Is algebraic topology a dying field?
This is a question that has been asked by many mathematicians over the years. There is no easy answer, as the field is constantly evolving and changing. However, there are some signs that algebraic topology may be in decline.
One sign is the decreasing number of papers being published in the field. A new study says that since the early 1990s, the number of papers published in algebraic topology journals has been going down. This could mean that fewer mathematicians are working in the area or that they are publishing their work in other places.
Another sign is the decreasing number of students who are studying algebraic topology. According to a recent survey the number of students who are majoring in mathematics and taking courses in algebraic topology has been declining since the early 2000s. This suggests that there is less interest in the field among young mathematicians.
But it’s important to remember that these are just trends. They don’t necessarily mean that algebraic topology is dead. There are still a lot of mathematicians working in the field and making important contributions. Not only that, but young mathematicians are still very interested in the field.
It is only a matter of time before we know if algebraic topology will continue to fall or if it will rise again.
What are the most important open problems in algebraic topology?
There are many important open problems in algebraic topology. Some of the most famous include:
- The Poincaré conjecture: This conjecture states that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere. It was finally proven by Grigori Perelman in 2002.
- The Hopf conjecture: This conjecture states that any map from the n-sphere to the (n-1)-sphere is null-homotopic. It was proven by Heinz Hopf in 1931.
- The Kervaire invariant problem: This problem asks whether there exists a non-trivial element in the stable homotopy groups of spheres in dimension 62. It was solved by Michael Atiyah and Isadore Singer in 1964.
- The Adams conjecture: This conjecture states that the stable homotopy groups of spheres are generated by elements of order 2. It was proven by J.F. Adams in 1964.
- The Novikov conjecture: This conjecture states that the higher signatures of a closed manifold are topological invariants. It was proven by Sergei Novikov in 1967.
These are just a few of the many important open problems in algebraic topology. There are many other problems that are still unsolved and there are many new problems that are being discovered all the time.
What are the best resources for learning algebraic topology?
There are many great resources for learning algebraic topology. Some of the most popular include:
- Algebraic Topology by Allen Hatcher: This is a classic textbook that covers all the basics of algebraic topology. It is a great resource for beginners.
- A Concise Course in Algebraic Topology by J.P. May: This is a more concise textbook that covers the same material as Hatcher’s book. It is a good choice for students who want to learn the material quickly.
- by Joseph J. Rotman: This is another classic textbook that covers the basics of algebraic topology. It is a good choice for students who want a more comprehensive treatment of the subject.
- Algebraic Topology by William S. Massey: This is a more advanced textbook that covers more advanced topics in algebraic topology. It is a good choice for students who want to learn more about the subject.
- The Stanford Encyclopedia of Philosophy: The SEP has an excellent article on algebraic topology that covers the history of the subject, the basic concepts, and some of the most important open problems.
These are just a few of the many great resources for learning algebraic topology. There are many other books, articles, and online resources available. With a little effort, you can find the resources that are right for you.
What are the career prospects for algebraic topologists?
Algebraic topologists are in high demand in academia, industry, and government. There are many opportunities for algebraic topologists to work in research, teaching, or applied mathematics.
Some of the most common career paths for algebraic topologists include:
- Professor: Algebraic topologists can work as professors at universities and colleges, where they teach courses in mathematics and conduct research.
- Researcher: Algebraic topologists can work as researchers at government labs, private companies, or independent research institutions. They conduct research in algebraic topology and related fields.
- Applied mathematician: Algebraic topologists can work as applied mathematicians in a variety of industries, including finance, insurance, and technology. They use their mathematical skills to solve real-world problems.
The job outlook for algebraic topologists is excellent. The Bureau of Labor Statistics projects that employment of mathematicians and statisticians will grow by 16% from 2020 to 2030, much faster than the average for all occupations. This growth is due to the increasing demand for mathematicians in a variety of industries, including finance, insurance, and technology.
If you are interested in a career in algebraic topology, there are many opportunities available. With a strong foundation in mathematics and a passion for the subject, you can have a successful career in this field.
maths is more than simply sums
The second of my interviews for the Defining topology through interviews series in with Carmen Rovi. Carmen was getting her PhD in math at Edinburgh while I was in college. She is now a Postdoctoral Fellow at Indiana University in Bloomington. She works in the exciting world of algebraic surgery theory!.
1. What would your own personal description of “topology” be?
Topology studies the geometric properties of spaces that are preserved by continuous deformations. Anything that bends or stretches an object won’t change its topological properties for a topologist because they focus on much more important factors. This is why people often say that a topologist can’t tell the difference between a coffee cup and a doughnut! Manifolds are the most common type of topological space. They have the local properties of Euclidean space in any dimension n. They are the higher dimensional analogs of curves and surfaces. For example, a circle is a one-dimensional manifold. Balloons and doughnuts are examples of two dimensional manifolds. It is clear that a doughnut and a balloon are not the same topologically because they can’t be continuously deformed into each other. If two topological spaces are topologically equivalent, they both have the same invariant. This is called an invariant of the topological space. In this case, the Euler characteristic tells us that the essential topological difference between the balloon and the doughnut is 2, while it’s 0 for the doughnut.
2. What do you say when trying to explain your work to non-mathematicians?
Most of the time, when people who aren’t mathematicians ask me this, I say, “I do topology, which is a branch of math.” Unfortunately the conversation usually ends there when the other person says “mathematics… that’s difficult. I had a hard time in school with that”. If I’m lucky, they’ll want to know more, and that’s when I can tell them that my work is in a field called “surgery theory.” Like real surgeons, a mathematician doing surgery also does a lot of cutting and sewing. There are, of course, very strict rules for this, just like there are for real surgery! In the picture below, we do surgery on a two-dimensional sphere. The first step is to cut out two patches (discs) from the sphere. This leaves two holes in the shape of two circles. We can then sew on a handle along those two circles. If we are careful on how we sew things together, we obtain a torus:
This shows how surgery can be used to describe how surfaces are grouped: sphere, torus, and torus with two holes. Of course, surgery can be used in much more complex situations, and it is a very useful tool for studying how to group things in topology. In order to classify complicated topological objects, you need to describe invariants that tell you something about the objects you are trying to classify. As we saw with this year’s Nobel Prize winners in physics, these kinds of invariants can then be used in “real life” to get amazing results.
3. How does your work relate, if at all, to the Nobel prize work?
One thing that this year’s physics prize winners were trying to figure out was how electricity moves through a layer of condensed matter. In tests done at very low temperatures and strong magnetic fields, the material’s electrical conductance took on very precise values, which doesn’t happen very often in these kinds of tests. They also noticed that the conductance changed in steps, when the change in the magnetic field was significant. The strange behaviour of the conductance changing stepwise made them think of topological invariants. Of course, it’s a long way between having this kind of intuition and being able to explain in words which topological invariant applies here. They use something called “Chern numbers” as their main invariant, which surprises me because I use this invariant a lot in my own work.
