2022 William Lowell Putnam Math Competition

Nineteen UML students spent six hours on December 3, 2022 competing in the 83rd annual William Lowell Putnam Mathematics Competition. Every year, thousands of students in the United States and Canada enter the competition. In each of two 3-hour sessions, they work on six difficult problems without the aide of any technology. One of the problems this year was

Let n be an integer greater than or equal to 2. Over all real polynomials p(x) of degree n, what is the largest possible number of negative coefficients of p(x)2?

Students working in the morning session.

Student solutions are graded by a committee of the Mathematical Association of America and results will be announce in a few months. They will appear on the Putnam Archive: https://kskedlaya.org/putnam-archive/. Professor Kenneth Levasseur supervised the competition at UML.

Abiti (Adil) Adili

We are happy to welcome Professor Abiti Adili to the UML Mathematical Sciences faculty. He has been hired at the rank of Assistant Teaching Professor.

Prof. Abiti Adili

Adil’s scholarship has concentrated on identification of new wave phenomena due to subwavelength resonances and multiple scattering as well as developing numerical methods for solving related eigenvalue problems. His current research focuses on accurate multi-scale analysis of heterogeneous media with non well separated length scales. Before coming to UML, Adil worked at Franklin & Marshall College. He received his Ph.D  from Louisiana State University.

2021 William Lowell Putnam Math Competition

Students working in the morning session of the 82nd annual William Lowell Putnam Math Competition

Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?

This is just one of twelve problems posed to 24 UML undergraduates as part of the 82nd annual William Lowell Putnam Math Competition sponsored by the Mathematical Association of America on December 4. Thousands of students throughout the United States and Canada simultaneously worked on these problems with no technology other than a pencil and paper. There were two 3-hour sessions with six problems in each session.

The Putnam returned to campuses after an unofficial virtual competition in early 2021. Professor Kenneth Levasseur supervised the competition at UML.

By the way, the solution to the problem above is

    \[ \frac{6}{\pi} \]

You have to show your work! All problems and solutions are available at the Putnam Archive.

New Faculty: Joris Roos

We are happy to welcome Dr. Joris Roos to the faculty of Mathematical Sciences.  Joris’ research area is harmonic analysis on Euclidean spaces, where he works on a variety of problems involving maximal functions, oscillatory integrals, singular integrals, or all of them combined.

Photo of Dr. Joris Roos
Dr. Joris Roos

Joris is also interested in discrete analogues of problems in real-variable Fourier analysis and applications of harmonic-analytic techniques to related areas such as arithmetic combinatorics, number theory, dispersive PDE and ergodic theory. Recently, he also became interested in assisted and automated theorem proving.

Prior to joining UML, Dr. Roos was a Van Vleck Visiting Assistant Professor at the University of Wisconsin-Madison. He has a doctoral degree from the University of Bonn.

Dr. Roos’s Website.

Zooming into a Flipped Class

This the third in a series of posts related to teaching mathematics during the Covid-19 pandemic.


There have been many different approaches to Emergency Remote Teaching. Here is one that, in my opinion, has reasonably approximated face-to-face meetings. I make no claims that this is superior to any other approach — it just seems to work with a the majority of my students. I welcome comments on variations you might be trying.

First let me describe how the class had been run before spring break. And then describe how it has changed since then. I am describing a section of Discrete Structures I, with 32 active students at the time of spring break.

Face to face format

Here is how class meeting n was conducted. About a week before the class was to meet, usually the date of class n-2, I gave the students a reading assignment. This normally consisted of reading one or two sections from our text. I used Piazza to communicate these assignments to the students. Piazza was a fairly small part of the course. I used it to assign the reading assignments and to answer questions students may have had on them. After doing the reading, students were asked to answer a question about the material, and usually one or two short exercises. The assignment was due on the day of class meeting n -1. Most student turned in there work on a single sheet of paper. I graded the assignments as soon as possible after that class, and returned the work at the start of class n. More on grading later.

In class n, I made comments on the reading assignment and answered questions. This tended to take 10 to 15 minutes. Then the bulk of the class is spent working on more challenging problems on class n topics. We meet for one hour 15 minutes, so for about 45 minutes students work in groups of 4-6 on these problems. I circulated around the room answering questions and giving hints. When a group had a correct solution to any problem they would share it. We were in a TEAL classroom, so students would usually use the virtual whiteboard or display their computer screen with solution. Occasionally I would interrupt the class to show them a solution but, normally we would wait for the last 15 minutes to come together and discuss solutions.

This format seemed to work pretty well. We had one hour exam during the face to face portion of the course. The questions were comparable to prior semesters’ exams, and the grades were considerably higher this semester. Most students also seem to enjoy the format.

The sudden change to a virtual classroom

We had about a week to decide how to the respond to the crisis, and here is how things changed. There has been no need to change the rhythm of assignments. Most students routinely turn in their assignments on time. I’ve made the adjustment of requesting their work by noon on the day of the class as opposed to at the class meeting and submissions are all done by email. Why email? More on that later. Naturally , the big change has been in the actual class meetings. We meet on Zoom for our 75 minute meetings. Again, in the first 10 to 15 minutes I make comments and respond to student questions. In face to-face meetings, attendance was good and students sat at “pods” that had a capacity of six students. In our virtual meetings attendance has been lower. Typically, there have been 20-24 students who attend. To maintain continuity, I distribute students into three Breakout rooms according to the pods they sat in during the face-to-face meetings to maintain some consistency. Should I teach online in this format again I would just distribute them into smaller groups, using Zoom’s randomization. Over the course of 45 minutes, I drop into each breakout room 2-3 times to answer questions and check students’ progress.

Piazza now plays an important role in the class. Once a pod has settled on a solution, one of the students posts it on Piazza as a response to a question posed to the pod. Other students can refine the solution, if needed. During the breakout sessions, I monitor the solutions in real time. When a correct one appears, I endorse it, pin it (posts can be pinned to single them out ), and make the post available to the whole class. By the end of the breakout session, some problems will typically be completed. In the last 10 to 15 minutes of class time students come back from groups and we discuss some of the problems. Students are encouraged to settle any remaining problems later. They are also encouraged to check out solutions that have been pinned.


Piazza.com offers course sites for free. The courses are categorized according to university so that faculty and students who are in more than one course that uses Piazza can access all courses with one login. In Piazza, students and instructors communicate through postings that are categorized as announcements, questions, or polls. Every posting is characterized according a system of folders. For a class that will meet on a given date, the reading assignment prior to that class and the day’s questions are all classified in a folder with that date. This makes filtering of postings during any given class easy to do since the mass of postings can be a bit confusing. Questions posed by either the instructor or a student can be responded to by any student in the class if it is globally available, or for any student in a group if it is a group posting. In setting up Piazza for the virtual classroom I created groups according to pairs of pods to match the pod structure within the class. I’ve only used polls a few times to get basic information like connection availability and students’ views on pass/fail, which had been debated at the time.


Grading of reading assignments would appear to be a daunting task since each student submits work for each class. However, I am following suggestions from [1] and treat the assignments as formative assessments. Essentially if a student puts in a reasonable effort even though their answers are generally incorrect they get full credit. Many of the incorrect answers reveal misconceptions that can be addressed quickly at the beginning of class. Occasionally I will create a short video using Camtasia to deal with misconceptions. Students get a grade of “Check,” which is full credit in most cases. Roughly 10% of submissions have been deemed to be subpar and get a grade of “check minus,“ which is half credit. I don’t tell students, but as long as they pass in a piece of paper or, when online, a photo of their work, they get half credit for almost anything. The stakes are quite low and most papers take a very short time to assess. When the class was face-to-face I would do this immediately after class and it would take less than an hour.

Online grading and email

In virtual mode, I have found that the most efficient way to receive and assess students’ work it is through email. Although it creates a lot of email messages, I have developed a routine that makes it far more efficient to do this as opposed to using something like a Dropbox in Blackboard. Here are the steps I take using my iPad.

  • Opening a message from a student, I press on the photo or PDF and this brings up the option “Markup and Reply“.
  • With a combination of text and virtual marker, I make quick comments including the check or check minus grade.
  • I then select “done“ which gives me the option to reply to the email message. Usually, I send with no additional comments in the body of the message. Just the marked up photo/PDF.
  • I then archive the student’s message, and immediately record the grade in a spreadsheet.

The whole process can take as little as 30 seconds. I rarely need more than two minutes for any submission. I do have to resist longer comments and reserve them for class discussions and/or quick videos which will cover all of the comments that I would normally make individually. Students get their work back faster in the virtual format than face-to-face where I would normally pass the assignment back at the beginning of class n.


Is this virtual format as effective as the face-to-face version? Probably not, but I think it’s a reasonable approximation. I certainly prefer meeting face-to face. Roughly one quarter of the students who were active and passing the course before we moved online are marginal participants in the class activities. I’ve been in contact with several who say that they have had difficulty with the situation for various reasons. Around half of the no-shows in Zoom sessions are still keeping up with submissions of reading assignments, The only value added dimension of the online format is better archiving of student work during problem sessions using Piazza. In theory, the technology embedded in the TEAL classrooms could be used to capture some of this, but that wasn’t happening.


David Pengelley, “From Lecture to Active Learning: Rewards for All, and Is It Really So Difficult?,” The College Math Journal, January 2020, 51 no. 1, 13–24, doi.org/10.1080/07468342.2020.1680228

A Simple Model for Epidemics

This is the second in a series of posts related to teaching math during the Covid-19 pandemic.

Around 15 years ago, when I was teaching Calculus II, I used an ice-breaker on the first day of class that simulated the spread of an epidemic. The activity certainly is not something that we could do now, but the results are instructive. I’ve added a another simulation that illustrates what we’ve all been hearing about the effects of social distancing.

A Mathematica Notebook can be downloaded if you would like to experiment with this model – I’m sure the programming could be streamlined.

I’d be interested to hear if anyone can come up with a way to to adapt this ice-breaker to online teaching.

The Classroom Simulation

Introduction: We’re going to model an epidemic. Here is how it will work. At the start of the process, one of you will be “sick” and we will go through a series of “days” Each day everyone will do the following: you will make contact with exactly one person by tapping him/her on the shoulder, telling whether you are sick or not, and then you can introduce one another. If you are sick and the person you tapped on the shoulder was well, then he/she becomes sick. If you are well and you tapped a sick person on the shoulder, you don’t become sick. You may be tapped on the shoulder more than one time or not at all in one day, but you should tap exactly one person on the shoulder. As the days pass, you may tap the same person on the shoulder more than once — ideally you should choose people randomly.

Before we get started, take out a sheet of paper. Draw what you think might be the graph of the number of people who get sick verses the days. This will actually be a discrete graph, like a bar graph, but you can draw a continuous curve to indicate the shape of the graph

You will get a “Post-it.” If it has the number 1 on it, you are the lucky person who is sick and starts the epidemic. When you get sick, make a note of the day on which you get sick on the Post-it

On the first day, the “patient zero” gets sick and we assume that day 0 has passed; so we start with day 1.

After 10 days, see if anyone is well. Continue until almost everyone is sick.

Stack your Post-it notes in the column marked by the day you got sick.

Computer Simulation

A computer simulation using Mathematica produces a similar result to the classroom activity – the code can be seen in the Mathematica Notebook. Here we assume 500 individual and find the the number of days before everyone is infected doesn’t change all that much with a larger “class.” Here are graphs of the cumulative infections and the daily new cases.

Spread of an epidemic among 500 individuals with no “social distancing.”

A second version of the model, which I added in 2020 allows for selecting a level of social distancing. Here we assume that a random 70% of individuals are quarantined on any day. Here are the results.

Spread of an epidemic among 500 individuals with 70% “social distancing.”

The two sets of plots have the same aspect ratio so you have to pay attention to the scales. Alternatively, here are the daily exposures for the two models plotted together.

Daily new infections with and without social distancing.

Here, you can see the flattening of the curve we’ve all heard about.

Online Math Teaching – Textbooks

This is the first in a series of posts that will discuss issues relating to online teaching during the covid-19 pandemic.

Covid-19 virus – a mathematical object?

There are countless issues that confront your students and one of them is access to textbooks. Although current courses have already adopted texts, student may not have their texts at home. You might want to give your students an alternative reference as you start working online. For mathematics, there are two collections of texts that contain virtually every topic we currently teach. In these sites, the books that are listed all offer free pdf’s and in most cases, online versions. The pdf versions are useful for students who don’t have high-speed internet access. The companion online versions use a bit more bandwidth, but have the advantage of having a variety of interactive features.

  • The American Institute of Mathematics has an Open Textbook Initiative with curated list of open source textbooks.
  • The PreTeXt project has a catalog of texts that have been developed with PreTeXt, an XML application that facilitates publishing a single source in multiple formats, including pdf, html, and Braille.

Even if you intend to keep using your current text, you might want make your students aware of these alternative resources.

The online versions frequently include editable code such as this page from a differential equations by Thomas Judson, which is listed on the PreTeXt site.

New Faculty: Elisa Perrone

We are happy to welcome Dr. Elisa Perrone to the faculty of Mathematical Sciences. Her research spans from mathematical statistics to applied statistics. She is mainly interested in multivariate statistics and dependence modeling, with particular emphasis on copulas and their geometric properties. This work includes the development of copula-based approaches used in optimal experimental design and environmental sciences such as hydrology and weather forecasting.

Dr. Elisa Perrone

Prior to joining UMass Lowell, Dr. Perrone was a postdoc at the Massachusetts Institute of Technology and the principal investigator of the project ‘Geometry of discrete copulas for weather forecasting’, funded by the Austrian Science Fund (FWF). Elisa has a doctoral degree from the Johannes Kepler University Linz (Austria).

Dr. Perrone’s faculty website.

New Faculty: Amanda Redlich

We are happy to welcome Dr. Amanda Redlich to the faculty of Mathematical Sciences.   Her research is in probabilistic combinatorics and randomized algorithms.  In her combinatorial work she looks for patterns in big random structures, like social networks.  In her algorithmic work she looks for efficient ways to solve big problems, like analyzing genetic code.  Her most recent work has been on randomized allocation algorithms and biological random graphs.

Dr. Amanda Redlich

Before coming to UML, Amanda worked at Bowdoin College and Rutgers University.  She has a bachelor’s degree from University of Chicago and a PhD from MIT.

Dr. Redlich’s faculty web page