Algebraic geometry is a fascinating yet complex field of mathematics. As an interviewee you can expect to face some tricky algebraic geometry questions that will test your conceptual knowledge and problem-solving skills.
In this article, I’ll share the 27 most common algebraic geometry interview questions and sample answers to help you prepare and succeed. I’ll explain key concepts in simple terms, providing tips and insights from my experience as an interview coach.
Whether you’re a student gearing up for college admissions or a professional aiming for your dream algebra role read on to master the art of tackling algebraic geometry problems.
1. Explain the Principle of Duality in Projective Geometry
The principle of duality is a powerful concept in projective geometry. It states that we can interchange points and lines in a geometric figure without altering its basic properties.
For example given a theorem about points on a line, the dual theorem will be about lines passing through a point. This duality allows us to translate difficult problems into potentially easier ones by switching between the dual settings.
In algebraic geometry, this manifests as the dual variety, obtained by taking all hyperplanes tangent to a given variety. Duality provides insight into the algebraic-geometric interplay, helping solve problems from different viewpoints.
2. Define a Scheme in Algebraic Geometry
A scheme in algebraic geometry generalizes algebraic varieties by using sheaves and commutative algebra. It is a topological space along with a sheaf of rings, forming a locally ringed space.
Schemes extend algebraic geometry by allowing more general geometric objects than varieties over algebraically closed fields. They provide a flexible framework to study solutions of polynomial equations over any field using the language of sheaf theory.
3. Implement Buchberger’s Algorithm in Python
Buchberger’s algorithm generates a Gröbner basis for an ideal generated by a set of multivariate polynomials. Here is an outline for implementing it in Python:
- Initialize Gröbner basis G with input polynomial set
- Compute pairs of polynomials in G
- For each pair, calculate S-polynomial and reduce using G
- If reduction is non-zero, add pair to basis
- Repeat until no new pairs reduce to non-zero
- Return G
The key steps are constructing S-polynomials from pairs and reducing them using the basis until no new basis elements get added. This incrementally builds a Gröbner basis for the ideal.
4. Explain Smoothness and Singularity of Algebraic Varieties
Smoothness and singularity describe local geometric properties of algebraic varieties.
A point on a variety is smooth if its tangent space has the same dimension as the variety. This ensures ‘well-behaved’ geometry.
Singular points do not satisfy this, leading to irregular local behavior. Based on the dimension and multiplicity of the tangent space, singularities are classified into ordinary and non-ordinary types.
Smoothness underpins many important results in algebraic geometry. Singularities lead to technical difficulties but also provide insights into a variety’s global structure.
5. What is Intersection Theory and its Significance?
Intersection theory provides a framework to algebraically quantify intersections between subvarieties of an ambient space. It associates intersections with multiplicities to account for tangential overlaps.
This allows complex calculations impossible geometrically, like self-intersections. Intersection theory has deep connections with algebra through the intersection product operation.
Its significance lies in enabling enumerative geometry problems to be solved using algebraic methods. This is fundamental to varied fields including string theory, physics and number theory.
6. Apply Riemann-Roch Theorem with an Example
The Riemann-Roch theorem relates the dimension of sections of a line bundle L on a curve C to its degree and the curve’s genus.
For instance, let C be a smooth projective curve of genus g over an algebraically closed field. Let L be a line bundle on C with degree d.
Then, Riemann-Roch states:
l(L) – l(Kc – L) = d + 1 – g
Here l(M) denotes the dimension of the space of sections of a line bundle M, and Kc is the canonical class of C.
7. How Does Algebraic Geometry Apply to String Theory?
Algebraic geometry is pivotal in string theory physics. It provides tools to describe extra compact dimensions through algebraic varieties called Calabi-Yau manifolds.
The geometry of these spaces governs the behavior of strings propagating through them. This geometry is linked to observables like particle masses and symmetries.
Algebraic techniques help construct and analyze Calabi-Yau shapes. Mirror symmetry of these manifolds has profound physics implications that are studied using algebraic geometry.
8. Explain the Role of Homological Algebra in Algebraic Geometry
Homological algebra is essential for studying algebraic objects through associated topological constructs like cohomology groups.
Cohomology theories like sheaf and etale cohomology use homological algebra tools extensively. These theories extract algebraic data from geometric objects like varieties and schemes.
Concepts like derived functors and spectral sequences help compute cohomology groups, which contain information like the dimension and smoothness of the underlying variety.
9. How Do You Resolve Singularities of Algebraic Varieties?
One method for resolving singularities is through successive blowups. The idea is to replace a singular point with a manifold structure while preserving the variety’s properties.
First we use blowups to introduce exceptional divisors at singular points. This partially resolves the singularities. The blowup process continues recursively, with each step reducing the complexity of singularities.
Hironaka’s theorem guarantees this process terminates in characteristic zero. The end result is a non-singular variety birational to the original singular one.
10. Explain Étale Cohomology and its Role in Algebraic Geometry
Étale cohomology is a variant of sheaf cohomology suited to studying algebraic varieties over imperfect fields like Q. Locally, it resembles singular cohomology.
Unlike classical methods, etale cohomology can capture intricate topological data like Galois representations. This makes it invaluable for geometrically analyzing problems from number theory.
For example, it provides proofs of the Weil conjectures on counting points on varieties over finite fields. Etale cohomology also links arithmetic geometry and algebraic geometry through concepts like Albanese and Picard varieties.
11. How Do You Use Moduli Spaces in Algebraic Geometry?
Moduli spaces parameterize geometric objects like curves, surfaces or varieties up to isomorphism. Their points correspond to equivalence classes of objects.
Studying moduli spaces allows us to understand global structural properties. Techniques like intersection theory on moduli spaces reveal information about the objects themselves.
Moduli problems also motivate central concepts like stacks. Construction of moduli spaces remains a highly active area of research in algebraic geometry.
12. Why Are Stacks Important for Studying Moduli Spaces?
Algebraic stacks provide an essential tool for rigorously handling moduli problems. They form a natural setting for objects with extra automorphism data.
Stacks overcome issues like lack of fine moduli spaces and coarse moduli space singularities. They are equipped with tangent spaces capturing deformation theory.
Techniques like quotient stacks and Artin’s axis allow systematic construction of moduli stacks. The rich theory of stacks furnishes foundations for many important moduli problems.
13. Discuss the Taniyama-Shimura-Weil Conjecture
The Taniyama-Shimura-Weil (TSW) conjecture connects two seemingly disparate objects – elliptic curves and modular forms.
It proposes that every elliptic curve over Q is associated with a modular form, obtaining a precise dictionary between these classes.
This bold claim spurred decades of intense research culminating in Wiles’ proof of a pivotal special case – Fermat’s Last Theorem.
Though initially baffling, the eventual proof demonstrated the profound unity in mathematics suggested by the TSW conjecture. Its methods forged new links between algebraic geometry, number theory and analysis.
14. Relation Between Elliptic Curves and Algebraic Geometry
Elliptic curves are foundational examples of algebraic curves defined by cubic equations in two variables. Their geometric structure as genus 1 curves exhibits a natural abelian group law.
Algebraic geometry provides the bedrock for analytically investigating these curves through invariants like genus, canonical class and point counting. This leads to ties with number theory and arithmetic geometry.
Concepts like morphisms, divisors and intersection theory applied to elliptic curves yield deep results like Faltings’ theorem and the Mordell-Weil theorem on rational points.
15. Application of Algebraic Geometry in Coding Theory
Algebraic geometry principles are widely applied in coding theory for error correction and detection. These codes leverage the geometric structure of algebraic varieties to provide efficient signal processing.
Reed-Solomon codes are an excellent example. They construct codewords using points on curves, allowing recovery from multiple erasures. The error correction capability stems from the algebraic-geometric properties.
Codes like Goppa’s utilize divisors on curves and places on function fields. Decoding algorithms rely heavily on algebraic geometry techniques and theorems.
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12 Answers 12 Sorted by:
Perhaps youre going about this the wrong way. It’s better to talk about a problem or set of problems that are concrete and easy to understand than to try to explain what most algebraic geometers do today. I think there are plenty of topics to choose from. Youve already mentioned solving systems of equations. Some algebraic geometers don’t even try to find solutions; they just want to know when there are solutions (Nullstellensatz) and how many parameters are needed. Another important reason in history is the study of elliptic and abelian integrals: you can go from adding laws for integrals to adding laws on curves and Jacobians.
Abhyankars book Algebraic Geometry for Scientists and Engineers doesnt give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of algebraic equations.
In conversations like this, I usually lead with a concrete example of a hard problem. To find complete intersections, first notice that two surfaces in three-space usually meet in a curve. Then, ask yourself if, given an algebraically defined curve, it is always the intersection of two algebraically defined surfaces. How do you tell the difference between those that are and those that aren’t? This gives you a chance to talk about how important it is to use both geometric intuition and algebraic calculations.
Now generalize to higher dimensions. Now, if they still want more, you can get into more specifics, such as the difference between a real complete intersection and a set-theoretic complete intersection. Or give a sequence of increasingly challenging specific cases. Et cetera.
Ive also — though this is sort of cheating — used the example of classifying vector bundles. In the case of topology, this is easy to explain: let’s say you have a circle and you want to connect a line to every point on the circle in a straight line. You can make a cylinder, or you can make a Mobius strip. What else can you make? When should you think of two of these things as “the same”? Now keep in mind that the answers to these questions depend on the rules you set for how you’ll build your objects and when you should think of two things as “the same.” If you want to study topology, everything has to be continuous. If you want to study algebraic geometry, everything has to be continuous. It’s pretty easy to see that all vector bundles of a certain rank are topologically equivalent if the base space is also a vector space. But it’s not so easy to see this in the algebraic case. Et cetera.
This is along the lines suggested by @DonuArapura: “describe a problem [. ] that is reasonably concrete and accessible, and go from there. “.
An engineer would enjoy this puzzle: what kinds of bent wire can go through a pinhole in a plane without moving at all? These kinds of curves are called threadable curves. 1.
Deciding whether a given planar algebraic curve $C$ is threadable depends on the number of bitangents. For a curve of degree $d$, this number is $O(d^4)$, a result of Schubert. See the MO question, Number of bitangents to connected algebraic curve.
1J.ORourke and Emmely Rogers, “Threadable curves,” Proc. 30th Canad. Conf. Comput. Geom., Aug 2018, 328—333. (arXiv abstract).
If you only want to explain what algebraic geometry is and not try to persuade the engineer that it’s important to study, a good place to start is to talk about the different types of conic sections (ellipse, parabola, and hyperbola) and how algebraic geometers try to group the different outcomes that can happen when there are more degrees, equations, or variables.
Another “disconnect” between engineers and mathematicians is that they don’t understand PDEs the same way you do. Engineers often just want to solve PDEs. Mathematicians are interested in solving PDEs too but are also interested in other questions. It might be easier to explain the idea of wanting to understand the qualitative features of a solution in terms of PDEs. Then you can say that the situation in algebraic geometry is similar.
I have had this conversation a few times. I start by explaining the idea of abstract classification of objects. I show them how a mechanical arm that can turn in a circle and has another mechanical arm at the end that can also turn in a circle is, in a way, the same thing as a torus, even though they look different. That being said, the question is how to group things that are defined by rules that look different but are really the same?
I then explain that in A. G. Polynomials are often used as constraints, which lets people use ideas and points of view that wouldn’t work with completely general constraints.
I believe a good explanation should show how algebraic geometry can help us understand what a general point on an irreducible variety really means.
I could use Gerstenhaber’s Theorem as an example. It says that the number of pairs of commuting complex matrices is uncountable. A generic pair is two commuting diagonalizable matrices with different eigenvalues. I think one could give a good idea of this without (explicitly) using the group action or topology. The engineer may already know that rotations of $mathbb{R}^3$ can be diagonalized over $mathbb{C}$. If that’s the case, I would tell them that these matrices aren’t quite general because of the $1$ eigenvalue.
If they want to know more, I’d say that the variety diminishes as the number of matrices increases, which means there is no good reason to think of a generic tuple.
One of the things engineers are very familiar with is integration. “What kind of substitution should I make to explicitly find antiderivative?” is a very natural question. And, you know, Algebraic Geometry sometimes helps with that.
My favorite answer for this kind of question is to start with an integral of something in $mathbb{R}(sin(t),cos(t))$. To write the point $(sin(t),cos(t))$ in terms of $tan(t/2)$, I then use a unit circle and stereographic projection. This leads to the substitution that solves the antiderivative. I will also briefly explain how you make connections between algebraic expressions and geometric objects that help you understand the algebraic object.
I would start by showing them to how find rational points on a conic. If you have a rational point then you can draw lines and find more. If they understand the geometric side, you could stress the “rationality” part of the construction by saying something like, “Look, both the slope and the y-intercept are rational. If one point of intersection is rational, then the other one is too.”
This building has just the right number of logical steps for the engineer to be able to check them if they want to. g. rationality, getting all of them, the necessity of finding a point to start off the process.
Then you could let them play with a quadratic extension (!) to “see what happens.”
This is a good example because both the algebra and geometry are easy enough for your audience to understand.
I’m not even going to talk about elliptic curves.
There are several good answers already so I cannot hope to add much. Still, another way to get engineers’ attention could be to use the well-known subject of linear algebra, especially the idea that solving systems of linear equations is similar to solving systems of polynomial equations.
Start by quickly going over linear systems and why there must be zero, one, or an infinite number of solutions. Show the usual pictures of lines and planes crossing in $mathbb{R}^2$ and $mathbb{R}^3$, such as two planes crossing in a line and a plane and a line crossing in a point. For example, you could use this to start a conversation about the size of solution sets for polynomial equations, the Hilbert Nullstellensatz, and Bezout’s Theorem.
I would say that it’s a subject where you do geometry and think about geometry, but you write about it in a way that sounds like algebra and use algebra, which can be pretty hard to understand. I have an example from Weibel’s Introduction to Homological Algebra, page 119. Hartshorne first used this idea.
Let $R = mathbb{C}[x_1, x_2, y_1, y_2]$, $P=(x_1,x_2)R$, $Q=(y_1,y_2)R$, and $I = P : cap : Q $.
Regular sequences make up $P$, $Q$, and $m = P Q = (x_1, x_2, y_1, y_2)R$. This means that the outside terms in the Mayer-Vietoris sequence are also regular sequences.
$H^3_P(R) bigoplus H^3_Q (R) rightarrow H_I^3(R) rightarrow H^4_m(R) rightarrow H^4_P(R) bigoplus H^4_Q (R) $
vanish, which tells us that $H^3_I(R) simeq H^4_m(R) eq 0$. (The cohomology is local cohomology here).
A normal engineer might find this hard to understand, but if you take away the words and jargon, it means that the union of two planes in $mathbb{C}^4$ that meet at a point can’t be described as the solutions of only two equations $f_1 = f_2 =0$. This is a real geometric fact.
There is an answer by D. Mumford to biologists, valid also for engineers: Can one explain schemes to biologists, blog post (2014) (link).
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How do I prepare for a math interview?
Preparing for Math Interview Questions Preparing for math interview questions requires a combination of theoretical knowledge and problem-solving skills. Here are some key steps to help you prepare effectively: Review the Basics: Brush up on essential mathematical concepts, including arithmetic, algebra, geometry, and statistics.
What math skills do you need for a job interview?
Brush up on your mental math skills, including calculations with fractions, decimals, percentages, and powers. Quick and accurate mental calculations can save you valuable time during interviews. Algebraic questions can range from solving simple equations to dealing with complex algebraic structures.
What does a math interview look like?
While academic exams test your knowledge and understanding of course material, math interviews focus on your problem-solving abilities, adaptability, and logical reasoning. You’ll often encounter open-ended questions that require you to think on your feet and explain your thought process step by step.
What are math interview questions?
If so, you’ll likely encounter math interview questions designed to assess your mathematical knowledge, problem-solving abilities, and analytical thinking. These questions are an essential part of the hiring process, as they allow employers to gauge your ability to think critically and apply mathematical concepts to real-world scenarios.
