Hilbert's 15th Problem: Schubert Calculus | Infinite Series
PBS Infinite Series・11 minutes read
Herrman Schubert's innovative methods in geometry were both questioned for rigor and praised for their ability to simplify complex problems, with puzzles aiding in the visualization and computation of intersections, and Dedekind cuts used to understand transcendental numbers. Steven recommended the graphic novel "Logicomix" as a valuable resource on the history of logic and math.
Insights
- Schubert's innovative approach to geometry problems involved simplifying computations by strategically positioning lines, despite facing skepticism about the rigor of his methods, akin to landing a jumbo jet blindfolded.
- The introduction of puzzles with colored sides not only aided in understanding geometry but also led to the development of conjectures about relationships between edges and triangles, which were proven using variables N and K, showcasing a unique and practical method for tackling complex geometry concepts.
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Recent questions
Who was Herrman Schubert?
A mathematician fascinated by geometry problems.
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Summary
00:00
Schubert's puzzles revolutionize complex geometry computations.
- Herrman Schubert, a mathematician in the late 1800s, was fascinated by intersection problems in geometry.
- Schubert computed complex geometry problems, such as the number of twisted cubics tangent to 12 quadrics.
- Schubert simplified problems by moving lines into specialized positions to ease computation.
- Schubert's methods were questioned for rigor, with his work being compared to landing a jumbo jet blindfolded.
- Puzzles involving equilateral triangles and parallelograms with colored sides were introduced to aid in understanding geometry.
- Observations from the puzzles led to conjectures about the relationships between edges and triangles of different colors.
- Variables N and K were introduced to aid in proving the conjectures about the triangles in the puzzles.
- The shrinking process of the puzzle helped prove the conjectures about the number of edges and triangles.
- The puzzles were used to compute intersections of lines by imposing multiple conditions.
- The puzzles allowed for the visualization and computation of lines intersecting multiple conditions, leading to a better understanding of complex geometry problems.
14:23
"Dedekind cuts and transcendental numbers"
- Dedekind cuts involve dividing rationals into two sets where one set's elements are all greater than the other set's elements. For example, the square root of 2 lacks a rational number immediately above or below it. Dedekind cuts can be used to obtain transcendental numbers, which are not algebraic numbers, by splitting rational numbers into two sets, with most of them being indescribable like the square root of 2. Steven recommended the graphic novel "Logicomix" on the history of logic and math's foundation.
