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Craig M. Johnson, Ph.D., Chair

Publications	Courses I Teach	Presentations
Student Research	ARML Math Team	Student Math Club

I am a professor of mathematics at Marywood University in Scranton, PA and the current chair of the department. My research interests are primarily in topology (see publications below) and in the connections of mathematics to music. Other activities include:

Coach of the Northeast Pennsylvania ARML (American Regions Math League) team for talented high school math students
Direction of undergraduate student research
Textbook writing

Office: 319 Science; Phone: (570) 348-6291;

E-mail: johnsonc@marywood.edu

Spring 2011 Schedule:

MWF: 9-10 am Physics 112 (Astronomy)
MWF:1-2:00 pm Math 430 (Real Analysis)
TTh: 1-2:15 pm Math 120 (Math in Liberal Arts)

Office Hours: MWF 10-12 pm

Publications

Introducing Group Theory through Music, Mathematics Teacher,Vol. 103, Issue 2, September 2009, pp. 116-123.
- Abstract: Musical inversions are one of several compositional techniques used by the classical composers. A strict inversion reflects notes across a particular pivot note and this changes the key of a piece of music in a computable way. Different pivot notes produce different key changes and the set of distinct inversions displays an identifiable structure. This article demonstrates that the set of inversions is a coset of a particular subgroup of the dihedral group of order 24. The musical analogs of the other cosets are also explored. Examples are given from Bach, Brahams, and Beethoven.
An Obstruction to Embedding a Simplicial n-Complex into a 2n-Manifold, Topology and Its Applications ,Vol. 122, Issue 3, August 16, 2002, pp. 581-591.
- Abstract: Let K be a finite, connected, simplicial n-complex (dimension n >=3) and M a 1-connected, smooth, orientable 2n-manifold without boundary. If f: |K| --> M is an arbitrary map, we define a first obstruction g(f) in H²ⁿ(J*K; Z), where J*K is the reduced deleted product of K and show that the vanishing of this obstruction is necessary and sufficient for f to be homotopic to an embedding.
Functions of Number Theory in Music, Mathematics Teacher, Vol. 94, No. 8, November 2001, p. 700-707.
- Abstract: This paper shows connections of fundamental concepts of modular algebra to music. It also explores free and strict inversions and develops a number-theoretic function that determines the change in key of a musical composition after it has been strictly inverted across a pivot note. Namely, if x is the original key (0<=x<=11), then after inversion across pivot note p, the new key is given by I_p(x) = [ 2p - x - 4 ] mod 12 where [ ] mod 12 indicates a pitch class.
Mathematics Without Words (Sum of a Geometric Series), The College Mathematics Journal, Vol. 32, No. 2, March 2001, p. 109.
Computing Distances to the Stars, PCTM Magazine (Pennsylvania Council of Teachers of Mathematics), Winter 2000, Vol. XXXVIII, No. 2, pp. 19-21. (Typesetting errors occurred in print. The corrected version appears on the PCTM website.)
The Derivative of the Inverse Sine, The College Mathematics Journal, Vol. 29, No. 4, September 1998, p. 313.
A Computer Search for Free Actions on Surfaces, The American Mathematical Monthly, Vol. 105, Number 2, February, 1998, pp. 144-153.
- Abstract: Define surfaces U^k = RP² # . . . # RP² and T^k = T # T # . . . # T, where T = S¹ x S¹. For a group G with |G| = 2n, let b: p₁U^r+2 --> G be an epimorphism. Define N_b to be the image under b of the even words in p₁U^r+2. We show that [G : N_b ] = 2 if and only if ker b is isomorphic to p₁T^nr+1. Now, ker b = p₁T^nr+1 iff there exists an exact sequence 1 --> p₁T^nr+1 --> p₁U^r+2 --> G --> 1 and this, in turn, is associated with an orientation-reversing free action by G on T^nr+1. Given a finite group G, a computer program checked all possible epimorphisms b from p₁U^r+2onto G and found those for which the associated N_b had index 2 in G. This resulted in the determination of upper bounds for the numbers of o.r. free actions on T^k by small finite groups and values for n and r for when the cyclic group Z_2n acts freely o.r. on T^nr+1 .
Look, Experiment, Think: LET Students Learn Mathematics. Using Technology to Motivate Connections in Calculus, PCTM Magazine , Fall 1996, Vol. XXXV, No. 1, pp. 13-15.
Using Astronomy to Motivate Pre-Calculus Math Concepts, CMC3 Newsletter (CA Math Council of Comm. Colleges), Fall, 1989.

Courses I Teach

MATH 120 Mathematics for the Liberal Arts	MATH 201 Calculus I	MATH 202 Calculus II	MATH 203 Calculus III	MATH 204 Calculus IV
MATH 311 Differential Equations Solutions to 1st/2nd order eqns and systems, direction fields, numerical methods, phase planes, Laplace transform	MATH 321 Abstract Algebra Groups, subgroups, rings, fields, ideals, homomorphisms, isomorphisms	MATH 322 Linear Algebra Systems, matrices, determinants, finite-dimensional vector spaces, linear transformations, eigenvalues	MATH 323 Number Theory Congruences, quad. reciprocity, cryptology, number-theoretic functions	MATH 324 College Geometry Axiomatic systems, Euclidean, hyperbolic, elliptic Geometries
MATH 425 Topology Metric spaces, continuous functions, connectedness, compactness, homotopy, fundamental group	MATH 430 Real Analysis Functions, epsilon-neighborhoods, open and closed sets, completeness, sequences, limits, continuity, Riemann integral	PHYS 112 Astronomy Sky cycles, planetary motion, eclipses, electromagnetic spectrum, telescopes, matter + energy, prop. of stars, black holes, galaxies, cosmology	PHYS 112L Astronomy Lab Solar rotation rate, phases of moon, mass of Jupiter, stellar spectra, Hubble redshift, telescopes, transit of Venus, photometry, structure of universe	MATH 170 Mathematics for Biology Functions, graphing, limits, derivatives, applications, integration

Presentations

Musical Transformations and the Dihedral Group
- MAA Session on Mathematics and the Arts, MAA/AMS Joint Meetings in New Orleans - Jan. 9, 2011
Introduction to Group Theory using Music
- NPCMC Dinner and Colloquium, Marywood University - April, 2009
- MAA Session on Mathematics and the Arts, MAA/AMS Joint Meetings in San Diego, CA- Jan. 8, 2008
- Colloquium Talk at University of Scranton, Scranton, PA - Oct. 25, 2007
- Math Explorers Club, Ithaca High School, Ithaca, NY - May, 2006
- Undergraduate Math Club, Cornell University - Oct. 24, 2005
A Computer Search for Free Actions on Orientable Surfaces
- Geometry/Topology Seminar, Cornell University - Sept. 27, 2005
- Invited mathematics colloquium talk at Lehigh University, Bethlehem, PA - March 11, 1998
Maps from Simplicial Complexes to Manifolds which are Homotopic to Embeddings
- Geometry/Topology Seminar, Cornell University - April 4, 2006
Functions of Number Theory in Music
- NPCTM Fall Meeting at Marywood University, Scranton, PA - October 26, 2000
- MAA/AMS Joint Meetings at Washington, DC - January 20, 2000
- Invited lecture at University of Scranton, Scranton, PA - April 21, 1999
- PCTM conference at Split Rock Resort, Lake Harmony, PA - March 19, 1999
An Obstruction to Embedding a Simplicial Complex
- Geometry and Topology Conference at Lehigh University, Bethlehem, PA - June 6, 1998
- Colloquium talk at University of Scranton, Scranton, PA - November 5, 1997
- MAA/AMS Joint Meetings at San Francisco, CA - January 17, 1991.
Music, Mathematics, and Cryptology
- MAA/AMS Joint Meetings at San Diego, CA - January 11, 1997
Mathematical Modeling in the Liberal Arts Class
- MAA/AMS Joint Meetings at Orlando, FL - January 12, 1996
- PCTM annual conference at White Haven, PA - March 1996
Studying Astronomy in Liberal Arts Mathematics
- PCTM Annual Conference at Pittsburgh, PA - March 17, 1995
- Colloquium talk at University of Scranton, Scranton, PA - March 31, 1995
Orientation-reversing Free Actions on Orientable Surfaces by Finite Groups
- Colloquium talk at University of Scranton, Scranton, PA - April 21, 1995
- Colloquium talk at California State University, Fresno, CA - March 25, 1991
- MAA/AMS Joint Meetings at Phoenix - January 11, 1989
Using Astronomy as a Vehicle in Liberal Arts Mathematics
- CMC3 Annual Conference at Monterey, CA - December 1993
- AMATYC Annual Conference at Boston, MA - November 1993
Eyes on the Sky
- Public astronomy talk, College of the Sequoias, Visalia, CA - September 1991
Neptune Night
- Public astronomy talk, College of the Sequoias, Visalia, CA - August 1989

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Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@marywood.edu

Last update: Jan. 12, 2011