Learn the formula for studying math and physics

What research papers and U of T professors say on how to best study quantitative subjects

Learn the formula for studying math and physics

Is there such a thing as having a ‘math brain’?

It may seem like most people in your classes were born with a profound knowledge of vector space axioms and the ability to do impeccable epsilon-delta proofs in their sleep. This might be true to some extent — research from John Hopkins University suggests that there are in fact people who are able to grasp mathematical concepts quicker, something the researchers attribute to an innate “number sense.” But not having this ability doesn’t mean you can’t become proficient at math.

Dr. Alfonso Gracia-Saz, an associate professor at U of T’s Department of Mathematics, compared developing your mathematical abilities to training for a marathon. Just like “almost all healthy adults [would] be able to run a marathon with training and discipline,” he wrote to The Varsity, “almost all healthy adults are able to excel at and enjoy math.”

I can personally testify to this. I used to be a quintessential social sciences girl who dreamed of attending the Munk School of Global Affairs and Public Policy, and eventually work on Parliament Hill; I was convinced that math and physics were way beyond my reach.

The environment I was in wasn’t helping either — my grade 10 science teacher preferred talking about his homegrown tomatoes and missionary trip to East Africa over actually teaching.

Fast forward three years, and I’m a math major.

In her book Mindset: The New Psychology of Success, Dr. Carol Dweck, a professor at Stanford University’s Department of Psychology, argues that our beliefs about whether our abilities are completely innate or able to be improved can play a pivotal role in our success.

The “growth mindset is based on the belief that your basic qualities are things you can cultivate through your efforts,” she wrote. Realizing that your math and physics skills can be boosted with effort and practice may help you change your relationship to encountering challenges and setbacks while learning.

Rethink your learning

There is no doubt that grasping unfamiliar concepts or solving difficult math problems can be incredibly challenging and frustrating, but researchers argue that struggling and making mistakes is a crucial part of the learning process. In other words, it may actually be beneficial for your brain and help you become a better learner.

Cristina Peter, a learning strategist at the Koffler Student Services Centre, spoke to The Varsity about difficulties with learning material. She elaborated that “most challenging concepts are built on basic fundamentals,” and encouraged students “to unravel the basic concepts that the challenging concept is built upon.”

Gracia-Saz agrees with the importance of embracing the struggle, adding that being patient with your own learning is also crucial to getting to the bottom of a new concept. Some approaches to solving unfamiliar problems may not work at first, but eventually you will find the one that works, and this process of trying and failing “will help you develop good intuition and solidify your understanding.”

A few days after my first midterm of PHY151 — Foundations of Physics I, an introductory course to calculus-based physics, my professor Dr. Stephen Julian told us an anecdote from his undergraduate career, which he kindly recounted in an email to The Varsity.

“I was pretty ignorant when I arrived at university. Probably many of my classmates had properly learnt their high-school math and physics, but for me it was all new. When the first set of midterm exams came up… the subject was special relativity, and I thought that the way to study was to read a book (it wasn’t even our textbook) about special relativity the night before the exam.”

“Of course, when I got into the exam I couldn’t actually do anything and I failed miserably,” he continued. “It was only in my third year that I really came to understand that in order to do well you need to learn the material as you go along (again, after every lecture you need to review and think about the material), and you need to practice solving problems.”

“Learning the concepts without the math is not going to work,” he added.

Why physics is so hard to learn, and how to handle it

Unfortunately, the prevailing view of physics is that it’s a notoriously hard subject which only a lucky few can master. Even physics students who love the subject often struggle to get to the core of new material.

Dr. Edward Redish, a professor at the University of Maryland’s Department of Physics explained in a research paper on the cognitive science of teaching physics that one of the characteristics of studying cognitive science is that we tend to organize our observations and experiences of a particular subject into mental models or patterns, which we have to build upon to maximize our learning. This usually means that passively reading and re-reading the textbook, and going to lectures may not be enough to build a solid understanding of the subject.

Peter suggests taking a moment after class to reflect on what you’ve just learned and identify gaps in your understanding.

Other researchers also suggest testing yourself repeatedly on each topic and making a set of criteria for applying new concepts to ensure that you know when and how to use your knowledge. It’s also beneficial to verbalize your thoughts to check if you truly understand the material — you might find it helpful to pretend you’re talking to a younger sibling — and to try to relate it to your prior knowledge or real-life examples.

Julian highlighted the importance of giving yourself time to get your head around a new concept — he recommended “thinking about the concept as soon as it is introduced in lectures (or indeed the day before if you are assigned pre-class readings).”

He also emphasized that “you need to keep going back to it over the following days and weeks. With time, and patience, the concept will take root and grow in your mind.”

Solve problems like a pro

The expert-backed approaches to solving math and physics problems in this non-exhaustive list may help you power through your next problem set with a lot more ease.

If you find yourself stuck on a question, Julian recommends thinking “of a simpler version of the same question, and keep simplifying until you find something that you can solve, [then trying] to work your way back to the original question.”

According to Gracia-Saz, looking at many solved examples is a “trap” that won’t replace actually going through the problem-solving process yourself, which is a valuable learning tool. He also stressed the importance of perseverance in learning math — “nothing is as satisfying as solving a good math problem after hours of struggling, except perhaps chocolate.”

Peter recommends rationalizing every step of solving a problem, such as creating a chart with your steps on one side and the explanation of your actions on the other.

George Pólya’s famous problem solving method is also worth looking into.

Getting help

Your professors and teaching assistants will most likely be happy to help you with the course material. When taking advantage of this, Peter recommends planning your questions in advance to maximize your learning.

If you are interested in learning more about time management, note-taking, and problem-solving strategies, you can book an appointment with a learning strategist at your faculty or college, or visit them during their drop-in hours at the Sidney Smith Commons. You can also check out on-campus math and physics resources, such as the Math Learning Centre, Vic Peer Tutors, the Statistics Aid Centre, and college-specific math help centres.

If you’re looking for a community to learn with, you should also consider joining or starting a study group, where you can make new friends and review concepts.

U of T shines at 55th annual Canadian Undergraduate Physics Conference

Students, alumni share their knowledge and passion in Montréal

U of T shines at 55th annual Canadian Undergraduate Physics Conference

The 55th Canadian Undergraduate Physics Conference was held from November 7–10 at McGill University and was attended by students from across the country, including those from U of T. While their research interests were diverse — ranging from imaging the brains of zebrafish to modelling galactic interactions with black holes — all shared the same burning passion for physics.

During the three-and-a-half days of the conference, visitors attended keynote lectures by top scientists and presentations by their fellow colleagues, as well as participated in panel discussions.

Keynote lectures

Dr. Ania Kwiatkowski, a keynote speaker at the conference, discussed her research of ion trapping techniques. These techniques can help measure masses, separate existing isomers, and discover new ones.

Kwiatkowski works at TRIUMF, Canada’s national particle accelerator that is co-managed by 21 universities, which includes U of T. The applications of her research are valuable to the field of astrophysics: since star formation is brought about by nuclear reactions and decays, scientists can model them in an accelerator to better their understanding of how stars are formed and destroyed. This may ultimately even help us understand why the universe was formed in the way that it was.

Another noteworthy keynote lecture was by Dr. Donna Strickland, who received the Nobel Prize in Physics for her research into nonlinear optics last year. In 1985, Strickland and her supervisor Gérard Mourou invented chirped pulse amplification, a technique for creating extremely intense pulses of light much more efficiently than before. Applications of the invention include laser eye surgery and high-precision laser cutting.

U of T’s student talents

During the poster session, undergraduate students from all over Canada presented their research on topics ranging from tiny low-energy atoms and thermal conductor microscopes to distant pulsating stars. The Varsity spoke with four U of T undergraduates who presented their posters at the conference.

Hanzhen Lin investigated a novel way of cooling atoms using a special cooling chamber and magnetic fields. Cold atoms are essentially very low-energy atoms that have low speeds and momenta, which means that they exhibit stronger wave-like properties than ‘regular’ atoms. The applications of cold atoms include quantum computing and quantum simulation, as well as creating exotic states of matter that cannot be observed in nature.

Grace Li set up an apparatus with a one-dimensional crystal at one end and a series of weights and sensors all along it. She then recorded the resonance frequencies of the sensors while varying the number of weights attached and the frequency of the oscillations, which allowed her to ‘translate’ quantum mechanics into classical mechanics.

Rica Cristina Cruz looked into the trends in methane emissions in Toronto. She found that, although the levels of methane emissions seem to be decreasing over time, it is difficult to say whether the city is going in the right direction, since the emissions of other greenhouse gases may very well be increasing.

This summer, Lucy Ma and her team set about building a portable rubidium atomic clock that would not lose its accuracy in a weaker gravitational field due to special relativity, unlike regular clocks. The team members travelled to Timmins, Ontario, to launch the clock they had built on a hot air balloon, where they were then able to confirm that the clock was still accurate further away from the planet.

Diversity in physics

During a panel discussion on equity, diversity, and inclusivity in physics, all four panelists acknowledged that there are still massive disparities in the representation of marginalized groups. This includes racialized peoples, Indigenous peoples, women, and gender and sexual minorities — even as people nowadays are much more aware of the importance of diversity in science.

U of T alum Dr. Kevin Hewitt, who is now a professor at Dalhousie University, has been involved in outreach programs for Black Canadian junior high and high school students in Nova Scotia for almost two decades. His goal has been to bridge the achievement gap and encourage more Black youth to get involved in STEM subjects. He spoke with panelists Drs. H. Cynthia Chiang, Marie-Cécile Piro, and Michael Ramsey-Musolf.

Hewitt also agreed with the other speakers concerning the significance of role models for children looking into science as a career option. “You cannot be what you cannot see,” he stressed.

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How do you fit 14 billion years of cosmic history into a 30-minute talk?

A theoretical astrophysicist tells the story of the universe in AstroTour talk

How do you fit 14 billion years of cosmic history into a 30-minute talk?

Dr. Patrick Breysse, a postdoctoral researcher at the Canadian Institute for Theoretical Astrophysics, dragged our minds back in time, painted a picture of the universe, and explained how laughably inadequate the term “Big Bang” is at capturing the birth of our reality, in a public talk delivered at U of T on August 1.

How early conceptions of the universe evolved over time

The talk, titled “A Brief History of Everything,” was attended by around 300 audience members at the Bahen Centre for Information Technology. Breysse opened with an introduction to the works of American astronomers Henrietta Swan Leavitt and Edwin Hubble in the early 1900s.

At that time, philosophers believed that the Milky Way was at the centre of the universe and was surrounded by a sea of unchanging, unmoving stars. But the discovery of massive swirling clouds called spiral nebulae threw a wrench in that model. 

“There was a great debate around [1908 about] whether these were little gas clouds in our galaxy or enormous things outside of it,” explained Breysse.

Leavitt noticed the blinking of special stars called Cepheid variables, and she figured out that the speed of this blinking could tell us how bright the star is. Hubble used this to figure out how far bright stars are. The giant cluster of stars they quantified eventually became known as the modern galaxies. 

Hubble also showed that these clusters of stars are separate from the Milky Way. The closest galaxy to us, Andromeda, is a whopping 24 quintillion kilometres away — that’s eighteen zeros!

“If I say Andromeda is as far as the moon is away from us, as the moon is from the earth, then the earth would have to be the size of a virus,” noted Breysse.

Hubble concluded that everything in the universe was moving away from us, and so there must have been a point when everything was together, and that an explosion resulted in this scattering of matter.

Decades of investigation later, we know that we are not at the centre of the universe, but are instead surrounded by massive constellations of massive galaxies and cosmic phenomena. 

“Astronomy is, if anything, good for telling you, you are not special in any comprehensive way!”

How theoretical physics underpins our current understanding

From the centre-of-the-universe conception, Breysse brought us up to date with the Lambda Cold Dark Matter model of the universe. This model is, in essence, the Big Bang Theory – the universe started out in a “hot dense state, and then nearly 14 billion years ago,” something happened and here we are.

Astronomers study the origin of the universe by constructing images of cosmic webs of matter and empty space using powerful telescopes and satellites. These researchers glean knowledge from ergo-planetary discs, and learn about the birth of galaxies from nebulae. 

Despite all this, there is a two-billion-year gap in our knowledge. Breysse and his research team use microwaves and intensity mappings to construct an idea of what our cosmic past might have looked like – the universe’s “baby pictures.”

“Every telescope is secretly a time machine,” said Breysse. “Telescopes look at light, and light travels at a constant speed.”

As light travels at nearly 300,000 kilometres per second, images captured from the distant reaches of the universe represent moments taken from the past. This is accounted for by the time it takes for the light to reach our instruments.

The further out we take observations from, the closer we get to the moment the universe originated.

Reactions from audience members

Audience members were given the opportunity to ask questions and express their appreciation at the end of the evening. 

“It was a fun and informative talk. He made a topic that’s quite difficult to understand simple, and did it in an entertaining way,” said Maeesha Mahbub, a third-year Fundamental Genetics and its Applications specialist, to The Varsity

The audience was also, surprisingly, filled with children from the neighbouring communities accompanied by their families. One brave nine-year-old, Leffe Monette, enthusiastically asked Breysse questions during the talk.

“I really enjoyed the talk! I liked how he showed us how far the other galaxies are,” said Leffe, a student of St. Michael’s College School. “I want to be a [space] scientist!”

Following the talk, audience members were given the chance to view Saturn and Jupiter through telescopes from U of T’s Department of Astronomy and Astrophysics atop McLennan Physical Libraries.

AstroTours also set up a free planetarium show for audience members following Breysse’s talk, conveying further information on the vast expanse of the universe to interested attendees.

“Wrong side of history”: U of T criticized for involvement in Hawaiian telescope project

U of T faculty, students in solidarity with Native Hawaiian protests to protect sacred site

“Wrong side of history”: U of T criticized for involvement in Hawaiian telescope project

Protests in Hawaii against the construction of the Thirty Meter Telescope (TMT) on the Mauna Kea — a sacred mountain that Native Hawaiians, known as Kānaka Maoli, regard as their origin site — have made their way to U of T. The university is a member of the Association of Canadian Universities for Research in Astronomy (ACURA), an organization that has funded the astronomy project.

U of T faculty and students criticized U of T’s involvement in the project, in solidarity with peaceful Kānaka Maoli protesters who have been occupying the site since construction began on July 15.

Astronomy’s rising star?

The TMT is a project over 10 years in the making, with the promise of enabling astronomers to look far into the past of stellar and galactic evolution. With an area nine times bigger than any existing visible-light telescope, the TMT is designed to identify images with unprecedented resolution, surpassing even the Hubble telescope.

The profound sensitivity of the TMT boasts the potential for observational data to answer questions about “first-light” objects, exoplanets, and black holes in the centre of galaxies.

This potential for furthering astronomy and astrophysics is what makes the TMT astronomy’s rising star.

Why is the TMT being protested?

In July 2009, the Board of Governors for the TMT chose the Mauna Kea as its location. Mauna Kea has long been an astronomical hotspot, serving as the location for 13 observatories. The TMT would be the 14th, standing as the biggest telescope on the mountain.

Mauna Kea is a sacred ancestral mountain, a place imbued with both natural and cultural resources. It is the location of many religious rituals conducted by the Kānaka Maoli, as well as a burial ground of sacred ancestors. Additionally, its ecological value is profound, housing esoteric ecosystems and providing water to the residents of Hawaii.

For these reasons, native kia’i (guardians) and kūpuna (elders) have resisted industrialization on Mauna Kea ever since the first telescope was built in 1968.

Subsequently, the TMT has attracted significant protests, serving as the Leviathan of telescopes. Dr. Uahikea Maile, Assistant Professor of Indigenous Politics at U of T, describes the TMT as a “unique beast” because of its size and location.

The project requires eight acres on the northern plateau of the mauna, which is currently untouched. Maile asserts that the corporation backing the TMT tempts the State of Hawaii into “valuing techno-scientific advances and alleged economic benefits over Native Hawaiian rights and the environment.”

Hence, ever since 2014, kia’i have attempted to halt the construction of the TMT by forming blockades at the base of the summit.

A brief space-time log of events

On July 10, Hawaiian Governor David Ige announced that construction of the TMT would begin on July 15, 2019. Five days later, hundreds of peaceful protestors stood together to form a blockade that would prevent construction crews from ascending Mauna Kea to begin constructing the TMT.

Located at an elevation of 6,000 feet, the blockade is logistically supported by the Pu‘uhonua o Pu‘uhuluhulu, a place of refuge providing resources and infrastructure to sustain all those involved in the blockade, wrote Maile. All people at the pu‘uhonua have access to free housing, food, health care, child care, and transportation.

Maile, who is of Kānaka Maoli descent, spent two and a half weeks at the protests. He recounted that the kia’i were “constantly prepared for the risk of police force and violence.” On the second day of protests, Governor Ige deployed the National Guard, militarizing the once peaceful site of protest.

On July 17, police arrived at the scene carrying riot batons, tear gas, guns, and a Long Range Acoustic Device, according to Maile. The elder kūpuna, many of whom were in their 70s or 80s, formed the central blockade, while they requested the kia’i to stand at the sides of the road.

Thirty-eight people were arrested at the scene, most of whom were kūpuna, but after hours of negotiations “a deal was struck and all police left.”

Numerous sources maintain that U of T’s statement on the Thirty Meter Telescope (artist’s depiction pictured) are not reflective of the views of all faculty members and students.
Courtesy of TMT Observatory Corporation

University of Toronto responds

U of T, a member of ACURA, is involved in the TMT. ACURA has served an advisory role in the estimated $1.5 to $2 billion project. Its members and other Canadian astronomers are planned to receive access to 15 per cent of the telescope’s viewing time.

It is important to note that U of T is not directly invested in the TMT. Nonetheless, Professor Vivek Goel, a board member of ACURA and Vice-President, Research and Innovation, and Strategic Initiatives at U of T, published an official statement explaining that he has been “watching closely the recent events at the construction site.”

He continued by writing that U of T “does not condone the use of police force in furthering its research objectives,” and noted that the university’s commitment to truth and reconciliation impels it to consult with Indigenous communities.

Lack of consensus amongst faculty members

U of T’s official statement has received backlash from numerous sources who maintain that it is not reflective of the views of all faculty members and students.

For instance, Dr. Eve Tuck, an Associate Professor in the Department of Social Justice Education, has written three letters to U of T President Meric Gertler, criticizing the statement for not going far enough in taking action against the TMT.

In an email to The Varsity, Tuck wrote that while the university has no direct funding in the TMT, there are still ways to divest. “There is more than money that can and should be withdrawn in this situation, including support, endorsement, affiliation, reputational backing, approval, and advocacy for the project.”

She believes that it is imperative for U of T to prevent the TMT’s construction, and if it does not do so, it “is on the wrong side of history.”

Moreover, protesters of the TMT have found an unexpected ally in some astronomers who, perhaps counterintuitively, oppose the project. For instance, Dr. Hilding Neilson, an Assistant Professor at U of T’s Department of Astronomy & Astrophysics, wrote that “the statement from the university doesn’t say a whole lot.”

He specifically questioned the statement’s assumption that astronomy has a “moral right” to the mountain because it is a scientific field, which supposedly seeks to benefit the accumulation of knowledge for all of humanity.

Power to graduate students

An open letter authored by astrophysics graduate students at the TMT’s partner institutions reinforced this opposition from U of T astronomy professors. The letter, published online, called on the astronomy community to “denounce the criminalization of the protectors on Maunakea” and to remove the military and police presence from the summit.

Two signatories, Melissa de los Reyes and Sal Wanying Fu, wrote to The Varsity that it is “imperative for the astronomy community to denounce [the arrests of kūpuna] and take a stand against the further use of violence in the name of science.”

Reyes is a second-year graduate student at the California Institute of Technology, while Fu is an incoming graduate student at UC Berkeley. Both are National Science Foundation graduate fellows.

The open letter was published despite the risk that it could potentially impact the signatories’ research careers. The signatories include graduate students, postdoctoral researchers, and professors.

Signatories from U of T include professors Hilding Neilson and Renee Hlozek, Postdoctoral Fellow John Zanazzi, Sessional Instructor Dr. Kristin Cavoukian, PhD students Fergus Horrobin, Fang Xi Lin, Marine Lokken, Adiv Paradise, and Emily Tyhurst, and undergraduate students Yigit Ozcelik, Andrew Hardy, and Rica Cruz.

Jess Taylor, the Chair of CUPE 3902 and a writing instructor in the Engineering Communication Program at U of T, was also a signatory.

The signatories Reyes and Fu hope that the discussion prompted by the letter causes academic astronomers to “reckon with the ways in which social systems are inextricably linked with the way we do science.”

Neilson commended the bravery of its signatories, writing that “for students to come out and do this, potentially not only against their own research, but against their supervisors’ and departments’ requires standing up to power.”

Activism by undergraduate students

The University of Toronto Students’ Union (UTSU) and the Indigenous Studies Students’ Union (ISSU) also published a joint statement on August 29 condemning the construction of the TMT at Mauna Kea.

The UTSU represents full-time undergraduate students at the St. George campus, while the ISSU’s membership includes students who are enrolled in the Indigenous Studies program or are taking at least one Indigenous Studies course.

The unions called upon U of T to “cease construction” of the telescope and to relocate it to an “area where its construction would not infringe upon the sacred land of Indigenous peoples or damage land that is environmentally protected.”

Eclipsing Indigenous knowledge

It is important to recognize that the Kānaka Maoli protests are not against science. Rather, they are against a Western ideology of economic development that — in the name of science and objectivity ­­— has historically propagated mechanisms of colonization, slavery, and incarceration. Following centuries of colonial and postcolonial development, the scientific industry today undermines and maligns Indigenous knowledge systems — associating it with primitivity.

Meanwhile, Neilson draws attention to the value of Indigenous knowledge, stating that “a lot of the tensions between Hawaiians and TMT come from the fact that a lot of us are ignorant of Hawaiian knowledge, and what it means for Mauna Kea to be sacred.”

Ultimately it is not a question about science versus culture, but about whether development under the guise of science reinforces a certain hierarchy of culture. It is evident that there is a need for a scientific Big Bang, one where Indigenous cultures is no longer at the bottom of this hierarchy.

Editor’s Note (September 9, 3:26 pm): The article has been updated to reflect that ACURA has funded the TMT, according to a 2013 ACURA report, but does not own a 15 per cent stake. Canadian contributions collectively have a 15 per cent share in the TMT project.

Casual setting, critical thinking

BookLab podcast explores ideas from modern physics to genetic testing

Casual setting, critical thinking

By listening to BookLab, a podcast by science journalists Amanda Gefter and Dan Falk, I better understood what makes science writing engaging and how to think critically about what I read, even if the authors are experts in their field.

BookLab describes itself as a series “that puts science books under the microscope.” Though Gefter and Falk specialize in physics reporting, they don’t shy away from exploring topics from a diverse range of fields.

Their discussions include books on how medicine may drastically change due to technology, unexpected implications of modern physics, issues with genetic testing, and much more.

Gefter and Falk integrate their experiences as science journalists into their discussions, exploring how certain sections of a book relate to recent discoveries they’ve reported on, and how their personal experiences can serve as examples of a key concept.

They bring in their experiences as longform writers, exploring how writers successfully distil complex ideas into accessible forms, and how poetic language can be distinctively used to capture an idea into words.

Each episode starts off by introducing the broad theme explored by a book under review. Gefter and Falk then provide a brief overlay of the writing, discuss particular parts of the book that stood out to them, and observations about the writings and arguments of the authors.

The episodes conclude with a brief segment called “What’s on your nightstand?”, in which each journalist describes a science book they’ve been reading recently, and what they liked about it.

It’s rare to find a podcast hosted by science journalists in a casual setting, as opposed to researchers or journalists in a professional setting. BookLab fits that niche and is engaging for scientists and science enthusiasts alike.

A duel of theories: quantum mechanics, general relativity — or both?

Talks by researchers during Science Rendezvous festival illuminate theories of physics

A duel of theories: quantum mechanics, general relativity — or both?

From the Department of Physics, Professor A. W. Peet and post-doctoral fellow Aharon Brodutch delivered two related yet different talks about crucial theories of physics to a wide range of attendees of the May 11 Science Rendezvous street festival on the St. George campus.

Gravity causes black holes to exist

Even if you dozed off in high school physics lectures, there’s almost no way you haven’t heard of black holes. Just over a month ago, NASA published the first ever image of a black hole, which left the world in utter awe.

But what causes black holes to exist? To answer, Peet began by explaining the theory of gravity.

We are all familiar with the force of gravity: you drop a tennis ball, and it falls downwards. Gravity is the invisible force that is responsible for the attraction of all objects to each other. Furthermore, the strength of the force is directly related to the object’s mass. The larger the mass, the larger the force of attraction between the objects.

However, gravity not only attracts mass, but it also pulls on light — despite the fact that light is composed of massless particles.

How can gravity ever be strong enough to trap massless light?

Consider black holes: Einstein’s theory of general relativity — which provides an alternative explanation of gravity as a property of space and time — anticipated that when a massive star dies, the remnant it casts off has three times the mass of the Sun, and a black hole is produced.

Peet gave an alternative definition of the phenomenon: “an object is called a black hole if it is dense enough to be contained within its own event horizon.”

The event horizon can be thought of as the point-of-no-return: if you fall into it, escape is impossible, regardless of your rocket power. This also applies to light. Inside this radius, the gravity is so strong that not even light can escape.

Yet despite their attractive force, black holes still emit radiation

Peet mentioned that “at [a black hole’s] heart, there exists a singularity, where its curvature becomes infinite.” With an infinite radius, Einstein’s theory of general relativity fails to demonstrate any results at the singularity of a black hole, since all the equations render infinity as the solution.

Meanwhile, with the employment of quantum theory into Einstein’s theory of general relativity, physicist Stephen Hawking was able to prove that black holes actually do emit radiation — therefore they do not appear completely black after all.

Two theories that cannot exist under the same roof

Einstein’s theory of general relativity describes the physics behind very heavy objects — such as planets, stars, and moons. Quantum mechanics, on the other hand, is the physics relating to extremely small particles.

Now, considering the two, you would think that their incompatibility is not truly problematic, since nothing can be both very heavy and very small. However, when has science ever been that simple?

“There are two things that we care about: black holes and the Big Bang!” added Peet. They continued by saying that in order to be able to effectively and accurately analyze these two mysteries, we would need a theory that could be applied to both massive and small objects.

String theory mends the two clashing theories

String theory predicts that inside the elementary particles — irreducible particles previously thought to be point-like — are actually one-dimensional vibrating strands of energy known as strings. The fact that strings are versatile demonstrates the ease with which they can interact, and thus solves the problems arising from the theory of general relativity.

Peet also mentioned that string theory can predict possible extra dimensions of space, explaining that “the strings could wrap around those hidden dimensions.”

Multiple worlds at once?

Now that we have reached a better understanding of black holes, let us consider other realms and dimensions through our understanding of quantum theory. In 1935, physicist Erwin Schrödinger came up with a world-changing theoretical experiment known as the Schrödinger’s cat paradox.

“He placed a cat in a steel chamber with a Geiger counter, a vial of poison, a hammer, and some radioactive substance,” explained Brodutch.

This process is not one that naturally comes to mind, yet it does make physical sense. With the decay of the radioactive substance, the Geiger counter would prompt the hammer to fall on the vial, releasing the poison and consequently killing the cat. Seems pretty straight forward, right? Then, what is the paradox about?

What Schrödinger wanted to demonstrate, explained Brodutch, was that we would not know whether the cat was dead or alive until we opened the chamber. Thus, in order to be theoretically accurate, we would have to assume that while the steel box is still closed, the cat is both dead and alive simultaneously in two different worlds.

With the help of the Schrödinger’s cat paradox, we are able to somewhat understand the possibility of the existence of more than a single world at once.

The worlds can interfere

This paradox was employed in order to account for the wave function of a particle: the particle could be in any allowed position at a certain instant, yet you could not know exactly where unless you directly saw it.

Brodutch further said that “each world will be one term in the equation, and as the branched worlds keep going, the equation becomes longer and longer.”

Thus, maintaining control over these quantum systems becomes the main concern. Applying quantum theory to modern-day technological advances, Brodutch explained that it enables quantum computers “to factorize really, really fast,” and as a result, “keep [security-sensitive] transactions very secure.”

Physics has a diversity problem

An age-old issue brought back to life following Jocelyn Bell Burnell’s Breakthrough Prize award

Physics has a diversity problem

The Breakthrough Prize — a $3 million award bestowed to researchers in Life Sciences, Fundamental Physics, and Mathematics — recently recognized astrophysicist Jocelyn Bell Burnell for her discovery of pulsars.

When Bell Burnell was a researcher at the University of Cambridge in 1967, she noticed a signal that repeated every second or so through a radio telescope. Bell Burnell and her advisor Antony Hewish weren’t sure what to make of it.

These signals turned out to be pulsars, or fast spinning neutron stars that emit electromagnetic radiation. 

Though Bell Burnell was the first to discover pulsars, the Nobel Prize in Physics was awarded to Hewish and his colleague Martin Ryle in 1974.

Bell Burnell’s receipt of the Breakthrough Prize is historically significant, as women have not traditionally blazed through male-dominated fields like physics. When Bell Burnell came to Calgary on September 19, she told CBC News that the prize money would go to supporting underrepresented graduate students in science.

Though attitudes toward women in math and science have changed since the 1960s, there is still progress to be made.

According to U of T’s Professor AW Peet in the Department of Physics, one reason could be that social and cultural aspects cause participation rates in math and science to vary from country to country.

Many Eurocentric countries like Canada, the US, and the UK, have, in fact, lower participation rates of women in math and science compared to countries like Lebanon or Iran.

Eight per cent of the physics faculty in US universities with PhD-granting departments have no representation of women.

The statistics in Toronto aren’t promising either.

Of 14 faculty members who teach subjects relating to physics at UTM, only one is female.

While the University of Toronto has seen a significant improvement in female representation in physics, the number of female graduate and undergraduate students still remains low compared to women in biology or chemistry.

In 2012, 24 per cent of undergraduate students enrolled in Applied Science & Engineering, which includes studies in physics, were female. Of graduate students in the faculty, 26 per cent were female. 

These statistics have improved after five years. In 2017, 33 per cent of undergraduate students enrolled in Applied Science & Engineering programs were female; of graduate students, 27 per cent were female.

In contrast, 65 per cent of students pursuing undergraduate Biological Sciences and 57 per cent of students pursuing graduate studies in Biological Sciences were female in 2017.

These statistics do not account for students who identify as nonbinary, and though they reflect an improvement in female participation in the sciences, particularly physics, they are still worrisome.

Organizations like the International Union of Pure and Applied Physics have brought  delegations from different countries together to compare representation in physics and become more cognizant of representation in physics.

Moreover, in Canada, the Canadian Association of Physicists (CAP) has taken initiatives to narrow the gap.

One of its initiatives, according to Peet, who is also the former Chair of CAP, is an annual conference for women in physics known as the Canadian Conference for Undergraduate Women in Physics.

However, if real progress is to be made in the sciences, changes at the institutional level, like promoting women into roles such as Canada Research Chairs, and at societal levels, like providing better support for women on maternity leave, are crucial.

Farewell, Professor Hawking

U of T remembers the renowned astrophysicist

Farewell, Professor Hawking

I first learned of Dr. Stephen Hawking from Star Trek: The Next Generation. The episode originally aired in 1993, this brief foray on screen saw Hawking playing poker with Einstein, Newton, and android Lieutenant Commander Data. Physics jokes were made, Hawking won the hand, and the cameo was over.

Given the endless index of extraordinary events that made up Hawking’s incredible life, which came to a sombre end this March 14, I could have easily chosen a more important event to begin this tribute, but I chose this memory because I remembered my precise childhood reaction to that scene: ‘I have no idea who this person is, but if he’s next to Einstein and Newton, he must be awesome.’

I strongly believe that that cameo will prove prophetic, and that Hawking will be remembered with the likes of Einstein and Newton.

An extraordinary mind, he made great strides working on the fundamental problem of physics: finding a unified theory to reconcile the vastly differing physics of the small, quantum mechanics, with the physics of the large, relativity. Along the way, he revolutionized astrophysics and cosmology with a plethora of theories, including the much-lauded Hawking radiation.

“Stephen combined Einstein’s general relativity of spacetime with quantum mechanics, two of the biggest developments in physics in the first half of the 20th century, to investigate the basic laws which govern the universe,” wrote Dr. AW Peet, a professor in the Department of Physics and a 25-year acquaintance of Hawking, in an email to The Varsity.

“He discovered Hawking Radiation, showing that black holes are not completely black: they can emit weak radiation and eventually evaporate completely. His Black Hole Information Paradox posed over forty years ago is still a very active field of research today,” continued Peet.

Dr. J. Richard Bond, a professor at the Canadian Institute for Theoretical Astrophysics, also noted Hawking’s ubiquity in cosmology.

“Everything I have been working on lately on sabbatical at Stanford has had Hawking discussion arising about it: Hawking temperature, Gibbons-Hawking entropy, black hole evaporation, [and] wave function of the universe,” said Bond.

It is clear that Hawking’s brilliance cannot be overstated. Yet it wasn’t just his brilliance that netted his multiple television cameos, and it likewise is not only for his astrophysical acuity that the world mourns him today. Of equal measure to his mind was his incredible capacity to convey the most complex of ideas to a general audience in a humourous, straightforward, and engaging way.

“People whom I clearly recall coming up to me at cocktail parties to explain, with satisfaction, that they never could do chemistry, decided, instead, that their lives would be incomplete if they did not encounter Stephen Hawking. They were right,” said Nobel laureate Dr. John Polanyi, University Professor in U of T’s Department of Chemistry.

Polanyi made that statement 20 years ago, addressing a packed Convocation Hall, when Hawking came to visit U of T in April 1998. It carries no flippant embellishment.

With his popular science book A Brief History of Time selling more than 10 million copies since its publication in 1988, it is no exaggeration to say that Hawking has inspired generations of scientists.

At one end of the spectrum are long-time physicists like Peet.

“I first met Stephen in 1992 when I was a baby Ph.D. student, at a dinner party of theoretical physicists at Stanford,” wrote Peet. “I was inspired to work on the research topics I investigate partly by his deep theoretical physics insights.”

Hawking’s following only grew in the twenty-first century, rousing another wave of young scientists to explore the universe.

“I remember reading A Brief History of Time during my days as an undergrad,” recalled Matt Young, a PhD student in the Dunlap Institute for Astronomy & Astrophysics. “Instead of coming across as a dry lecture, the book told the story of the universe and all its fascinating physics at a level that was accessible by everyone.”

Even after his death, Hawking’s mammoth influence in the field continues to generate enthusiasm in the next generation of physicists.

“I still have my own well-worn copy of A Brief History of Time on my bookshelf from days of old,” said second-year Physics and Philosophy student Patrick Fraser. “It was arguably that introduction to physics that inspired my own journey, hoping to one day be a physicist myself.”

Although a physicist and cosmologist, Hawking always sought to promote not just a single field, but the attitude and spirit of science in general. Having worked tirelessly in his promotion of rational thought and public involvement in research, it is not only students of physics who answered his call.

“[He was] truly an inspiration,” said second-year Electrical and Computer Engineering student Tobias Rozario. “A Brief History of Time helped develop my passion in physics and engineering.”

Second-year Molecular Genetics and Biochemistry student Matthew Gene expressed similar sentiments. “Stephen Hawking was an inspiration to all — not just in his work, but also in the way he lived. Despite being diagnosed with a terminal disease, Hawking fought on, continuing to be… one of the most recognisable public faces in science. As a student in the Life Sciences, it’s the resolve of men like Hawking that makes me dream of a better future for medicine and humanity.”

Yet among the multitude of thoughtful sentiments, there is one fact that remains to be mentioned: the inevitable image of Hawking speaking in the familiar programmed voice of his omnipresent wheelchair. Fraser succinctly addresses this elephant in the room.

“It is true that he was a great scientist despite his physical limitations. However, what many people perhaps fail to realize is that he was a great scientist, period,” said Fraser.

Although iconic, Hawking’s amyotrophic lateral sclerosis is not what society should focus on when remembering him. Instead, we should remember one of the greatest minds of the twentieth and twenty-first centuries, who grappled with problems about the very underpinnings of the universe and left an academic legacy for the aeons.

We should remember a brilliant writer and unmistakable orator, who used his astonishing talent for communication to promote a better future for all of humankind.

We should remember, perhaps above all, an unbreakable human spirit, who once, in the words of Peet, “unexpectedly sped off down the steep driveway… for fun, with a huge grin on his face, enjoying the apparent consternation on the faces of non-disabled folks around him.”

Hawking — a Companion of Honour, Commander of the Order of the British Empire, Fellow of the Royal Society, Fellow of the Royal Society of Arts, and former Lucasian Professor of Mathematics — is an essential contradiction in the world. He was unable to physically perform and partake in so much that society fundamentally associates with humanity, yet one would be hard-pressed to find someone who lived a fuller life than Stephen Hawking.


“To have been on the leading edge of physics with such a disease over so many decades has to be one of the greatest triumphs of human will in the history of humankind. His life was a celebration of human spirit.”

— Dr. J. Richard Bond, Canadian Institute for Theoretical Astrophysics

“Always he insisted [that] mankind stop its pursuit of insane weaponry, hinting that our imaginations had become paralyzed. He will be remembered for centuries.”

— Dr. John Polanyi, Nobel Chemistry laureate, Department of Chemistry

“Stephen was a brilliant mind, a phenomenal researcher, a truly extraordinary scientist. He also had a magnificent sense of humour. For example, he once famously drove over Prince Charles’s foot while showing him some wheelchair tricks. Don’t refer to Stephen as “wheelchair bound” or “suffering from” ALS/motor neurone disease or other pity-based words to describe disability. His wheelchair and robot voice system didn’t constrain him – instead, they liberated him.”

— Dr. AW Peet, Department of Physics

“I think Stephen Hawking will be remembered with the likes of Newton [and] Einstein, people that revolutionised their fields and dedicated their lives to understanding the world around us. Much of [today’s] research on topics such as black holes is directly building on Hawking’s contribution to science.”

— Matt Young, PhD student, Dunlap Institute for Astronomy & Astrophysics

“The sheer magnitude of his accomplishments in the field was astounding, entirely irrespective of the difficulties of his daily life… In the history of science, [his] academic achievements have been matched by few… He will be missed, but his legacy will live on.”

— Patrick Fraser, Physics and Philosophy specialist student

“His death caught me off guard. I never really expected him to die. Much like the Queen, I saw him as a staple of culture that just ‘exists.’”

— Daniel Wardzinski, Computer Science major, Mathematics and Philosophy double minor student

“He showed me that success comes from how you think and not what people think of you. I guess a lot of people measure how successful they are by how people look at them, but Hawking didn’t care about that.”

— Jenoshan Sivakumar, Astrophysics specialist student

“His collaborations with his daughter Lucy Hawking to write children’s books on space and science were incredibly influential on my childhood. They played a part in why I decided to learn physics.”

— Abhinav Bhargava, Physics and Philosophy specialist student

“Great scientists are rare, and great explainers are possibly even rarer. But in [A Brief History of Time], Stephen Hawking wrote about complicated concepts so well that they seemed almost intuitive.”

— Cameron Davies, Mathematics specialist, Ethics, Society, and Law major student

“Dr. Hawking was a genius and a pioneer in physics, but what was most inspiring about him was his pursuit of passion and of life in the face of adversity.”

— Hansen Jiang, Astrophysics specialist, Computer Science minor student

“It’s safe to say that Stephen Hawking is half the reason we’re here. He inspired an entire generation to look to the stars and imagine the unimaginable. To say he’ll be missed would be a drastic understatement.”

— U of T Astronomy & Space Exploration Society