This is a conversation with Dr. Larry Rolen, a tenure-track professor in the Department of Mathematics at Vanderbilt University. Dr. Rolen's research interests include modular forms, integer partitions, and L-functions.

Dr. Rolen's homepage: https://math.vanderbilt.edu/rolenl/index.html

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Timestamps:

[ 1:30 ] Early education

[ 6:30 ] The Banach-Tarski paradox, fractals

[ 10:00 ] Measuring coastlines

[ 12:30 ] First exposure to number theory, switching colleges, research opportunities

[ 18:00 ] Introducing number theory, ubiquity of number theory in branches of math

[ 20:00 ] Integer partitions, p(5n + 4) = 5k (Ramanujan), physics applications of integer partitions

[ 24:00 ] Modular forms (hyperbolic complex functions), Fourier Series in music

[ 29:00 ] Symmetry of modular forms, growth rates of number sequences

[ 30:00 ] Connecting 0 to infinity

[ 32:00 ] Symmetries of Modular forms in physics

[ 35:00 ] Connecting distant mathematical fields with symmetry, slightly broken symmetry

[ 38:00 ] Fermat's Last Theorem (x^n + y^n = z^n, n > 2) proof from Wiles, Pythagorean triples

[ 45:00 ] The Weil Conjectures, Connecting elliptic curves to modular forms, mod p, encryption

[ 53:00 ] Lattice-based cryptography, Discrete Log Problem, RSA key crowding

[ 59:00 ] Unpacking elliptic curves, point adition on elliptic curves, unpredictability of elliptic curve sequences

[ 1:03:00 ] George Andrews, Rescuing Ramanujan's lost notebook, mock theta functions

[ 1:13:30 ] The Riemann Zeta Function, The Riemann Hypothesis, infinite sums, the critical strip

[ 1:18:30 ] Counting prime numbers, connecting proportions of primes to the zeros of the Riemann Zeta Function, burden of proof for the Prime Number Theorem

[ 1:26:00 ] Montgomery's pair correlation conjecture, random matrix models of the distribution of prime numbers

[ 1:28:00 ] Dr. Rolen's favorite number e^(pi * sqrt(163) and its connection to Hilbert's complex multiplication

[ 1:34:00 ] Combining LLMs and Sage for proof assistance

[ 1:38:00 ] Hobbies outside of math

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Music Credits:

Spring Flowers by Keys of Moon | https://soundcloud.com/keysofmoon

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