Independent Studies-Mathematics 199C-SPRING 2021  EVOLUTION ALGEBRAS AND APPLICATIONS  

Midweek meeting: Wednesday evening 7:30-8:15 PM (ZOOM)

Weekend meeting Saturdays 4-4:45 PM (ZOOM)
(syllabus-revised April 15)

FORUM (Questions from students and answers) Updated May 4 (click here)

MEETINGS

April 7 Structure constants, or What is an algebra? (1)(pp 3-11) (recording)

April 10 From calculus to algebra, or What is a derivation (on matrix algebras)?
(2) (pp10-35) (same as (1)(pp 32-45)) (recording)

April 14 (7:30-8:15 PM) (unfortunately, I failed to save the recording)
(a) Axiomatic approach to algebras, or
What is a Jordan algebra? (2)(pp 35-45) (pdffile)
(b) What is an evolution algebra? (click here)
(c) The zygotic algebra is not an evolution algebra
(includes HW 2, Exercise 1)
(click here)

April 17 (4:00-4:45 PM) (Bernie Russo has the day off)

April 21 (8:00-8:45 PM time changed for this day only)
(a) Every Jordan algebra is power associative.
Not every evolution algebra is power associative
(Includes HW2 Exercise 2) (pdffile)
(b) What is an evolution subalgebra?
What is an evolution ideal
What is the extension property?
(Includes HW2 Exercises 3,4,5) (click here) (recording)

April 24 (4:00-4:45 PM) Signed Digraphs and the Energy Crisis
Part I Pulse Processes (pdffile) (recording)

April 28 (7:30-8:15 PM) Discussion of some papers not yet chosen for projects, namely, I(5), I(7), III(1), III(2), III(3).
It was noted that III(4)(5)(8)(9) involve derivations (partial recording)
(I(n) and III(n) refer to the references for project reports below)

May 1 (4:00-4:45 PM) Signed digraphs and the energy problem.
Part II A dynamic model. Theorems 1,2,3 (click here) (recording)
Solutions to HW1 (for self check) (click here)

May 5 (7:30-8:15 PM) Signed digraphs and the energy problem.
Part III Stability. Theorem 4 (click here)
(unfortunately, I forgot to record the meeting)

May 8 (4:00-4:45 PM) Signed digraphs and the energy problem.
Part IV Stability. Theorem 5 (click here)
(recording)

May 12 (7:30-8:15 PM) Signed digraphs and the energy problem.
Part V Stability. Theorem 6 (click here)
Discussion of HW2 (click here)
(recording)

May 15 (4:00-4:45 PM) Pulse processes and evolution algebras
Part I unitization; multiplicative spectrum
(pp. 1-4 of click here) (p. 4 of click here) (pp. 5-6 of click here)
"The multiplicative spectrum and the uniqueness of the complete norm topology
(p. 476 of click here) (p. 4 of click here)
(recording)

May 19 (7:30-8:15 PM) Proposition 2.2 from
"The multiplicative spectrum and the uniqueness of the complete normed topology (pp. 5-8 of click here) (p. 477 of click here)
Pulse processes and evolution algebras Part II Theorem 6
Behavior of multiplicative spectrum under algebra isomorphisms.
(pp. 1-2 of click here) (p. 7 of click here) (p. 5 of click here)
(recording)

May 22 (4:00-4:45 PM) Proposition 2.2 (revised) from
"The multiplicative spectrum and the uniqueness of the complete normed topology (pp. 5-8 revised- click here) (p. 477 of click here)
Pulse processes and evolution algebras Part III Theorem 7
Multiplicative spectrum as eigenvalues of a matrix.
(pp. 3-4 of click here) (p. 7 of click here) (p. 5 of click here)
(recording) (transcript)

May 26 (7:30-8:00 PM)
Discussion of procedure for presentations of projects
(You can read the transcript or watch the recording for details)
(recording) (transcript)

May 29 (4:00-4:45 PM) Review of Theorems 1-5, and 7 from
"Pulse processes in networks and evolution algebras" (click here)
Evolution algebras and pulse processes-Section 4 of
"Pulse processes in networks and evolution algebras"
(3rd and 4th page of click here)
Discussion of procedure for presentations of projects
(You can read the transcript or watch the recording for details)
(recording) (transcript)

HOMEWORK ASSIGNMENTS

Homework #1: Exercises on algebras (#1-7)
(Due April 27 electronically) (pdffile)

Homework #2: (Due April 30 electronically) (pdffile)

COURSE MATERIALS

(1) An ALGEBRA (pp 3-11) and a DERIVATION (pp 32-45) (pdffile)

(2) Derivations on algebras of matrices (pdffile)

(3) Exercises on algebras (#1-7) (pdffile)

(4) Signed digraphs and the energy crisis
Fred S. Roberts, Thomas A. Brown (pdffile)

(5) Pulse Processes in Networks and Evolution Algebras
Julio Becerra, Maria Beltran, and M. Victoria Velasco (click here)

REFERENCES FOR PROJECT REPORTS

I. Some Research Articles on evolution algebras (click here)

II. More Research Articles on evolution algebras (click here)

III. Even More Research Articles on evolution algebras (click here)