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Ph.D. in Mathematics

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The Faculty of Mathematics and Natural Sciences is Norway's largest natural sciences faculty and has a long tradition of Ph.D. programs. The faculty is one of Europe's most attractive academic communities in mathematics, natural sciences, and technology.

The faculty's departments and centers cover a broad spectrum of interdisciplinary research. Ph.D. candidates are integrated into large, excellent research communities that play a leading role nationally and internationally. About 50% of Ph.D. candidates and 20% of academic staff come from foreign academic communities, which increases the strong international profile of the faculty.

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Mathematical Modeling

Simen Bræck, Associate professor

Simen Bræck obtained his Master and PhD in condensed matter physics from the University of Oslo. His current scientific interests include gravitational physics, condensed matter physics and quantum physics.

OsloMet webpage

Mihaly Andras Csirik, Postdoc

Mihaly received his PhD in applied mathematics at the Eotvos Lorand University at Budapest. He worked as a postdoc at the University of Oslo and the Wigner Research Centre for Physics in Budapest. After his initial work in nonlinear analysis and PDEs, he worked on coupled-cluster theory. He is a member of ERC project REGAL and he is currently focusing on the mathematical aspects of density-functional theory.

Bendik Steinsvåg Dalen, PhD candidate

Bendik Steinsvåg Dalen is a PhD candidate concerned with numerical simulations of quantum phenomena. In particular he focuses on the time evolution of quantum matrices, which has applications within many different areas in quantum physics including noisy quantum computing. He received his master’s degree within computational science at the University of Oslo, with a thesis titled «Characterization of Cardiac Cellular Dynamics Using Physics-informed Neural Networks

Sergiy Denisov, Professor

Sergiy Denysov  is a physicist (a theoretical one). He received his PhD in Ukraine in 2001 and then worked  at different research centers and universities in Israel and Germany. His main scientific interests are computational quantum physics and quantum computing. He also do some research on random walks and diffusion processes.

Personal webpage

Halvard Fausk, Associate professor

Halvard received his PhD from the University of Chicago in 2003.  His research has mainly been done in algebraic topology, especially topics related to equivariant homotopy theory.  He also has some background in theoretical physics. 

OsloMet webpage Personal webpage

Hugo Lewi Hammer, Professor

Hammer has a master in industrial mathematics from 2003 and a Ph.D. in computational statistics from 2008. His research interests include improvement, reliability, and transparency of machine learning, reinforcement learning and deep learning models by developing methods within model interpretation, uncertainty quantification, robust statistics and causal inference.

Nikolai Bjørnestøl Hansen, Assistant professor

Nikolai is a geometer/topologist working on a PhD from the University of Oslo. He has worked on multiple books on mathematics used by pupils in the Norwegian school system, and has been teaching mathematics in some form since 2008. His hobbies include recreational mathematics, programming, and games in all shapes and sizes.

John Haugan, Dosent Emeritus (equiv. to Professor)

John earned his cand. real. degree in physics from the University of Oslo.  After a period as senior research engineer in Det Norske Veritas (DNV) he has focused his R&D-effort on developing students’ learning and understanding of mathematics and science in engineering education. His current main interest is implementing constructive alignment in beginning courses in mathematics. John is author and co-author of several textbooks ranging from lower secondary to university level. In 2019 he was awarded the title Distinguished university teacher (Merittert underviser).

Kristoffer Herland Hellton, Associate Professor

Oslomet webpage

Kristoffer joined the Mathematical modelling group in August 2023.

Andre Laestadius, Associate Professor

Andre has a background from mathematics and theoretical chemistry. In his master’s thesis, he studied mathematical aspects of quantum chemistry, specifically time-dependent current-density-functional-theory (under supervision of Prof. Olav Vahtras). After that, he completed a PhD in applied mathematics with the thesis Foundations of Density Functionals in the Presence of Magnetic Field. He is currently leading the ERC project REGAL.

Marco Matassa, Associate professor

Marco Matassa is a mathematician who also has a background in theoretical physics. He received his PhD in Italy and has been a postdoctoral researcher at various universities in Europe. His main mathematical interests are non-commutative geometry and quantum groups, as well as their applications to physics.

Markus Penz, Researcher

Markus did his master and PhD studies in Theoretical Physics and Mathematics in Innsbruck, Austria. He was initially interested in the foundations of quantum mechanics, later moved to the more applied field of computational methods in many-particle quantum mechanics, primarily density-functional theory, but stayed focused on the mathematical and theoretical side. After a postdoc at the Max Planck Institute for the Structure and Dynamics of Matter in Hamburg, Germany, he is now a member of the ERC project REGAL at OsloMet.

Mira Randahl, Associate professor

Mira Randahl has a master in mathematics and a PhD in didactics of mathematics. Her PhD thesis was entitled “Engineering students approaching the mathematics textbook as a potential learning tool – opportunities and constraints”, and her research interests include differential equations, mathematical proof, engineering students’ learning and application of mathematics and calculus textbooks’ approach to concept formation.

Sølve Selstø, Professor

Sølve Selstø earned his PhD at the University of Bergen – in physics. While his research background has, for the most part, belonged to the area of atomic, molecular and optical physics, his scientific passion is really quantum physics itself – both in terms of phenomenology, theory, implementation and application. Numerical methods and scientific computing in math teaching is also close to his heart.

Martin Lilleeng Sætra, Associate professor

Martin Lilleeng Sætra earned his PhD in scientific computing at the Centre for Mathematics for Applications, University of Oslo. He has since been employed as a Scientist with the Norwegian Meteorological Institute. His research interests include accelerated and high-performance computing, numerical simulation, data assimilation, and scientific visualization.

Lars Tuset, Professor

Lars Tuset has a Ph. D. in mathematics from NTNU in 1995, and spent four years on postdocs in Rome and Cork. His area of research includes quantum groups, operator algebras and non-commutative geometry.

Leiv Øyehaug, Associate professor

Leiv Øyehaug earned his PhD in mathematical biology at the Norwegian University of Life Sciences in 2002. His research centers on application of mathematical models and methods to various real-world systems, most prominently to cellular systems in neural and heart biology. He also has a keen interest in teaching mathematics and its applications to bachelor, masters and PhD students.

Heine Olsson Aabø, Researcher

Heine Olsson Aabø is a PhD candidate concerned with quantum computing in the context of complex optimization problems. He received his Master’s degree entitled «Quantum Computing and Many-Particle Problems» at the University of Oslo in computational science focusing on quantum computing and many-body physics. He is currently working on research related to quantum computing.

Department of Mathematics

Main content.

At its core, mathematics is about the study of abstract structures. This may sound very theoretical, but it is a fascinating fact that the abstract structures that are studied in mathematics can be applied in an astounding multitude of practical problems. Mathematical sciences play an increasingly prominent role in the natural sciences and technology, and in society as a whole, highlighting the need for a high competence in mathematical sciences in order to succeed in the job market.

Our faculty members conduct research in a wide range of theoretical and applied mathematics, including statistics and computational mathematics. Pull-down the menus for more information about our study programs and research groups.

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The department of mathematics offers study programs in mathematics at bachelor's, master's and Ph.D. levels, as well as mathematics education.

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Using mathematics to fight disease

In June 2023 the project Math4SDG held its annual meeting in Bergen. We met up with five PhDs in the project to discuss how they will use applied mathematics to combat infectious disease and how the project aims to increase mathematical literacy in Tanzania and Uganda.

Four researchers receive Marie Curie grant

The University of Bergen has been awarded four EU-funded mobility grants through Marie Skłodowska-Curie Actions (MSCA). One grant goes to The Faculty of Humanities, and three will go to The Faculty of Mathematics and Natural Sciences.

Data science reveals universal rules shaping cells’ power stations

A UiB-led study published in Cell Systems reveals that all cellular power plants - our mitochondria, plant chloroplasts, and more - have evolved according to a universal set of rules.

Seminar by Gustavo Ponce

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Doctoral education at the Faculty of Mathematics and Natural Sciences

The Faculty of Mathematics and Natural Sciences offers doctoral education within the following disciplines: biology, chemistry, earth science, physics and technology, informatics, mathematics, molecular biology and geophysics. We also offer doctoral training in collaboration with several partner institutions. For more information click here

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PhD programmes

As a PhD candidate at OsloMet, you can expect a high-quality research education in a supportive academic environment.

At OsloMet, you will be a full-fledged member of our research community and will join a supportive and diverse community of academics. We have a large and growing number of international doctoral candidates and are committed to attracting talented aspiring researchers from around the world.

Many alumni of our PhD programmes pursue careers in academia, while others go on to high-level careers in the public and private sector—both in Norway and around the world.

PhD candidates in Norway are not students, but employees. Research fellows who are employed by a Norwegian university can expect some of the best working conditions in the world. As you work toward your doctoral degree, you will enjoy a good salary and a host of benefits, including five weeks of paid holiday.

You can also pursue a PhD at OsloMet while working for a different employer, either in Norway or elsewhere in the world.

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All of our PhD positions are publicly advertised. You can view current available PhD positions in our job portal. 

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Interested in learning more?

You can find detailed information about our PhD programmes, including application procedures and how to apply for funding, on our employee website. 

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Life in Oslo

Are you considering applying for a PhD or academic position at a Norwegian university? We asked the experts—our own international employees—for their advice on how to navigate the transition to working in Norway.

What defines quality of life in a city? Is it the availability of good restaurants, bars and entertainment options? Or is it easy access to unspoilt nature? In Oslo, you don’t have to choose between the two—we have plenty of both.

The sun doesn't set until close to midnight, and even then it never really gets dark. You can spend the evening watching the light change at an outdoor café with friends, or go for a relaxing swim in the fjord alone. Oslo in the summer is a pretty magical place.

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The PhD program

The PhD program at the Department of Mathematical Sciences usually lasts 3 years, for students enrolled with a Master's Degree, and 4-5 years for students enrolled in the integrated program with a bachelor's degree (or equivalent). The program consists first of all of an independent research project resulting in the end of a PhD thesis with publishable results.

It is also a part of the PhD program to follow some PhD courses, and our PhD students are expected to spend an extended period (up to 3-6 months) at a foreign research institution during the PhD program.

PhD students (except industrial PhDs) are required to teach, typically as teaching assistants. Our advanced courses are taught in English. Our PhD students must be able to teach in English and to follow courses taught in English.

A PhD in mathematics will make you qualified for a wide range of career possibilities including continuing an academic career or going into the private or the public sector.

PhD students who hold a stipend from the University of Copenhagen will receive funding as follows: Students enrolled in the 3-year program will receive a salary throughout the PhD period. Students enrolled in the integrated program will receive a number of state education grant portions until they acquire the master's degree 2 years before handing in their PhD thesis. For the remaining 2 years they will receive a salary.

Both stipends also include a travel allowance for participation in conferences and for research visits to foreign research institutions.

For more information about the PhD programme see https://science.ku.dk/phd/

The Department of Mathematical Sciences offers PhD degrees in pure and applied mathematics, statistics, insurance and financial mathematics within four research sections and two centres:

• Algebra & Geometry
• Analysis & Quantum
• Insurance and Economics
• Statistics and Probability
• Copenhagen Centre for Geometry and Topology (GeoTop)
• Centre for the Mathematics of Quantum Theory (QMATH)

The department has about 45 tenured faculty, 30 postdocs, and 60 PhD students.

Applying for a PhD

• After accepting a PhD stipend  
• Mini-guide on how to fill the online application form

For current PhD students

See MATHnet  (department intranet) for courses, economy, travel etc.

PhD students

Theodor Henningsen, PhD student

Silvan Vollmer, PhD student

Taro Spirig, PhD student

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Doctoral Positions at Mathematics Münster

Benefit from the stimulating atmosphere at the department and use the invaluable opportunity to learn from faculty and visitors, to collaborate and network.

The Mathematics Münster Graduate School at the University of Münster, Germany,  is seeking to fill up to nine positions for

Doctoral Research Associates Wissenschaftliche/r Mitarbeiter/in (salary level TV-L E 13, 75%)

to commence at the earliest possible date and no later than July 2024. Positions are available in all mathematical fields related to the work being undertaken at the Cluster of Excellence Mathematics Münster:  “Dynamics – Geometry –Structure” .

We are offering the fixed-term part-time positions (75% FTE) for 3 years. Some of the positions are funded by the German Research Foundation (DFG), for example through the Cluster of Excellence Mathematics Münster, and some by the mathematical institutes of the University of Münster. The financial terms of the positions remain consistent regardless of the funding source. The main selection criterion is academic excellence.

Your tasks: This position is tied to working towards a doctorate (Dr. rer. nat.) in any mathematical field related to the Cluster of Excellence Mathematics Münster. Successful candidates may be required to teach up to 4 hours per week during the semester (for full-time positions) depending on the source of funding.

Our expectations:

• A university degree with above average grades in mathematics or a related field that meets the qualifications required for a doctorate is essential.
• Research interests that align with or foster links between the research areas at Mathematics Münster are required.
• Above average mathematical ability, as well as intellectual and analytical strength are expected.
• A high level of motivation and enjoyment in scientific work, ambition and commitment is expected.
• Good English communication skills are necessary but German language skills are not a requirement.
• The ability to work independently and collaboratively in a team is expected.

Advantages for you: The successful candidates will profit from the amenities of the Cluster of Excellence and the Mathematics Münster Graduate School, including funding for conference travel or summer schools and opportunities to co-organise academic events and participate in numerous networking opportunities and events.

Application documents:

• Applications should contain a cover letter, CV, copies of degree certificates (bachelor’s and master’s) and full academic transcripts, as well as evidence of proficiency in English.
• Applicants are invited to include examples of their academic work if appropriate.
• Applicants should also nominate at least two potential local supervisors listed as investigators in a relevant research area at Mathematics Münster.
• Please arrange for at least two letters of recommendation to be sent directly to [email protected] by the referees before 1 November 2023.

The University of Münster strongly supports equal opportunity and diversity. We welcome all applicants regardless of sex, nationality, ethnic or social background, religion or worldview, disability, age, sexual orientation or gender identity. We are committed to creating family-friendly working conditions.

We actively encourage applications by women. Women with equivalent qualifications and academic achievements will be preferentially considered unless these are outweighed by reasons which necessitate the selection of another candidate.

If you have any questions, please contact [email protected] . Please also consult our answers to Frequently Asked Questions regarding the application process.

Are you interested? Then we look forward to receiving your application by 1 November 2023 via our online application form below (please note that we cannot consider other application formats).

Information on reference letters for referees

If possible, we kindly ask you to consider the following questions/points in your letter:

• Since when and in which capacity have you known the applicant?
• Please assess the applicant’s intellectual and analytic strength, their mathematical knowledge, their technical mathematical skills, their ambition and their independence.
• Please describe the applicant’s attitude towards academic work, their reliability and their ability for team work.
• Comment on the applicant’s proficiency in English.
• Would you accept this applicant as a graduate student at your own institution?

Letters of recommendation should be sent directly to [email protected] before the deadline. You will receive a confirmation of receipt.

The application phase is closed.

PhD position in mathematics or mathematical statistics, with focus on geometric deep learning

Job Information

Offer description.

The Department of Mathematics and Mathematical Statistics is opening a PhD position in mathematics or mathematical statistics, focusing on geometric deep learning. The position covers four years of third-cycle studies, including participation in research and third-cycle courses. The last day to apply is August 26th 2024.

Project description and tasks  Machine learning (‘artificial intelligence’) is having an immense impact on both society at large and research especially, and this impact is expected to increase. This boom is driven by so-called deep neural networks, a class of machine learning models proven incredibly powerful, versatile, and capable of solving many machine learning tasks. Mathematicians have taken huge steps towards theoretically understanding their empirical success, but many open questions remain.

A subfield within neural network theory is geometric deep learning. It concerns symmetries in the data or the learning task and constructing neural networks that react properly to them (equivariant networks). Examples of such symmetries are symmetries towards rotations of point clouds, translations of images or permutations of nodes in graphs. Combining the geometric/algebraic theory of (group) symmetries with the more analytical/statistical theory of machine learning allows for mathematically multifaceted research.

The project aims at deepening the mathematical theory of geometric deep learning. Exciting research questions include the development of new ways of constructing equivariant networks, describing the resulting models mathematically, and directly analyzing how symmetries affect the training of neural networks.

The project is affiliated with the AI/Math track within Wallenberg AI, Autonomous Systems and Software Program (WASP), and the PhD student will take part in the WASP graduate school.

Qualifications  The doctoral student will be admitted to study one of the two third-cycle programmes: Mathematics or Mathematical Statistics. To fulfil the general entry requirements, the applicant must have qualifications equivalent to a completed degree at second-cycle level or completed course requirements of at least 240 ECTS credits, including at least 60 ECTS credits at second-cycle level. 

To fulfil the specific entry requirements to be admitted for studies in either mathematics or mathematical statistics, the applicant is required to have completed at least 60 ECTS credits within mathematics or mathematical statistics, of which at least 15 ECTS credits shall have been acquired at second-cycle level. Applicants who have acquired largely equivalent skills in some other system, either within Sweden or abroad, are also eligible. 

Good programming skills (preferably Matlab or Python) and good written and spoken English knowledge are required. Documented knowledge and experience in machine learning, image analysis, probability theory, differential geometry, algebra, optimization, representation theory and functional analysis are merits. Note that you are not expected to have specialist knowledge in all of the above-listed fields. You will acquire knowledge as a part of your doctoral studies and be able to collaborate with others to complement your specific skill set.

You are expected to take an active role in this project and institutional work. You have a scientific mindset and are determined to continuously develop your skills and contribute to mathematical machine learning research. 

The assessment of applicants is based on their qualifications and ability to benefit from the doctoral study they will receive. 

About the employment  The employment is a full-time paid position, for a fixed term of four years full-time or up to five years when teaching part-time. The position is intended to result in a doctoral degree.  The position is intended to result in a doctoral degree. The main task of doctoral students is to pursue their third-cycle studies, including active participation in research and third-cycle courses and activities and courses at the WASP graduate school. The duties may include teaching or other departmental work, although duties of this kind may not comprise more than 20 per cent of a full-time post. Salary is set according to the salary ladder for PhD positions at Umeå University. Employment commences in the winter of 2024/2025 or by agreement. 

You will become a part of the young and growing research group Mathematical Foundations of Artificial Intelligence at the Department of Mathematics and Mathematical Statistics. Most junior and senior researchers in this group are connected to WASP and research topics similar to those with which this project is concerned. This provides opportunities for exciting collaborations, both internally and externally. For more information, see https://www.umu.se/en/department-of-mathematics-and-mathematical-statis…

The WASP graduate school is dedicated to providing the skills needed to analyze, develop, and contribute to the interdisciplinary area of artificial intelligence, autonomous systems and software. The graduate school actively supports forming a strong multi-disciplinary and international professional network between PhD students, researchers, and industry through an ambitious program that includes research visits, partner universities, and visiting lecturers. Read more: https://wasp-sweden.org/graduate-school/&nbsp ;

Application  Application should be made in our electronic recruitment system before August 26th 2024. Log in and apply using the button at the bottom of the page. The application must include the following documents written in English or Swedish: 

• a personal letter with a brief description of your qualifications and research interests. Motivate why you are applying for the position and how your qualifications and merits are relevant.
• a curriculum vitae.
• authenticated copies of degree certificates, diplomas or equivalent, including documentation of completed academic courses, received grades, and possibly other certificates.
• copies of relevant work such as a Master’s thesis or articles you have authored or co-authored. If the master’s thesis has not been completed before the application deadline, a summary of the master’s thesis project and current progress shall be included. The summary can be at most five pages, including figures and references.
• contact information for at least two reference persons.
• if available, GMAT, GRE, TOEFL or IELTS scores may be submitted.

Umeå University wants an equal environment where open dialogue between people with different perspectives lays the foundation for learning, creativity and development. The people working at the Department of Mathematics and Mathematical Statistics have diverse backgrounds and experiences, which we value. Work-life balance is essential for us: Employees are supported in caring for their mental and physical health, and we accommodate students taking parental leave during their studies – this is not uncommon. In this spirit, we encourage everyone to apply. 

Pursuant to Chapter 12 Section 2 of the Swedish Higher Education Ordinance (SFS 1993:100), the decision regarding the position cannot be appealed.

More information Further information is provided by Professor Jun Yu ( [email protected] ) or Assistant Professor Axel Flinth ( [email protected] ).

More information about the Department of Mathematics and Mathematical Statistics: https://www.umu.se/en/department-of-mathematics-and-mathematical-statis… ;

Wallenberg AI, Autonomous Systems and Software Program (WASP) is Sweden’s largest individual research program ever, a major national initiative for strategically motivated basic research, education, and faculty recruitment. The program addresses research on artificial intelligence and autonomous systems acting in collaboration with humans, adapting to their environment through sensors, information and knowledge, and forming intelligent systems-of-systems. The vision of WASP is excellent research and competence in artificial intelligence, autonomous systems and software for the benefit of Swedish society and industry. Read more: https://wasp-sweden.org/&nbsp ;

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Elaine Liu: Charging ahead

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MIT senior Elaine Siyu Liu doesn’t own an electric car, or any car. But she sees the impact of electric vehicles (EVs) and renewables on the grid as two pieces of an energy puzzle she wants to solve.

The U.S. Department of Energy reports that the number of public and private EV charging ports nearly doubled in the past three years, and many more are in the works. Users expect to plug in at their convenience, charge up, and drive away. But what if the grid can’t handle it?

Electricity demand, long stagnant in the United States, has spiked due to EVs, data centers that drive artificial intelligence, and industry. Grid planners forecast an increase of 2.6 percent to 4.7 percent in electricity demand over the next five years, according to data reported to federal regulators. Everyone from EV charging-station operators to utility-system operators needs help navigating a system in flux.

That’s where Liu’s work comes in.

Liu, who is studying mathematics and electrical engineering and computer science (EECS), is interested in distribution — how to get electricity from a centralized location to consumers. “I see power systems as a good venue for theoretical research as an application tool,” she says. “I'm interested in it because I'm familiar with the optimization and probability techniques used to map this level of problem.”

Liu grew up in Beijing, then after middle school moved with her parents to Canada and enrolled in a prep school in Oakville, Ontario, 30 miles outside Toronto.

Liu stumbled upon an opportunity to take part in a regional math competition and eventually started a math club, but at the time, the school’s culture surrounding math surprised her. Being exposed to what seemed to be some students’ aversion to math, she says, “I don’t think my feelings about math changed. I think my feelings about how people feel about math changed.”

Liu brought her passion for math to MIT. The summer after her sophomore year, she took on the first of the two Undergraduate Research Opportunity Program projects she completed with electric power system expert Marija Ilić, a joint adjunct professor in EECS and a senior research scientist at the MIT Laboratory for Information and Decision Systems.

Predicting the grid

Since 2022, with the help of funding from the MIT Energy Initiative (MITEI), Liu has been working with Ilić on identifying ways in which the grid is challenged.

One factor is the addition of renewables to the energy pipeline. A gap in wind or sun might cause a lag in power generation. If this lag occurs during peak demand, it could mean trouble for a grid already taxed by extreme weather and other unforeseen events.

If you think of the grid as a network of dozens of interconnected parts, once an element in the network fails — say, a tree downs a transmission line — the electricity that used to go through that line needs to be rerouted. This may overload other lines, creating what’s known as a cascade failure.

“This all happens really quickly and has very large downstream effects,” Liu says. “Millions of people will have instant blackouts.”

Even if the system can handle a single downed line, Liu notes that “the nuance is that there are now a lot of renewables, and renewables are less predictable. You can't predict a gap in wind or sun. When such things happen, there’s suddenly not enough generation and too much demand. So the same kind of failure would happen, but on a larger and more uncontrollable scale.”

Renewables’ varying output has the added complication of causing voltage fluctuations. “We plug in our devices expecting a voltage of 110, but because of oscillations, you will never get exactly 110,” Liu says. “So even when you can deliver enough electricity, if you can't deliver it at the specific voltage level that is required, that’s a problem.”

Liu and Ilić are building a model to predict how and when the grid might fail. Lacking access to privatized data, Liu runs her models with European industry data and test cases made available to universities. “I have a fake power grid that I run my experiments on,” she says. “You can take the same tool and run it on the real power grid.”

Liu’s model predicts cascade failures as they evolve. Supply from a wind generator, for example, might drop precipitously over the course of an hour. The model analyzes which substations and which households will be affected. “After we know we need to do something, this prediction tool can enable system operators to strategically intervene ahead of time,” Liu says.

Dictating price and power

Last year, Liu turned her attention to EVs, which provide a different kind of challenge than renewables.

In 2022, S&P Global reported that lawmakers argued that the U.S. Federal Energy Regulatory Commission’s (FERC) wholesale power rate structure was unfair for EV charging station operators.

In addition to operators paying by the kilowatt-hour, some also pay more for electricity during peak demand hours. Only a few EVs charging up during those hours could result in higher costs for the operator even if their overall energy use is low.

Anticipating how much power EVs will need is more complex than predicting energy needed for, say, heating and cooling. Unlike buildings, EVs move around, making it difficult to predict energy consumption at any given time. “If users don't like the price at one charging station or how long the line is, they'll go somewhere else,” Liu says. “Where to allocate EV chargers is a problem that a lot of people are dealing with right now.”

One approach would be for FERC to dictate to EV users when and where to charge and what price they'll pay. To Liu, this isn’t an attractive option. “No one likes to be told what to do,” she says.

Liu is looking at optimizing a market-based solution that would be acceptable to top-level energy producers — wind and solar farms and nuclear plants — all the way down to the municipal aggregators that secure electricity at competitive rates and oversee distribution to the consumer.

Analyzing the location, movement, and behavior patterns of all the EVs driven daily in Boston and other major energy hubs, she notes, could help demand aggregators determine where to place EV chargers and how much to charge consumers, akin to Walmart deciding how much to mark up wholesale eggs in different markets.

Last year, Liu presented the work at MITEI’s annual research conference. This spring, Liu and Ilić are submitting a paper on the market optimization analysis to a journal of the Institute of Electrical and Electronics Engineers.

Liu has come to terms with her early introduction to attitudes toward STEM that struck her as markedly different from those in China. She says, “I think the (prep) school had a very strong ‘math is for nerds’ vibe, especially for girls. There was a ‘why are you giving yourself more work?’ kind of mentality. But over time, I just learned to disregard that.”

After graduation, Liu, the only undergraduate researcher in Ilić’s MIT Electric Energy Systems Group, plans to apply to fellowships and graduate programs in EECS, applied math, and operations research.

Based on her analysis, Liu says that the market could effectively determine the price and availability of charging stations. Offering incentives for EV owners to charge during the day instead of at night when demand is high could help avoid grid overload and prevent extra costs to operators. “People would still retain the ability to go to a different charging station if they chose to,” she says. “I'm arguing that this works.”

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• Department of Mathematics

Related Topics

• Electrical engineering and computer science (EECS)
• Mathematics
• Laboratory for Information and Decision Systems (LIDS)
• Undergraduate Research Opportunities Program (UROP)
• Renewable energy
• Electricity
• Electric vehicles
• Undergraduate

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Softening the sharp edges in mathematics.

For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start — to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics — a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

“Mathematics should be flexible, conceptual, a place where we play with ideas and make connections," says Professor Jo Boaler. (Photo: Robert Houser Photography)

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced “How to Learn Math,” the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users. 

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer. 

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 — but the most common answer 13-year-olds gave was 19. The second most common was 21. 

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important? 

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mind-set. I think we all need a little more ish in our lives. 

You also argue that mathematics should be taught in more visual ways. What do you mean by that? 

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things — all of that contributes to our understanding of how it works. 

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking. 

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, ‘I’m in geometry class now, and I still remember that  sugar cube, what it looked like and felt like.’ His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing — elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

Click to enlarge: A depiction of various ways to calculate 38 x 5, numerically and visually. (Image: Courtesy of Jo Boaler)

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them. 

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

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'It was incredibly rewarding to tackle such difficult problems'

5/17/2024 A&S Communications

Andrew DiFabbio

Physics & Mathematics Brewster, N.Y.

Why did you choose Cornell?

Cornell has been host to a variety of discoveries and big names in my chosen field of physics, and it continues to be exceptional in that area. People like Kenneth Wilson, Richard Feynman, Freeman Dyson and Hans Bethe, number among the many great thinkers who have pushed the boundaries of physics at Cornell. This is not to mention Cornell’s excellence in experimental physics, having seen the discovery of such phenomena as superfluidity in 3He and hosting such facilities as the Cornell High Energy Synchrotron Source (CHESS) and the Cornell Center for Materials Research (CCMR), among others. As I see it, there is no better place for an undergraduate to study physics, including its many sub-disciplines.

What is your main extracurricular activity and why is it important to you? 

I am a member of the Big Red Marching Band. As part of the clarinet section, I have marched during football games and other events for the past three years. Having been a part of my high school’s marching band, this was one of the things that drew me to Cornell in the first place. More than anything, I appreciate the people in the band, and particularly those in my section, as a community that I can be a part of.

What Cornell memory do you treasure the most?  

During my freshman fall, when I took PHYS 1116 as my first physics course, I distinctly remember a bonus problem from the second problem set. It had to do with pursuit curves and described a scenario where one person chases another, with each moving according to the other’s position. Trying to solve the problem naively was very difficult, so the solution wasn’t immediately obvious. I remember going for a walk through the arts quad and looking up at the clock tower when it occurred to me that the problem admitted a simple geometric solution, and I went back to my dorm room on north campus to write it up. I could have chosen a number of different occasions like this, when a problem that seems impossible suddenly becomes trivial, but this was one of the first times I had that experience at Cornell, so I remember it fondly.       

What have you accomplished as a Cornell student that you are most proud of?

I was involved in research under two different faculty members, Professor Rob Thorne and Professor Carl Franck. My work with Rob Thorne concerned cryogenic electron microscopy, particularly the method of single particle analysis. Cryo-EM seeks to resolve the structure of protein complexes, typically to around 3Å resolution, and allows observing them in conformational states more faithful to their natural environments than is achievable with techniques like protein crystallography. Under Carl Franck, my work involved developing code to process data from a coincidence experiment; I vastly improved the performance of our data reduction algorithm and, along the way, developed a numerical method to characterize the time resolution of the detectors involved. I developed a variety of skills from these experiences, including cryo-EM sample preparation and familiarity with several numerical techniques. It was incredibly rewarding to be involved in the scientific process and to tackle such difficult problems.

How have your beliefs or perspectives changed since you first arrived at Cornell? 

On an intellectual level, I’ve always understood that sharing people’s company and perspectives is important for academia, just as it is in everyday life. However, my behavior didn’t always reflect this fact. In high school, it was fairly easy to be an island and work on everything alone; I’ve since realized the limits of this approach and understand the value in collaboration that simply can’t be achieved individually.

Every year, our faculty nominate graduating Arts & Sciences students to be featured as part of our Extraordinary Journeys series.  Read more about the Class of 202 4.

Diversity, empathy fuel national win for Men’s Fencing Club

'Applicable Algebra reshaped my understanding of math'

'I studied why miscommunication and ambiguity are so prevalent in human language'

'I've met a variety of people with a wide range of interests'

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2023 Spring EDUC 131AC 001 SEM 001

Why choose this programme?

Mathematics is useful. Math is fun. Maths sharpen the brain. Mathematics is part of our cultural heritage. Mathematics is beautiful.

About the programme

The Master's program in Mathematics is based on the bachelor's programmes Mathematics with Informatics and Mathematics and Economics. The master's programme gives you a basic introduction to mathematical axiomatic-deductive methodology. You learn to formulate stringent issues in a mathematical language suit, choose strategies for solving the problems, and provide logically accurate proof of your results.

The master's programme has two programme options:

• Mathematics
• Mathematics for Applications

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We know that students who are happy with their study environment are more likely to succeed with their studies.  Therefore we want you to enjoy being a student with us!

At the beginning of the study you have an opportunity to get to know both older and newer students and employees, in professional and social settings.  If you wonder about some practical aspect of your study, or need advice, please contact us.

The University of Oslo has a rich student life with more than 200 student associations. We encourage all our students to be socially active.  As a student you have access to pleasant study areas, both for studies and social activities, in near vicinity to the research environments.

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You can go on exchange through the agreements offered by the University of Oslo. We recommend that you go during the second, or maybe the third, semester. Another possibility is to spend time abroad during your thesis work.

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After the studies you will be well prepared for working in several fields both within the private and public sectors.  Many of our master's theses are directly related to the problems and activities of technological companies and institutions in Norway. Some of the master's students who obtain very good results choose to continue on a research carrier by taking a doctoral degree (PhD) at the University of Oslo or a university abroad.

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Are you academically ambitious? It is possible to supplement your master's degree with an honours certificate worth 20 credits. You take the certificate concurrently with the master's programme.

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Anders Brennhagen defended his PhD

Doctoral candidate Anders Brennhagen at the Department of Chemistry, Faculty of Mathematics and Natural Sciences, defended his thesis  " Bi metallates as conversion-alloying anodes for Na-ion batteries" for the degree of Philosophiae Doctor April 5th.

Trial lecture

April 5th, 10:15 AM, Auditorium 2, Chemistry building

Trial lecture title:

"Lithium vs. Sodium Batteries: Status, Challenges and Applications". 

Kreeringssammendrag/Conferral summary 

Natriumionebatterier kan være et billigere og mer bærekraftig alternativ til dagen litiumionebatterier, og kommer til å være en viktig del av fremtiden. Dette arbeidet har ikke skapt noen revolusjonerende gode batterimaterialer, men har utviklet metoder og kunnskap som kan fremme batteriutviklingen i årene som kommer.

Main research findings

Rechargeable batteries are essential for the green energy transition and Li-ion batteries (LIBs) are currently dominating the marked. Li and other elements used in LIBs are limited resources on the planet, which makes them relatively expensive. Na-ion batteries (NIBs) work similarly to LIBs, but with slightly lower capacities due to the higher mass of Na compared to Li. However, the much higher and widespread abundance of Na could make NIBs cheaper and more sustainable that LIBs, perfect for large scale stationary energy storage.

One of the major challenges with NIBs is to find a good anode material. In this work, we have not been able to do that. However, we have shown how a group of anode materials works during charge and discharge to a much greater detail than before, and understood why they do not perform so well. We used these results to improve the performance of the materials, but not enough to be of commercial interest. Nevertheless, the methods, ideas and knowledge developed in this thesis can be used for other materials and lead to a great improvements of NIBs in the future.

Candidate contact information

LinkedIn:  https://www.linkedin.com/in/anders-brennhagen-94276314b/

Email: [email protected]

Tel. +47 46947610

Adjudication committee

Professor Christian Masquelier University of Picardie Jules Verne, France

Professor  Dorthe Ravnsbæk Aarhus University, Denmark

Associate Professor Jonathan Polfus Department of Chemistry, University of Oslo, Norway

Supervisors

Professor Helmer Fjellvåg Department of Chemistry, University of Oslo, Norway

Professor Anja Olafsen Sjåstad Department of Chemistry, University of Oslo, Norway

Researcher David Stephen Wragg, Ph.D. Centre for Materials Science and Nanotechnology Chemistry, Norway

Associate Professor Vajeeston Ponniah Department of Chemistry, University of Oslo, Norway

R&D Senior Material Scientist Carmen Cavallo FAAM, Italy

Chair of defence

Professor Truls Norby Department of Chemistry, University of Oslo, Norway

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