# John Schmitt

## Associate Professor of Mathematics

jschmitt@middlebury.edu

work802.443.5952

Monday 10:00-11:30am, Wednesday 10-11:30am, Thursday 1:30-3pm and by arrangement

Warner Hall 311

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 (the study of networks), with a particular fondness for implementing the polynomial method. He has co-authored numerous research articles. His Erdös number is two. His research has been supported by the following external sources: National Science Foundation, National Security Agency and VT-EPSCoR. He has held visiting positions at the Institute for Pure and Applied Mathematics at UCLA and at Carnegie Mellon University.

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 2016 with the Perkins Award.

Away from academic life, he enjoys skiing with his family in the great outdoors of Vermont. Also, John has a passion for growing his own organic produce.

## Courses

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**

Fall 2016

##### MATH0122 - Calculus II

**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 ▲

**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

**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 2013, Fall 2014

##### MATH0247 - Graph Theory

**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**

Spring 2014

##### MATH0345 - Combinatorics

**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**

Spring 2015

##### MATH0500 - Advanced Study ▲

**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.

Spring 2013, Fall 2013, Winter 2014, Spring 2014, Fall 2014, Winter 2015, Spring 2015, Winter 2016, Fall 2016, Winter 2017, Spring 2017

##### MATH0704 - Senior Seminar

**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.

Spring 2015

##### 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).

Spring 2017

#### Publications

Copies of Professor Schmitt's publications may be found here.