Applicability of Algebraic Geometry (and all the rest of math that goes with it)

In summary, Algebraic Geometry is a highly useful and practical subject that has a wide range of applications in both mathematics and other disciplines, making it a valuable area of study for those interested in using math to solve real-world problems.
  • #1
donnylee
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Applicability of Algebraic Geometry (and all the rest of math that goes with it)

Hello fellow mathematicians and physicists. I’m at that crossword in my university life right now, fall semester of sophomore year, and I really could use some advise in my next load of classes. I have always wanted to be a math major. Apart from choosing a subject which to me carries a lot of mystic (maybe second to Quantum Mechanics – but hey, group theory is also a subset of that), I knew that one day, whatever math I learn will give me a new angle in attacking whatever physical problem be it in engineering, physics or economics. In that regard, I always seeks to learn math both in breath and depth.

Well, I starting to loose confidence in this path of mine. Unlike most college students who try to graduate with the highest GPA or join the honor society, my goal is somewhat unique: I want to come out of Duke university getting an ‘A’ in Algebraic Geometry. I understand that this class is perhaps the last class you’ll take in the list of graduate requirements but realistically speaking, I came into college having done multivariable calculus. So having 4 years to march through Real Analysis -> Algebra -> Topology -> Algebraic Topology -> Complex Analysis -> Commutative Algebra -> Algebraic Geometry (as the sequence in my school) seemed quite reasonable.

Now, here is where I need some help. I don’t have a clear understanding of what Algebraic Geometry is in particularly it’s usefulness in the other sciences. All I hear are phrases such as ‘The subject that leads us to the deepest waters of math’ or ‘Mathematicians learn Algebraic Geometry to conquer the world’. Sure, that sounds impressive and it surely adds to the mystic factor I so desire in learning math. Yet I need to know the answer to a more practical question, one which answers my intent of learning math in the first place: “Will I one day look at a engineering problem, say Aerodynamics or Control Systems, and say “Yes, let’s employ an Algebraic Solution””

The answer does depend on the level of advancement in Algebraic Geometry and this I totally have no idea of. Calculus has certainly matured where double integrals are undoubtedly fused with engineering. My question is that as of NOW or maybe in the next 2 years, does Algebraic Geometry already have its practical uses in Engineering.

If the answer is no or perhaps it’s usefulness will come in say 10 years later, then I would have missed the mark because with all the respect I have for Pure Mathematicians, I would love to learn math so as to use math and not prove a theorem which only exist in the abstract realm (which I sometimes do but that’s not where my main interest lie).

Thanks for listening and any advice is greatly appreciated.
 
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  • #2
Algebraic geometry is an important field of mathematics with a wide range of applications, both to mathematics in general and to other disciplines. In particular, it has found applications in areas such as number theory, cryptography, coding theory, algebraic topology, algebraic number theory, algebraic combinatorics, and even quantum computing. It has also been used to solve problems in the physical sciences, including problems related to aerodynamics and control systems. Algebraic geometry can be used to find solutions to problems that would otherwise be difficult to solve analytically, and it can also be used to study structures in a variety of physical systems. As such, it can be an invaluable tool for solving engineering problems.
 
  • #3


I understand your concerns about the practical applicability of Algebraic Geometry and other advanced math subjects. It is true that these topics may not have immediate applications in engineering or other sciences, but that does not diminish their importance or value.

Firstly, it is important to note that mathematics is a constantly evolving field. What may seem abstract and theoretical now, may have practical applications in the future. This has been the case with many branches of math, including calculus, which was once considered purely theoretical but is now widely used in engineering and other sciences.

Moreover, even if Algebraic Geometry may not have direct applications in engineering at the moment, the skills and problem-solving techniques you will learn in this subject will undoubtedly be useful in tackling complex problems in any field. The ability to think abstractly and logically, and to manipulate complex equations and concepts, are all valuable skills in any scientific discipline.

Additionally, it is important to keep in mind that the goal of higher education is not just to acquire practical skills, but also to broaden our understanding of the world and to challenge ourselves intellectually. Algebraic Geometry, and math in general, may not have immediate practical applications, but they are fascinating subjects that can lead to new insights and ways of thinking about the world.

In conclusion, while it is understandable to want to focus on practical applications of math, it is also important to appreciate the value of studying advanced topics such as Algebraic Geometry. These subjects may not have direct applications now, but they contribute to the advancement of knowledge and can lead to new discoveries and breakthroughs in the future. So I encourage you to continue pursuing your passion for math and to not lose confidence in your chosen path.
 

What is Algebraic Geometry?

Algebraic Geometry is a branch of mathematics that studies solutions to polynomial equations using techniques from algebra and geometry. It combines abstract algebra with geometric concepts to study the properties of solutions to polynomial equations.

How is Algebraic Geometry applicable in real life?

Algebraic Geometry has many real-life applications, such as in computer graphics, robotics, coding theory, and cryptography. It also has applications in physics, biology, and economics, where it is used to model and analyze systems and phenomena.

What are the main concepts in Algebraic Geometry?

The main concepts in Algebraic Geometry include varieties, sheaves, schemes, and cohomology. Varieties are geometric objects defined by polynomial equations, while sheaves and schemes provide a more abstract and general framework for studying these objects. Cohomology is a mathematical tool used to study the topology of these objects.

What other branches of math are involved in Algebraic Geometry?

Algebraic Geometry involves many other branches of mathematics, such as algebra, topology, differential geometry, and number theory. These fields provide the necessary tools and techniques for studying and understanding the behavior of algebraic varieties.

What are some common challenges in studying Algebraic Geometry?

One of the main challenges in studying Algebraic Geometry is the abstract nature of the subject, which can make it difficult to visualize and understand. It also requires a strong foundation in algebra and geometry, as well as knowledge of other related fields. Another challenge is the complexity of the mathematical concepts involved, which can make it challenging to apply in practical situations.

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