Research - Blake's Binder

Series & Rotations

Blake's Bender

An original problem · started in Calculus 2 and Linear Algebra

Everyone knows 1 + ½ + ¼ + … = 2. Now lay those terms down as arrows, head to tail, and turn each one by an angle θ from the one before it. The series bends, and its sum becomes a point in the plane. Geometric sequences trace circles; the harmonic sequence traces a tongue through ln 2 and π/4, and leads to Blake's Conjecture.

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Topological Data Analysis

Persistent Homology on Financial Time Series

University of Houston · April 2024

A 50-day window slides along the stock market's history. Each window's daily returns become a point cloud, a Vietoris–Rips filtration grows over it, and persistent homology records how long its loops survive. The area under the first persistence landscape (its L¹ norm) turns those loops into one number per day. That number climbed before the dot-com crash of 2000, and its swings grew in the year before both 2000 and 2008.

Follows Gidea & Katz (2018). The animation runs on made-up data shaped like the real thing; its persistence is computed for real.

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Diophantine Approximation

A Geometric Proof of Peck's Theorem

with Kavita Dhanda and Alan Haynes · University of Houston · in preparation

If 1, α₁, …, α_d is a basis of a number field, Peck (1961) showed that some n ≤ T approximates every α_i to within n^−1/d, with an extra factor of (log T)^−1/(d−1) on all but one coordinate. Our proof is a picture: multiply the trace-zero element of the dual lattice by a unit, and its approximation errors are the unit's imbalance read through a fixed linear map. One application of Minkowski's theorem, in the lattice of unit logarithms, finds a unit below any height that is balanced to within 1/log T. The same box, reshaped, attacks Peck's own conjecture about splitting the savings unequally.

The animation is the paper's Figure 1 in motion, for ℚ(∛2), with exact integer arithmetic.

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Financial Econometrics

Pricing Kernels in the Crude Oil Market

a paper by my friend Julian · finance PhD · my read of the math in it

Every option is a bet, and the second derivative of the call curve turns the market's bets into a probability density. It's the wrong density, inflated by fear, and the ratio to the true one is the pricing kernel: for crude oil it's U-shaped at five days and W-shaped at thirty, because a spike is as feared as a crash. The paper recovers the true density from a histogram of past surprises, boils it down to volatility, skewness and kurtosis, and asks, slice by slice with quantile regression, how yesterday's move changes today's fear.

The animation runs on a made-up market shaped like the paper's; every curve is computed, not drawn.

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Mean Field Games

Mean Field Games, Computed

Achdou & Laurière, lecture notes on arXiv (2020) · my read of the math in them

A crowd of identical, selfish agents each steer through the crowd they are part of. The value function of one agent runs backward in time, the density of the crowd runs forward, and neither can be solved alone. The paper is about computing that: an upwind scheme whose density equation is the transpose of its value equation, a convex dual that turns the pair into one minimization, multigrid that coarsens only the direction the smoother cleaned, Newton with continuation where there is nothing to minimize, and then two crowds in a hallway, an evacuation with a price of anarchy, and households saving against a borrowing limit.

The animation is a made-up game on a circle, solved by the paper's own finite-difference scheme; every curve is computed.

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Diophantine Approximation

Accumulation Points of Normalized Approximations

Kavita Dhanda and Alan Haynes · Journal of Number Theory 268 (2025) · my read of their paper

Multiply α by every integer q, take the error to the nearest integer vector, and stretch it by |q|^η. Where do all those values pile up? For a typical α they pile up everywhere, as long as η ≤ 1/d, and the paper measures exactly how the exceptions fail. For an algebraic α they pile up on something you can draw: a discrete set of norms on the line, or a nest of ellipses or hyperbola pairs in the plane, one ring per norm of the number field. The missing rings are the inert primes.

The animation is the paper's Figure 1 in motion, for ℚ(∛2): every dot is a real approximation with exact integer arithmetic, and ring 7 never comes.

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