# Mathematics

## Two planes are heading for the same destination 8000 miles away.

Plane A leaves at 1:00 pm, averaging 450 mph. Plane B leaves at 6:00 pm, averaging 400 mph. Which plane gets there first?

You’ve heard a problem like this before, but never outside a classroom. Because in the real world, it’s much less black and white.

That’s because life has invisible variables. For instance, it will depend on the accuracy of weather pattern predictions to account for delays. It will depend on assessments about the plane’s material parts and fuel efficiency. It will depend on the air traffic controllers issuing course corrections in real time. All these factors will contribute to when the planes will arrive.

**So what’s the answer?** The truth is that the answer is too small for the real question. The world depends on more than just calculations — it requires people who have the foresight and perspective necessary to keep the planes arriving on time.

The College of Science mathematics degree prepares students not only to solve for (x), but to see the whole equation.

# Applied Mathematics Masters

New applications of mathematics are constantly being discovered, and established techniques are being applied in new ways and in emerging fields. Northeastern’s Master of Science in Applied Mathematics is catered to students who are looking to enter or who are currently working in a variety of applied math careers, such as financial service and investment firms; data science and high-tech firms; computer information and software firms; and academic institutions and research institutes.

Develops problem-solving skills while simultaneously teaching mathematics concepts. Each unit centers on a particular applied problem, which serves to introduce the relevant mathematical topics.

Presents mathematical connections and foundations for art. Topics vary and may include aspects of linear perspective and vanishing points, symmetry and patterns, tilings and polygons, Platonic solids and polyhedra, golden ratio, non-Euclidean geometry, hyperbolic geometry, fractals, and other topics.

Traces the development of mathematics from its earliest beginning to the present. Emphasis is on the contributions of various cultures including the Babylonians, Egyptians, Mayans, Greeks, Indians, and Arabs.

## Co-Op Stories

Many math students choose to participate in the university’s signature co-operative education program because it offers excellent preparation and exposure to exciting careers. Here’s what our students are saying:

## News

### Recipients of the 2020 College of Science Excellence in Teaching Awards

The College of Science Excellence in Teaching Awards recognize and reward outstanding NU College of Science faculty for their significant contributions to student learning. We are pleased to announce this year’s winners.

__Oyindasola (Oyinda) Oyelaran__

Dr. Oyelaran Is a Teaching Professor in the Department of Chemistry and Chemical Biology. She joined Northeastern in 2011 and since then has taught more 35 sections of organic chemistry to hundreds of students from several majors.

Despite the difficulty of the material, Oyinda is overwhelmingly evaluated by students has highly effective, and is even described as inspirational and life-changing. She is praised for her classroom engagement and out of class support and mentoring.

She is a leader in the practice of evidence-based teaching, which includes conducting and presenting her own research. Oyinda demonstrates leadership as the Co-PI for a Research Experiences for Undergraduates grant from the National Science Foundation and for her service on University committees to facilitate the “One Faculty” initiative.

__Prasanth George__

Dr. George is an Associate Teaching Professor in the Department of Mathematics. He joined Northeastern in 2014 and in just the last three years has taught seven different classes, most of them multiple times. His effectiveness in teaching difficult material to large numbers of students from several majors is truly remarkable.

The words students often use to describe Prasanth include, “best”, “favorite”, and “incredible.” In 2017 he received the College of Engineering Outstanding Teacher of First Year Students Award.

Prasanth has also been a leader in the overall management and delivery of the mathematics undergraduate curriculum serving as the Undergraduate Director, Assistant Undergraduate Advisor, Mathematics Tutoring Center Director, Trainer of Teaching Assistants, and member of the College of Science Undergraduate Curriculum Committee.

### He Knows the Key to Unlocking the Secrets of the Universe

Ever feel overwhelmed by the chaos and variability of the day to day? Maybe you’re in the wrong field. Try studying representation theory—it lets you predict things that you might otherwise never see, like how individual microscopic particles spin.

If you’re “absolutely awed” by this hack, you’re in good company. As an undergraduate, **Valerio Toledano Laredo** didn’t even know the field of study existed, but he was a quick convert. Now a mathematics professor at Northeastern in his third decade of research, he’s helping others, through education and outreach, begin their own journeys.

*His* started with the electron.

“There’s a very complicated microscopic particle and these very complicated lab experiments to do to understand its behavior,” says Toledano Laredo, whose professor at the time put representation theory on his radar. “But you can just sit back in your chair with a paper and pen and, in half a page, actually predict some of these properties—just by pure thought.”

So what’s the secret sauce for learning about something you can’t see?

Symmetry.

The symmetries of an object, or the transformations you can make without fundamentally changing it (think rotating a square 90 degrees), manifest in different ways. Mathematicians call these manifestations “representations,” hence “representation theory.”

Here’s how to think about this theory in everyday terms: If I closed my eyes while you rearranged playing cards facedown, I’d have no way of knowing whether you even rearranged them. Similarly, a monochrome ball would look the same no matter how many times you spun it around.

This quality is considered a symmetry, and it shows up—in the form of representations—in very different objects. On the subatomic level, representations of this symmetry manifest in electrons and photons, two distinct kinds of particles.

Electrons and photons are fundamentally different particles because of how they behave. You don’t have to be a physicist to recognize this: We’ve all seen lasers (“social” photons piling on top of one another) and electricity (“asocial” electrons keeping their distance from one another and stretching a signal along).

Toledano Laredo says that knowing, through representation theory, that these particles are also somehow *similar* helps mathematicians and scientists understand how these tiny components of the universe behave—and prepares researchers to conduct new experiments without going in blind.

But it’s not always obvious when distinct objects share symmetries, which, Toledano Laredo says, is what makes representation theory so important to study.

Now he wants to give the next generation of researchers the tools to learn more about this complex theory, along with the related fields of algebraic geometry and mathematical physics.

As part of a team funded by an ongoing **research and training grant** from the National Science Foundation, he’s working to prepare students to excel in the science, technology, engineering, and math fields through programs such as **Bridge to Calculus**, which prepares high schoolers in Boston to succeed in their math classes.

Training the coming generations of mathematicians, he says, will enable our understanding of the universe to expand.

“There’s so much more to know,” says Toledano Laredo. “For me, for math, for physics—for the whole scientific community.”

This story was originally published on News@Northeastern on September 13, 2019.

### Northeastern Mathematics Hosts Quantum Structures Conference

The Northeastern Mathematics Department will be hosting a conference on Quantum Structures in Algebra and Geometry from August 26-30. The conference will focus on the interaction between Representation Theory (the mathematical study of symmetry), Algebraic Geometry (the study of curves and surfaces defined by polynomial equations, and their higher dimensional analogs) and Mathematical Physics.

It is sponsored by the National Science Foundation, through a $2.3M RTG grant for 2017-2022, MIT, the Clay Mathematics Foundation, the College of Science and the Mathematics Department.

A detailed program can be found at here.

### Looking Back on Bridge to Calculus 2019

*For the past 25 years, Northeastern University has been partnering with Boston Public High Schools to give rising junior and senior students the skills they need to succeed in their fall calculus classes through the Bridge to Calculus program. As the 2019 session, which started June 24th, draws to a close this Thursday, August 1st, we’d like to give a shoutout to this year’s talented group of Northeastern University PhD Graduate students (Cancan Zhang, Changchang Liu, Chen Li, Hiu Ying Man, Jiewei Feng, Jieying Jin, Lei Yang, Mohamed Elbehiry, Tomas Skacel, and Xuezhu Lu) and Boston Public High School Math teachers (Anthony Bernazzani of Boston Latin Academy, Phuong Cao of Boston Latin Academy, Noemi Famador of East Boston High School, Mike Sheehan of Boston Community Leadership Academy, and Juan Tapia of John D. O’Bryant School). Thank you for lending your time and expertise! We’re already looking forward to another great session next summer.*

*So why should you participate in the Bridge to Calculus program? We talked to two educators working with Bridge to Calculus, Noemi M. Famador, a Precalculus and AP Calculus teacher at East Boston High School and Michael Sheehan, who works at the Boston Community Leadership Academy, to find out.*

#### 1. Why are your participating in the Bridge to Calculus?

**Noemi Famador:** Teaching math in the regular school year, I have to deal with a lot of unmotivated students. I still do my very best since it’s precisely this group of students that I want to convince that math is not that bad and that with the right attitude and extra effort one can actually learn it and love it. This challenging work can be stressful and discouraging. This is also why I always look forward to teaching in the BTC summer program. Students in the summer program are really motivated to learn because they do it to be ready for their AP Calculus class in the fall. They do have their struggles- financial, work, family issues, health, etc – but they are all still willing to come to learn at 7:30 am everyday for six weeks.

**Michael Sheehan:** I have always loved teaching BPS students, as I feel we each have something to offer each other as a result of our different backgrounds. I especially love teaching BPS students when they truly want to learn. Bridge To Calculus students show up voluntarily, early in the morning. They listen and are not always trying to sneak in some text conversation on their phones like so many of my students during the school year. By putting in the kind of effort they do, these students get the maximum benefit from what I have to offer them. I also enjoy being able to treat them as adults; they can wear hats if they like; they can get up and use the bathroom when they see fit instead of having to raise their hands and ask for a pass. It’s just an all-around healthy and productive relationship.

#### 2. How did you come to know about the Bridge to Calculus?

**NF:** Summer of 2008, my math department head (Mr. Francisco Garnica) asked me to take his place in the bridge program. He used to teach it but because of his workload he couldn’t do it anymore. He also wanted me to get to know my AP Calculus kids earlier by teaching them in the summer program. It was a perfect arrangement and I got hooked since then.

**MS:** I came to know about Bridge To Calculus through its founder, Professor Bob Case.*

#### 3. How long have you been involved with the Bridge to Calculus?

**NF:** Since summer 2008.

**MS:** I have been involved with the program since its inception (roughly 1994 or 1995 if my memory is accurate). I was originally an assistant in the classroom for a few years, then taught a class, then a week, and then became a full time instructor.

#### 4. Why do you love Math?

**NF:** My love for math started when my Grade 3 teacher would bring me around our city (Cebu City, Philippines) to join math competitions. The training, although gruelling, was also very rewarding especially when I would get some medals and this made my parents really happy and proud. This love for math became deeper such that in college I chose a scholarship to take up BS in Math for Teachers instead of one for Engineering. At that time, there was already that desire in my heart to make other people see the beauty of math. I was thinking I could do that if I teach math. After teaching math in the Philippines for 20 years and then teaching math here in the US for another 20 years – I’m still on that mission. Everyday in class, I always try my best to make the students know more of the beauty of math so they become less afraid of it. Hopefully when they are less afraid, they can learn it better and thus love it a little bit more. I know this is challenging but if I can make a difference everyday in class, then it’s a worthy endeavor.

**MS:** I enjoy the challenge of math, and believe that developing discipline in math has benefits in other areas of life. Also, ever since I was a high school student myself, I have felt like math is one of the few areas in life where people from all countries and all walks of life agree on the same solutions to problems (assuming they all understand the problem!)

*the program was originally called “Boston Summer Advanced Math”