Dr. Mark Lynch Mathematics

B.S. Millsaps College (1976)

Ph.D. LSU (1985)

Note: I didn't get an M.S. degree (they wanted thirty bucks for it!).

I worked for the National Security Agency from August '85 to August '87.  However, a description of my duties is classified.

Topology (hyperspaces)

          In the text Hyperspaces:  Fundamentals and Recent Advances, the authors devote
          an entire section of the text to one of my theorems and call it "powerful".

          (20)  "A counterexample to Liouville's Theorem", College Mathematics Journal, Vol. 39, No. 4, September 2008, 300.

          (19)"Continuity and separation in symmetric topological spaces", with John M. Harris, iJMEST,  2007, Vol. 38, No. 1, 127-131.  We  generalize the result in paper (17) and give a new way to define topological spaces.

          (18)"A paradoxical paint pail", College Mathematics Journal, Vol. 36, No. 5, November 2005, 402-404.  Gabriel's Horn is unbounded and can not be filled with paint (in finite time) as the Calculus texts claim.  We give an example of a bounded paint pail which can be filled in finite time but which can not be painted since it has infinite surface area.

         (17) "A new characterization of continuous functions", iJMEST, 2005, Vol. 36, No. 5, 549-551.  We characterize continuity in terms of separated sets.  It makes rigorous the idea that 'tearing' is discontinuous.

         (16)"Compact, convex, and symmetric sets are disks", iJMEST, 2004, Vol. 35, No. 2, 268-269.  We classify the shapes of disks.

         (15)"A class of metrics with strangely shaped disks", iJMEST, 2003, Vol. 34, No. 2, 287-291.  In this paper, we define infinitely many            metrics whose disks have strange shapes (like octagons or part of the disk has straight edges and other parts are curved).

          (14)"Topological X-rays and MRIs", iJMEST, 2002, Vol. 33, No. 3, 389-392.  In this paper, we abstract the process of X-rays to compact sets in the plane and study which sets are determined by their topological MRIs.          

(13)"A topological characterization of the implication 'differentiability implies continuity'", iJMEST, 2001, Vol. 32, No. 4, 509-511.  In this paper, we give an example of a differentiable, nowhere continuous function by changing the metric on the reals (it's only different at 0).  And, we characterize the topologies where such functions can exist.

(12)"Limits, continuity, and a drunkard's bow tie: Rigor without epsilon or delta", iJMEST, 2000, Vol. 31, No. 5, 779-787. In this paper, we give a slightly weaker, but equally rigorous, definition of limit which does not involve epsilon or delta.  And we show that the geometry of differentiation is a drunkard's bow tie (much like the geometry of continuity is small rectangles.).

(11)"A path having all of its subpaths of infinite length", Int. J. Math. Educ. Sci. Technol., 1998, Vol. 29, No. 5, 782-784. As the title suggests, this path is infinitely long on every interval where it is defined.

(10)"A space filling curve", Int. J. Math. Educ. Sci. Tech., 1995, Vol. 26, No. 5, 767-769.  The construction is new and does not use uniform convergence.

(9)"A consequence of the nearness of rationals to reals", College Math. J., 1995, Vol. 26, No. 3, 221. we give a 'new' proof that one equals zero. Several masters level mathematics students could not find the error.  This paper has appeared in the text Fallacies, Flaws, and Flimflam.

(8)"Advanced calculus reform: Continuity and integration", Int. J. Math. Educ. Sci. Tech., 1994, Vol. 25, No. 4, 563-571. We give new definitions of continuous functions and the Riemann integral which shows more clearly how similar the concepts are.

(7)"A mapping theorem for compact metric spaces", Int. J. Math. Educ. Sci. Tech., 1993, Vol. 24, No. 3, 413-416. See the next paper.

(6)"A continuous, nowhere differentiable function", Amer. Math. Monthly, 1992, Vol. 99, No. 1, 8-9. This one gives the first example of a continuous, nowhere differentiable function which does not involve uniform convergence. The mapping theorem paper gives a general method for constructing continuous functions. These papers prompted a journal editor to suggest I write a text in Intermediate Analysis and Topology which "features these clever ideas".  He also offered to recommend me to a publisher.

(5)"Whitney levels and certain order arc spaces", Topology and its Appl., 38 (1991), 189-200.

(4)"Spaces of order arcs in hyperspaces of Peano continua", with D. W. Curtis, Houston J. Math., 15 (1989), 517-526.  This is my only joint publication.

(3)"Whitney properties for 1-dimensional continua", Bull. Pol. Acad. Sci., 35 (1987), 473-478.

(2)"Whitney levels in Cp(X) are AR's", Proc. Amer. Math. Soc., 97 (1986), 748-750.
This is the theorem that bears my name in the text Hyperspaces:  Fundamentals and Recent Advances.

(1)Remark: I also directed a student's (Melissa Richey) senior thesis which has been published. "Mapping the Cantor set onto [0,1]", Math. Mag., Vol. 70, No. 1 (1997), 57-58.

I've also been invited to speak at one international conference, one national conference, and one regional conference.

Syllabi

Personal Interests