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This is MATH 4800-001, the introduction to mathematics research for undergraduates class at the University of Utah, which focuses on fractal geometry and dynamics.
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Table of Contents:
00:00:00 1 of 2
00:00:42 recall: s-Hausdorff outer measure
00:09:11 exercise
00:10:07 exercise
00:11:55 exercise
00:13:14 exercise
00:15:02 alpha-Holder constant Lip_alpha(f) of f
00:17:26 exercise
00:17:50 exercise
00:18:16 exercise: snowflaking a metric, ultrametric, strong triangular inequality
00:21:07 exercise: any d_theta metric is an ultrametric
00:21:35 applications of ultrametrics; trees, fractals, Thompson groups
00:25:09 lemma: the s-Hausdorff outer measure of f(A) is bounded from above by Lip_alpha(f)^s times the s alpha-Hausdorff outer measure of A
00:27:53 proof of lemma
00:29:12 exercise
00:29:33 proof continued
00:35:47 more on lemma [correction: for f Lipschitz one still has a multiplicative constant possibly, but outer measures for the same s value are compared]
00:36:33 phase transition for H^s(A) for A fixed and s variable
00:45:51 Hausdorff dimension dim_H(A)
00:48:18 s-Hausdorff outer measure of A for s = dim_H(A) could be 0, infinity, or a finite positive number
00:50:55 exercise: Hausdorff dimension of f(A) for f alpha-Holder
00:51:25 exercise: Hausdorff dimension is invariant under bi-Lipschitz homeos
00:53:49 exercise: Hausdorff dimension bounds from above Lebesgue covering dimension, lower Minkowski dimension bounds from above Hausdorff dimension
00:54:30 exercise: upper bounds for the Minkowski and Hausdorff dimensions of attractors of B-IFSs
01:00:58 theorem: Moran-Hutchinson Dimension Formula; Bowen formula [pressure function to be corrected]
01:16:31 2 of 2
01:17:29 a correction: pressure function of a B-IFS
01:18:02 recap of Moran-Hutchinson
01:24:15 proof of Moran-Hutchinson; the upper bound
01:28:12 exercise
01:28:45 proof continued
01:42:29 machinery for Moran-Hutchinson; the lower bound: mass distribution principle and covering lemma
01:45:00 prop: mass distribution principle [first attempt at statement]
01:49:21 more on the Hausdorff dimension as a phase transition
01:54:35 proof of prop
02:08:32 prop: mass distribution principle [second attempt at statement]
02:12:13 prop: mass distribution principle [third attempt at statement]
02:14:53 covering lemma [first attempt at statement]
02:25:24 covering lemma in English
02:28:39 proof of covering lemma
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Links:
More on Thompson's Groups:
th-cam.com/video/F1cSHyXLSnk/w-d-xo.html
th-cam.com/video/m8NXIV2NduU/w-d-xo.html
th-cam.com/video/wYIb__FXuEc/w-d-xo.html
th-cam.com/video/GUjl-_BjD8I/w-d-xo.html
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th-cam.com/video/bAs-InHEUeU/w-d-xo.html
th-cam.com/video/wRp4Ud-DW0s/w-d-xo.html
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License:
CC BY-NC-SA 4.0
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Public License
creativecommons.org/licenses/by-nc-sa/4.0/
Alp Uzman
alpuzman.github.io/