# John Schmitt

## Professor of Mathematics

jschmitt@middlebury.edu

work802.443.5952

(Spring 2022): on sabbatical in U. Graz, Austria

75 Shannon 102J

John Schmitt (homepage) is 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: Fulbright U.S. Scholar Program, 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; he is currently visiting at Karl Franzens U. (Graz, Austria).

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.

John has a passion for growing his own organic produce. His favorite new sport is wake-foiling https://www.youtube.com/watch?v=uen1I045Bn8.

## Courses

Courses offered in the past four years.

▲ *indicates offered in the current term*

▹ *indicates offered in the upcoming term[s]*

##### MATH 0122 - 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. **DED**

Fall 2022

##### MATH 0200 - 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 2020, Fall 2020, Fall 2021

##### MATH 0223 - 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**

Spring 2018, Fall 2018, Fall 2019, Spring 2021

##### MATH 0247 - 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**

Fall 2019, Spring 2021

##### MATH 0345 - 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 2018, Spring 2020, Fall 2021

##### MATH 0500 - 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 2018, Fall 2018, Winter 2019, Spring 2019, Fall 2019, Winter 2020, Spring 2020, Fall 2020, Winter 2021, Spring 2021, Fall 2021, Winter 2022, Spring 2022, Fall 2022, Winter 2023, Spring 2023

##### MATH 0745 - 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).

Fall 2018, Fall 2020, Fall 2022

#### Publications

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