A counterintuitive guide to navigating academic mathematics using the Lindy Effect
Every mathematics graduate student eventually faces the same existential question: Should I work on classical problems that have captivated mathematicians for centuries, or chase the latest trends, be they in machine learning, quantum computing, or cryptography? The answer, surprisingly, might challenge everything you’ve been taught about mathematical importance.
The Lindy Effect and the Immortality of Ideas
The Lindy Effect offers a simple heuristic: the future life expectancy of ideas and technologies is proportional to their current age. Euclid’s Elements has survived 2,300 years and will likely survive another 2,300. Meanwhile, that hot new machine learning framework from last year? It might not make it to next semester.
For mathematicians, especially those trained in analysis, where we still teach Cauchy’s 19th-century epsilon-delta definitions, this principle feels intuitively correct. The Banach spaces you’re studying today will outlive any programming language you might learn. The measure theory giving you headaches will remain relevant long after current AI architectures become historical curiosities.
This suggests an obvious career strategy: focus on the eternal, ignore the ephemeral. Study classical analysis deeply. Work on problems that have stood for centuries. Build your career on permanent foundations.
But here’s where things get interesting, and counterintuitive.
The Shocking Historical Pattern
Look at the mathematicians whose work best exemplifies the Lindy Effect today. Galois created group theory not by pursuing eternal abstractions, but by attacking a trendy early-1800s problem about solving polynomials, the machine learning of his era. Riemann gave us the most famous unsolved problem in mathematics while chasing fashionable extensions of Cauchy’s recent complex analysis work.
The pattern repeats throughout history: mathematicians working on their generation’s “trendy” problems often create the next generation’s permanent mathematics. Those consciously trying to work on “eternal” problems? They usually produce respectable but ultimately marginal refinements.
Why? Because truly important eternal problems often disguise themselves as temporary trends. They first appear as specific, narrow, “applied” questions. Only later do we recognize them as windows into deep, permanent mathematical structures.
The Analysis Student’s Dilemma
This creates a particularly acute challenge for those trained in analysis. Your education emphasized rigor, foundations, and classical techniques. You’ve spent years mastering Lebesgue integration, functional analysis, and partial differential equations. The culture of pure analysis can be dismissive of “applications,” they lack elegance, require messy computations, and seem philosophically confused.
Yet historically, many breakthroughs in analysis came from physicists and engineers forcing mathematicians to make sense of their “unrigorous” methods. Distributions, Fourier analysis, and even calculus itself emerged this way. Today’s messy machine learning mathematics might be tomorrow’s elegant theory.
Consider what’s happening right now in neural network theory. Researchers are discovering connections to kernel methods, random matrix theory, and statistical physics. They’re being forced to develop new frameworks for infinite-dimensional optimization, non-convex landscapes, and high-dimensional probability. This isn’t just application, it’s analysis being born.
A Strategic Framework for Modern Academic Success
So how can you leverage these insights for your career? Here’s a practical framework:
1. The Foundation-Application Bridge Strategy
Position yourself as a translator between classical analysis and contemporary applications. Can you rigorously formalize what machine learning practitioners are doing with gradient flows? Can you apply optimal transport theory to modern problems? This rare bilingual ability commands both respect and resources.
Many successful analysts now run dual research programs: proving theorems about PDEs (for reputation and tenure) while applying functional analysis to data science (for funding and impact). The key is ensuring these programs cross-pollinate.
2. The Second-Wave Advantage
Don’t be first into a trendy field, be second. Let others establish that quantum machine learning or topological data analysis has genuine substance. Then arrive with your analytical toolkit to:
- Prove the first real convergence theorems
- Establish when the methods actually work
- Connect the field to classical mathematics
This second wave often receives more recognition than the pioneers. You get credit for “making it rigorous” while avoiding the career risk of working on something that might evaporate.
3. The Problem Selection Matrix
Choose problems that satisfy multiple criteria:
- Immediate payoff: Can you get papers out quickly?
- Method development: Will you create tools others can use?
- Classical connection: Does it relate to established mathematics?
- Future potential: Could this define a new field?
For instance, working on mathematical foundations of deep learning hits all four: constant publication opportunities, new analytical techniques, connections to approximation theory and dynamical systems, and potential to define how we understand artificial intelligence.
4. The Teaching Arbitrage
Create courses that connect classical analysis to modern applications: “Functional Analysis for Machine Learning,” “PDE Methods in Image Processing,” or “Measure Theory for Data Science.” These courses:
- Attract graduate students from multiple departments
- Generate lecture notes that become standard references
- Position you as indispensable to your department
- Build networks across pure and applied communities
5. The Collaborative Power Play
Pure analysis training is becoming rarer, making it increasingly valuable. Partner with computer scientists, physicists, or engineers who need rigorous foundations for their work. You provide the theoretical backbone; they provide problems, funding, and impact.
These collaborations often yield surprising mathematics. The computer scientist’s algorithm might secretly be a discretization of a gradient flow. The physicist’s approximation might suggest a new class of inequalities. The engineer’s heuristic might hide a deep variational principle.
The Deeper Game
The ultimate insight is that mathematics doesn’t progress through conscious attempts at creating eternal truths. It advances when mathematicians are forced to develop new tools for urgent problems. The urgency provides motivation and constraints; the mathematical response, if deep enough, becomes permanent.
This means the optimal strategy isn’t choosing between classical and trendy, but creating feedback loops between them:
- Use classical analysis to understand modern problems
- Extract new principles from contemporary solutions
- Formalize these principles using rigorous foundations
- Apply the formalized theory to even newer problems
A Personal Research Philosophy
As you build your career, consider adopting this philosophy: Be deeply grounded in classical analysis, know your Banach spaces, your Sobolev embeddings, your spectral theory. But deploy this knowledge opportunistically on problems that matter now.
Your measure theory background might revolutionize understanding of neural network initialization. Your knowledge of harmonic analysis might crack problems in signal processing. Your facility with variational methods might formalize reinforcement learning.
The mathematicians creating tomorrow’s classical analysis aren’t necessarily the ones working on yesterday’s classical problems. They’re the ones forced by “temporary” challenges to invent new permanent tools.
The Final Paradox
The Lindy Effect itself, this ancient observation about the persistence of ideas, predicts its own limitation. Most attempts at creating permanent mathematics will fail, whether aimed at eternal or ephemeral targets. But this same principle suggests a strategy: maximize your shots on goal.
Working on trendy problems with classical tools gives you two chances at permanence: you might solve the immediate problem (short-term success) or accidentally create lasting mathematics (long-term immortality). Working solely on classical problems gives you only one chance, and it’s a much harder shot.
So embrace the paradox. Learn analysis deeply, not to work only on classical problems, but to have the tools ready when the next trendy problem reveals itself as an eternal truth in disguise. The measure theory you’re struggling with today might seem abstract and removed from reality. But when the right application appears, you’ll be the one who recognizes it as a measure theory problem, and that recognition might define a new branch of mathematics.
Your career doesn’t have to choose between the eternal and the ephemeral. With the right strategy, you can pursue both, and maybe, just maybe, discover that they were the same thing all along.
Remember: The mathematicians we revere today mostly didn’t set out to create eternal mathematics. They set out to solve the urgent problems of their time with the best tools available. Do the same, and let eternity sort itself out.