smoke, and other things.

Besides the obvious representations of geometry like triangle and quad meshes, or bezier and other analytical shapes, there is also a lesser known concept of the signed distance field which can be used for both 3d modeling as well as 2d vector graphics.  In both cases, when you zoom into these shapes they have inifinite detail - such as being completely smooth, or having a perfectly sharp corner or whatever else.

Here's a video about one way subdivision surfaces can be achieved in real time graphics, shown at SIGGRAPH 2016, regarding the call of duty game:
https://www.youtube.com/watch?v=PTGTviYwolE

If I were in your shoes, part of what I would research would be what the "demo scene" is up to.  By demo scene I mean the people who make 64KB executables that when you run them, display photo realistic movies and have music and audio as well.  Those folks have some very creative ways of doing things that often ties into cutting edge research topics.

Shadertoy.com is a place where you can write webgl pixel shaders and share them with others.  There is a lot of really interesting and exotic graphics stuff there, including signed distance field rendering, ray marching, ray tracing, path tracing, and more.
https://www.shadertoy.com/

Here's a demoscene website:
http://www.pouet.net/

Lastly, if I may humbly point you towards a paper I wrote, I came up with a technique for getting the GPU texture sampler to calculate Bezier curve points of arbitrary dimension for you as part of a texture lookup, using the non programmable N-dimensional texture sample linear interpolator.

I believe it can be expanded to support Bezier surfaces as well, but haven't yet looked into the details of that.

http://blog.demofox.org/2016/02/22/gpu-texture-sampler-bezier-curve-evaluation/