John Schmitt (homepage) is an Associate Professor of Mathematics and has been at Middlebury College since 2005. He received his B.A. from Providence College, his M.S. from the University of Vermont, and a Ph.D. from Emory University.
His research interests are in combinatorics (the art of counting) and graph theory. As such, he has co-authored over ten research articles, most dealing with extremal graph theory, and has a similar number of co-authors. His Erdös number is two. His research has been supported by the following external sources: National Science Foundation, National Security Agency and VT-EPSCoR. In addition to having given invited lectures at many universities, most recently he was a Visiting Fellow at the Institute for Pure and Applied Mathematics at UCLA.
Professor Schmitt enjoys teaching students how to count and play (combinatorial) games – things they thought they already knew. Students are frequently involved with his research. His efforts in teaching were recognized by students in 2008 with the Perkins Award.
Away from academic life, he enjoys spending time with his family in the great outdoors of Vermont.
Courses offered in the past four years.
▲ indicates offered in the current term
▹ indicates offered in the upcoming term[s]
FYSE1314 - The Mathematical Gardner ▲
The Mathematical Gardner
In this course we will have an “orgy of right-brain tomfoolery” as inspired by the writings of Martin Gardner. For several decades Gardner's contributions to Scientific American, in the form of his column “Mathematical Games,” bridged the divide between professional mathematicians and the general public. He shared with us like no other, introducing or popularizing topics such as paper-folding, Hex, polyominoes, four-dimensional ticktacktoe, Conway’s Game of Life, the Soma cube --- the list goes on seemingly forever. We will examine these mathematical curiosities for pure pleasure. 3 hrs. sem. CW DED
MATH0122 - Calculus II ▲
A continuation of MATH 0121, may be elected by first-year students who have had an introduction to analytic geometry and calculus in secondary school. Topics include a brief review of natural logarithm and exponential functions, calculus of the elementary transcendental functions, techniques of integration, improper integrals, applications of integrals including problems of finding volumes, infinite series and Taylor's theorem, polar coordinates, ordinary differential equations. (MATH 0121 or by waiver) 4 hrs. lect./disc. DED
Spring 2013, Fall 2013, Fall 2016
MATH0200 - Linear Algebra ▹
Matrices and systems of linear equations, the Euclidean space of three dimensions and other real vector spaces, independence and dimensions, scalar products and orthogonality, linear transformations and matrix representations, eigenvalues and similarity, determinants, the inverse of a matrix and Cramer's rule. (MATH 0121 or by waiver) 3 hrs. lect./disc. DED
Spring 2013, Spring 2014, Fall 2014, Spring 2017
MATH0223 - Multivariable Calculus
The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications. (MATH 0122 and MATH 0200 or by waiver) 3 hrs. lect./disc. DED
Fall 2012, Fall 2013, Fall 2014
MATH0247 - Graph Theory
A graph (or network) is a useful mathematical model when studying a set of discrete objects and the relationships among them. We often represent an object with a vertex (node) and a relation between a pair with an edge (line). With the graph in hand, we then ask questions, such as: Is it connected? Can one traverse each edge precisely once and return to a starting vertex? For a fixed k/, is it possible to “color” the vertices using /k colors so that no two vertices that share an edge receive the same color? More formally, we study the following topics: trees, distance, degree sequences, matchings, connectivity, coloring, and planarity. Proof writing is emphasized. (MATH 0122 or by waiver) 3 hrs. lect./disc. DED
MATH0345 - Combinatorics
Combinatorics is the “art of counting.” Given a finite set of objects and a set of rules placed upon these objects, we will ask two questions. Does there exist an arrangement of the objects satisfying the rules? If so, how many are there? These are the questions of existence and enumeration. As such, we will study the following combinatorial objects and counting techniques: permutations, combinations, the generalized pigeonhole principle, binomial coefficients, the principle of inclusion-exclusion, recurrence relations, and some basic combinatorial designs. (MATH 0200 or by waiver) 3 hrs. lect./disc. DED
Fall 2012, Spring 2015
MATH0500 - Advanced Study ▲ ▹
Individual study for qualified students in more advanced topics in algebra, number theory, real or complex analysis, topology. Particularly suited for those who enter with advanced standing. (Approval required) 3 hrs. lect./disc.
Fall 2012, Winter 2013, Spring 2013, Fall 2013, Winter 2014, Spring 2014, Fall 2014, Winter 2015, Spring 2015, Winter 2016, Fall 2016, Spring 2017
MATH0704 - Senior Seminar
Each student will explore in depth a topic in pure or applied mathematics, under one-on-one supervision by a faculty advisor. The course culminates with a major written paper and presentation. This experience emphasizes independent study, library research, expository writing, and oral presentation. The goal is to demonstrate the ability to internalize and organize a substantial piece of mathematics. Class meetings include attendance at a series of lectures designed to introduce and integrate ideas of mathematics not covered in the previous three years. Registration is by permission: Each student must have identified a topic, an advisor, and at least one principal reference source. 3 hrs. lect./disc.
MATH0745 - Polynomial Method Seminar ▹
The Polynomial Method
A tutorial in the Polynomial Method for students who have completed work in Abstract Algebra and at least one of Combinatorics, Graph Theory, and Number Theory. We will study Noga Alon’s Combinatorial Nullstellensatz and related theorems, along with their applications to combinatorics, graph theory, number theory, and incidence geometry. Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. Fulfills the capstone senior work requirement for the mathematics major. (Approval required; MATH 0302 and one of the following: MATH 0241, MATH 0247, or MATH 0345).
MATH1007 - Combinatorial Gardner
The Combinatorial Gardner
It has been said that the Mathematical Games column written by Martin Gardner for Scientific American turned a generation of children into mathematicians and mathematicians into children. In this course we will read selections from three decades of this column, focusing on those that deal with combinatorics, the “science of counting,” and strive to solve the problems and puzzles given. An example problem that illustrates the science of counting is: what is the maximum number of pieces of pancake (or donut or cheesecake) one can obtain via n linear (or planar) cuts? (MATH 0116 or higher; Not open to students who have taken FYSE 1314). DED WTR
M. Ferrara and J. Schmitt, A General Lower Bound for Potentially H-Graphic Degree Sequences, SIAM J. Discrete Math., 23 (2009) 1, 517-526.
M. Ferrara, M. Jacobson, J. Schmitt and M. Siggers, Potentially H-Bigraphic Sequences, to appear in Discuss. Math. Graph Theory
M. Ferrara, R Gould and J. Schmitt, Using Edge Exchanges to Prove the Erdos-Jacobson-Lehel Conjecture, Bull. Inst. Combin. Appl., 57 (2009), 73-80.
O. Pikhurko and J. Schmitt, A Note on Minimum K2;3-saturated Graphs, Austral. J. of Combin., 40 (2008), 211-215.
Guantao Chen, M. Ferrrara, R. Gould and J. Schmitt, Graphic Sequences with a Realization Containing a Complete Multipartite Subgraph, Discrete Math., 308 (2008) 23, 5712-5721.
J. Yin, Gang Chen and J. Schmitt, Graphic Sequences with a Realization Containing a Generalized Friendship Graph, Discrete Math., 308 (2008) 24, 6226-6232.
R. Gould and J. Schmitt, Minimum Degree and the Minimum Size of K^t_2-saturated graphs, Discrete Math., 307 (2007) 9-10, 1108-1114.
M. Ferrara, R. Gould and J. Schmitt, Graphic Sequences with a Realization Containing a Friendship Graph, Ars Combin., 85 (2007), 161-171.
R. Gould, T. Luczak and J. Schmitt, A Constructive Upper Bound for Cycle Saturated Graphs of Minimum Size, Electron. J. Combin., 13 (2006) R29, 19pp.
Conference organizer of Discrete Mathematics Days of the Northeast in September 2007. This conference, sponsored by the National Security Agency and Middlebury College, brought together world-class mathematicians, researchers, and graduate and undergraduate students.
Selected Invited and Sponsored Talks
Dartmouth College Combinatorics Seminar, October 2008, "A dual to the Turán problem".
University of Vermont Mathematics Colloquium, April 2007, "Extremal Problems on Bipartite Graphs".
Horizons of Combinatorics, Lake Balaton, Hungary, July 2006, "On a Relationship of Two Extremal Functions". (Support from the European Mathematical Society.)
6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Prague, Czech Republic, July 2006, "An Erdös-Stone Type Conjecture". (Partial support from DIMATIA of Charles University.)