Dr. Jeff Rolland's Homepage

"All [STEM] is either [Geometric Control Theory] or stamp collecting." - Ernest Rutherford #Science #Physics
A picture of me  "If you can dream it, you can do it!" 

"I only hope that we never lose sight of one thing: that it was all started by a [drawing of a] mouse." - W. Disney #ItsUhTheStuffThatDreamsAreMadeOf


Partial List of Surfaces (2D Closed Manifolds)

Boy's Surface
2-Sphere


Klein bottle





Genus 1 2-Torus
 RP^2   Sonnect Sum Symbol Genus 1 2-Torus         


Genus-2 2-Torus

Klein bottleSonnect Sum SymbolGenus 1 2-Torus Genus 1 2-TorusSonnect Sum SymbolGenus-2 2-Torus
Boy's SurfaceSonnect Sum SymbolKlein bottleSonnect Sum SymbolGenus 1 2-Torus
Genus-2 2-TorusSonnect Sum SymbolGenus-2 2-Torus
\(\mathbb{R}P^2\) Boy's Surface
(Euler Characteristic) \(\chi = 1\)
Non-Orientable
1
\(\mathbb{R}P^2\)


(If \(M = \#_{i=1}^g \mathbb{R}P^2\) is the non-orientable connect sum of \(g\)
\(\mathbb{R}P^2\)'s, \(\chi(M) = 2-g\).)

(The Euler characteristic \(\chi\) and orientability/non-orientability of a surface uniquely characterize it up to diffeomorphism.)
\(S^2\) 2-Sphere
\(\chi\ = 2\)
Orientable
Orientation Double-Cover of
\(\mathbb{R}P^2\)

(If 2\(M\) is the double-cover of \(M\), then \(\chi(2M) = 2\chi(M)\).)


(The 2-sphere acts as an identity elements for connect sum: \(M \# S^2 = M\).)
\(K^2\) Klein Bottle
\(\chi = 0\)
Non-Orientable
Connect sum of 2
\(\mathbb{R}P^2\)'s
Fundamental Relation 1 (FR1)
\(T^2_1\) Genus-1 2-Torus
\(\chi=0\)
Orientable
Orientation Double-Cover of
2 \(\mathbb{R}P^2\)'s = 1 \(T_1^2\)

(If \(M = \#_{i=1}^g T_1^2\) is the orientable connect sum of \(g\) \(T^2_1\)'s, \(\chi(M) = 2-2g\).)
Connect Sum of 3 \(\mathbb{R}P^2\)'s
\(\chi = -1\)
Non-Orientable
Connect Sum of 3 \(\mathbb{R}P^2\)'s = Connect Sum of \(\mathbb{R}P^2\) and \(T^2_1\)
Fundamential Relation 2 (FR2)
\(T^2_2\) Genus-2 2-Torus
\(\chi = -2\)
Orientable
Orientation Double-Cover of 3 \(\mathbb{R}P^2\)'s = Connect Sum of 2 \(T^2_1\)'s
Connect Sum of 4 \(\mathbb{R}P^2\)'s
\(\chi = -2\)
Non-Orientable
Connect Sum of 4 \(\mathbb{R}P^2\)'s = Connect Sum of \(K^2\) and \(T^2_1\) (using FR2 then FR1)
\(T^2_3\) Genus-3 2-Torus
\(\chi = -4\)
Orientable
Connect Sum of 3 \(T^2_1\)'s
Connect Sum of 5 \(\mathbb{R}P^2\)'s
\(\chi = -3\)
Non-Orientable
Connect Sum of 5 \(\mathbb{R}P^2\)'s = Connect Sum of \(\mathbb{R}P^2\), \(K^2\), and \(T^2_1\) (using FR2 then FR1) = Connect Sum of \(\mathbb{R}P^2\) and 2 \(T^2_1\) (using FR2 twice)
\(T^2_4\) Genus-4 2-Torus
\(\chi = -6\)
Orientable
Orientation Double-Cover of 5 \(\mathbb{R}P^2\)'s = Connect Sum of 4 \(T^2_1\)'s
Goal:
1) Find a Machine for Every Even Euler Characteristic Surface with that Surfaces as the C-Space for the Machine
2) Create Model of Each
Machine in OnShape
3) Find de Caro-Rolland Table Entry for Each Machine from OnShape Model
4) Automate Construction of Lagrangian from de Caro-Rolland Table for Each Machine in MATLAB
5) Automate Contstruction of Inertial Riemannian Metric from Lagrangian
for Each Machine in MATLAB
6) Automate Computation of Riemannian Sectional Curvature of Each C-Space in MATLAB
7) Adjust OnShape Models so Each C-Space is a Space Form (Riemannian Manifold with Constant Sectional Curvature)

Geometric Control Theory Projects:
1) Optimal Geometric Control Theory and Pontryagin's Maximum Principle: https://deadbeatjeff.sdf.org/mathjax/PMP.html
2) A Smooth (\(C^{\infty}\)) Rise-Fall Function for a Cam-Follower System: https://deadbeatjeff.sdf.org/mathjax/riseFunction01.html
3) Flat Genus-1 Torus C-Space of a 2R Robot Arm: https://deadbeatjeff.sdf.org/mathjax/covering_transformations.html, in progress Genus 1 2-Torus
4) Hyperbolic Genus-2 Torus C-Space of a 5-Bar Mechanism: https://deadbeatjeff.sdf.org/mathjax/genus-2-torus.html, in preparation Genus-2 2-Torus
5) Flat Klein Bottle C-Space of a Mechanism: https://deadbeatjeff.sdf.org/mathjax/klein-bottle.html, in preparation Klein bottle
6) 2-Sphere Configuration Space (C-Space) of a Pendulum: https://deadbeatjeff.sdf.org/mathjax/2-sphere.html, in preparation 2-Sphere
6+i) (I am unaware of a mechanism that might have Boy's surface AKA \(\mathbb{R}P^2\) as its c-space) Boy's Surface
7) A Crank with an Internal Slider Mechanism with Coriolis Acceleration: http://deadbeatjeff.sdf.org/mathjax/coriolis01.html, in preparation


Design Projects:
A. SolidWorks
1) My ME 379 Mechatronics project, a home CNC machine, designed with my design team, in progress
2) My first toy project in SolidWorks, a block with a hole.

Block with Hole SolidWorks

B. OnShape
1) The first link in a 6D robot arm https://cad.onshape.com/documents/737c9e74eeac1c4869a626cd/w/6f8cb5b972292076fdd226a4/e/37c182b71c24e71c21f26d2c
6D Robot Arm 01


2) A 2D robot arm https://cad.onshape.com/documents/fa6e8c4e48a0b023ac4fe14e/w/c08bad50356cf8ed12928f31/e/ff987bbfdbd9fe7208668065?renderMode=0&uiState=65d3428171bf0a08158594f7
2D Robot Arm A 3D print of a 2D robot arm



C. CREO
1) My ME 270 Computer-Aided Design project, the Boll Aero toy airplane engine, designed with my design partner, in progress
Boll Aero Engine Assembly 01 
The Full Engine Assembly


Boll Aero Crankshaft Subassembly
Boll Aeor Fule Subassembly

Boll Aero Piston Subassembly


The crankshaft subassembly The fuel system subassembly The piston subassembly

2) My ME 111 Fundamentals of Engineering II project, an outlet lock, designed for  Master Lock, Inc., with my design team (Faceplate 3D printer file) (Cover 3D printer file) Feb 2023 - May 2023

Faceplate and Cover 04  Faceplate and Cover 05  Faceplate 02  Cover 02  Cover 3

Final Design

outlet_lock_faceplate_and_cover_01.jpg

Design 3

outlet_lock_faceplate_and_cover_02.jpg

Design 2

Outlet Lock Cover  Outlet Lock Faceplate 3D Print Outlet Lock Cover  Outlet Lock Cover 3D Print   Outlet Lock Faceplate and Cover 3D Print

Design 1

CREO Thermal Model

 My first thermal model in CREO.

School:
I am matriculated in a BSE program in Mechanical Engineering at UWM.

Work:
I had a full-time co-op for Fall 2023 - Summer 2024 at Copland Corporation (formerly a division of Emerson) in Sidney, OH working in Data Science.
I had a full-time internship for Summer 2023 at AngleLock by Controlled Dynamics in Grafton, WI working in Mechanical Engineering.

GitHub: hhttps://github.com/DeadbeatJeff?tab=repositories

Research:
(Google Scholar https://scholar.google.com/citations?user=3IUdT2EAAAAJ)
(Orcid ID https://orcid.org/0000-0003-0167-670X))
(Math Genealogy Project https://www.mathgenealogy.org/id.php?id=190149)

Research Interests: My Ph.D. is in manifold topology under Prof. Craig Guilbault (UWM's Math Department's Chairman). However, I am moving my research interests into an application of manifold topology, specifically geometric control theory (control engineering on manifolds). I have a blog related to my learning experiences in these new areas of STEM (for me), This Week's Finds in Second-Order Ordinary Differential Equations on Manifolds and Geometric Control Theory.

Publications:
An Introduction to de Rham Cohomology for Engineers with Applications to Machines' Configuration Spaces using Python (book), toying with the idea (excerpt)
"Configuration Spaces of 6R (nR) Robot Arms are Calabi-Yau", in preparation
"\(\mathcal{EZ}\)-Structures for Group Extensions of Fundamental Groups of Nonpositively Curved Space Forms and Locally Symmetric Spaces", on hold
"Some Examples of Pseudo-Collars on High-Dimensional Manifolds with Non-Superperfect Kernel Groups", on hold
"Some Examples of Pseudo-Collars on High-Dimensional Manifolds with Non-Split-Exact Group Extensions", on hold
"A Necessary and Sufficient Condition for a Self-Diffeomorphism of a Smooth Manifold to be the Time-1 Map of the Flow of a Differential Equation" https://arxiv.org/abs/2110.12806, unpublished
"More Examples of Pseudo-Collar Structures on High-Dimensional Manifolds", Topology and its Applications, Vol. 270, February 2020 https://arxiv.org/abs/1804.02597 https://www.sciencedirect.com/science/article/abs/pii/S0166864118305716
"A Geometric Reverse to the Plus Construction and Some Examples of Pseudo-Collars on High-Dimensional Manifold", Michigan Mathematical Journal, Vol. 67 No. 3, August 2018 https://arxiv.org/abs/1508.03670 https://projecteuclid.org/journals/michigan-mathematical-journal/volume-67/issue-3/A-Geometric-Reverse-to-the-Plus-Construction-and-Some-Examples/10.1307/mmj/1522980163.short
"Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds", doctoral dissertation, May 2015 http://arxiv.org/abs/1502.04338 https://dc.uwm.edu/etd/916/

Presentations:
UWM Topology Seminar 2021 (UW-Milwaukee)
UWM Topology Seminar 2020 (UW-Milwaukee) [Revised - Original slides]
Workshop in Geometric Topology 2016 (Colorado College)
Dissertation Defense (01/2015) (UW-Milwaukee)
Workshop in Geometric Topology 2014 (UW-Milwaukee)
Workshop in Geometric Topology 2009 (UW-Milwaukee)

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Jeffrey Rolland.
rollandj@uwm.edu  My Twitter BlogSpot Logo LinkedIn GitHub Repo