Hiking and fishing in the Rocky Mountains were part and parcel of 10th HLF attendee Rachel Masters’ childhood. Nowadays, she taps into these experiences while developing the best possible immersive and therapeutic nature-based virtual reality (VR) experiences – forest landscape escapes through which people can take quick mental breaks. Read more

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>*The Heidelberg Laureate Forum has a single purpose: To provide some of the brightest minds in mathematics and computer science with the space and time to make connections and find inspiration. The HLFF Spotlight series shines a light on some of the brilliant young researchers attending the event, their background and research, as well as their expectations for the HLF.*

Hiking and fishing in the Rocky Mountains were part and parcel of 10th HLF attendee Rachel Masters’ childhood. Nowadays, she taps into these experiences while developing the best possible immersive and therapeutic nature-based virtual reality (VR) experiences – forest landscape escapes through which people can take quick mental breaks.

The energetic 23-year-old’s VR-based “forest bathing” project forms part of her PhD studies at Colorado State University (CSU) in the USA. It is funded through a National Science Foundation grant.

Rachel explains the rationale behind her studies as such: “Forest bathing is a therapeutic technique involving immersion in nature to reduce stress and restore mental resources. Unfortunately, it’s not readily available to city dwellers, for instance. They need a uniquely immersive, accessible platform that can provide benefits similar to nature. VR is a promising supplement as a virtual stress relief tool, especially since portable VR headsets are becoming more accessible and affordable.”

Besides programming and designing the virtual landscapes, she will – as part of her PhD studies – test the elements and level of design detail crucial to providing an overwhelmingly relaxing nature-based VR experience that improves someone’s mental health. This will include experimenting with how many different plant species need to be included in a virtual landscape for it to be realistically soothing.

“Since VR cannot fully replicate reality, it is critical to understand the essential components for creating effective virtual nature environments, or VNEs,” adds Rachel. “There are still gaps in the literature and many unanswered questions. We need to understand what is restorative about nature before we can understand what is important when designing VR environments. VR isn’t reality. It operates a little bit differently, and people perceive things in different ways. We need to understand how they understand real nature so that we can translate this to VR and apply it to different populations.”

To this end, she was the lead author of a paper on the topic at the 2022 ACM Symposium on Applied Perception, titled “Virtual Nature: Investigating The Effect of Biomass on Immersive Virtual Reality Forest Bathing Applications For Stress Reduction”.

As part of her PhD research, Rachel will also run experiments to establish the degree to which VR software can help office workers and students destress amid the strains of a working day.

“People often feel burnt out in the middle of the day. Having a ten-minute long recovery experience in your office instead of a power nap could help you push through the rest of the day,” she hopes.

Rachel admits that the biggest challenge so far has been to recruit enough study participants, and to ensure that they remain part of the programme until all data has been collected.

Ideally, this entrepreneur at heart would like to see the VR landscapes she is creating through her PhD become commercially available. She is already one step closer to this dream, after winning the Michael Best Most Promising New Venture Award at her university’s CSU Business Showcase pitch competition.

This star student, who received her undergraduate degree *magna cum laude* earlier this year, received an honorable mention in the National Center for Women & Information Technology’s Collegiate Awards, and a Goldwater Scholarship in 2022. She also has accolades such as the 2022 Best in Class Intern Award from Hewlett Packard Enterprise at Fort Collins, an Achievement Rewards for College Scientists Scholarship, and a Colorado State University Honors Cilento Scholarship to her name.

Rachel was first introduced to the VR field thanks to a hackathon hosted at Colorado State in 2019, during her first year as a computer science student. Her team was tasked with developing a music-based intervention for people suffering from the degenerative brain disorder Alzheimer’s disease.

The challenge spoke to her belief that computer science holds endless possibilities to address or at least alleviate a vast array of problems facing the world and its inhabitants.

“I thought it was super cool, almost like a miracle, to have the opportunity to make an impact on people who live with Alzheimer’s,” remembers Rachel, who has seen the devastating impact that the disease can have on people’s mental capacity.

Importantly, the hackathon introduced her to the research of Francisco Ortega and the Natural User Interaction Lab (NUI) at CSU. She was hooked and left no stone unturned to introduce herself to the man who today is her PhD supervisor.

She has since become a valued member of the NUI Lab, and she has been involved in numerous projects that have subsequently informed her PhD project. For her honours project, for instance, she researched the psychology of drug addiction and the potential use of VR to help people overcome their drug habits.

When not working on her own VR forest bathing ideas, Rachel oversees projects in the NUI Lab on how the aesthetics of a VR nature landscape can help reduce users’ stress levels, especially older people, and how VR can form part of a package of solutions to help people cope with post-traumatic stress disorder (PTSD).

She co-authored a workshop position paper presented at CHI 2023 along with Dr Ortega and Dr Victoria Interrante that advocates for more research to be done on the use of smell and temperature technologies to enhance virtual reality nature immersion.

She is also assisting with a project that looks into the use of augmented reality to improve the training of drone pilots.

“VR can be a great educational tool. It gives you the chance to practice things firsthand, without actually doing it firsthand,” she says.

Given her research focus, Rachel would have loved to have made a beeline to an HLF regular, laureate and computer graphics pioneer Ivan Sutherland, come the 10th HLF. Sutherland, who co-designed the first virtual reality headset, the “Sword of Damocles”, in 1968, and won the Turing Award in 1988, will however unfortunately not be able to attend this time round.

“He is an inspiration to the field I am in,” Rachel acknowledges.

She is excited about listening to the laureates as they tell stories about their own contributions, and to learn from them insights on how she too can make the most meaningful contributions to her field.

“I hope to leave inspired by all of the great research going on and take that inspiration back to my own research to help innovate in the field,” adds Rachel, who is looking forward to meeting innovators in computer science and mathematics at the 10th HLF.

“I am especially excited to hear from Whitfield Diffie and Martin Hellman about their work on asymmetric public-key cryptography, as computer security and innovation in secure systems pique my curiosity.”

Rachel describes herself as someone who is “passionately curious” about her research field. However, there’s more to her than just programming, virtual worlds, and screen time. She still prefers an actual nature hike to donning a VR headset.

She’s a hands-on fiber artist who likes crocheting, fiber dyeing, weaving, and felting. She is also a seamstress of note who is not scared to take up the challenge of making or altering a wedding dress or evening gown. The undergraduate business management minor and certificate in entrepreneurship she earned previously come in handy in the management of this sideline business.

All the while, her endeavours are fueled by her personal mission statement: “Strive to exceed your personal best in body and mind.”

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>Ghanaian PhD student Francis Saa-Dittoh is a ferocious reader of sci-fi books. His reading habits have taught him that technology-wise almost anything imaginable is possible. However, his practical involvement in the field of information and communication technology for development (ICT4D) has taught him that what is possible isn’t always appropriate. Read more

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>*The Heidelberg Laureate Forum has a single purpose: To provide some of the brightest minds in mathematics and computer science with the space and time to make connections and find inspiration. The HLFF Spotlight series shines a light on some of the brilliant young researchers attending the event, their background and research, as well as their expectations for the HLF.*

Ghanaian PhD student Francis Saa-Dittoh is a ferocious reader of sci-fi books. His reading habits have taught him that technology-wise almost anything imaginable is possible. However, his practical involvement in the field of information and communication technology for development (ICT4D) has taught him that what is possible isn’t always appropriate. He therefore believes that those designing technological options to solve some of the world’s pressing developmental issues must always take note of the needs, abilities, and resources of the communities they aim to serve and support.

“Too many projects have failed because people have not followed this simple rule,” notes the 41-year-old Francis, who works part-time on a PhD in Computer Science, with a specific focus on ICT4D, through the Vrije Universiteit Amsterdam in the Netherlands.

“If a community, for instance, generally uses mobile phones to make voice calls, it doesn’t help to develop apps for smartphones that need a good internet connection. It will fail,” he says by way of example.

As early as primary school, Francis knew that the IT world would be his future. He encountered his first computer in his father’s university office. He used the opportunity of subsequent visits to teach himself the basics of programming, with the help of whatever manual was at hand. He has since become the go-to IT guy for his family and friends.

After school, Francis completed a BSc degree in Computer Science in 2006 at the Kwame Nkrumah University of Science and Technology in Ghana. One of his undergraduate projects saw him co-design a 3D game with Eyram Tawia of the Ghanaian gaming studio Leti Games. A gaming historian later listed it as the first of its kind in Africa.

Earlier this year, Francis was appointed as a lecturer in the new Department of Computer Science at the University for Development Studies (UDS) in the Northern Region province of Ghana. In this role, he has helped develop a new BSc curriculum that introduces students to core concepts around ICT4D.

“UDS is one of the first African universities to have such a course,” Francis notes.

His relationship with UDS stretches back to 2007, when he was hired as an IT specialist. He has since supported the endeavours of many research groups in some way involved in developmental work – be it in agriculture, education, medicine, or ICT.

Ghana’s northern provinces are recognized as some of the least developed areas in this West African country of 34 million people.

“The average literacy rate for Ghana is well over 80%, but in the northern parts, where UDS is situated, this drops to around 32%,” he explains. “A lot still needs to be done to bring the rural areas in the north closer to the national norm.”

Francis’ own formal foray into the world of ICT research came through his completion of a MSc in Computer Science (Internet & Web Technologies) through the Web Alliance for Regreening Africa research group at the Vrije Universiteit Amsterdam in the Netherlands in 2013. He is now working within the same research group on his PhD. He is investigating digital information access for resource-constrained environments.

He is currently the principal investigator of the TIBaLLi (which means “our language” in Dagbani) Project. The effort uses artificial intelligence (AI) and machine learning to bring locally relevant information about farming practices, weather conditions, commodity prices and such to local communities. It also forms part of efforts to “decolonize the internet” as the aim is to provide locally relevant information in terms of its contents as well as the technology and languages used.

According to Francis, such work is needed, as many local Ghanaian languages are so small that they rarely feature on the priority list of major software companies’ computerized translation services.

The TIBaLLi Project will provide pre-recorded local content information to rural communities speaking the Dagbani language in the semi-arid Sahel region of northern Ghana. Pre-recorded local radio broadcasts are used, because community members in this isolated area have ready access to radios. Interaction with the system by farmers in these rural communities will be achieved using AI (and Natural Language Processing in particular) as well as a domain-focused speech vocabulary. The latter has already been crowd-sourced among Dagbani speakers as part of TIBaLLi’s experimental field work.

“Electronic delivery of solid information in speech and local language to communities beyond the current boundaries of the internet is new. This project will point to internet information beyond the current internet,” he explained in a recent article about TIBaLLi on the website Perspectives on ICT4d.org.

In it, he pointed out that the internet is not always the most important or appropriate vehicle through which to solve the so-called “digital divide” of the Global South. Issues of language and the relevance of content must be addressed.

Earlier this year, the TIBaLLi project received a $100,000 research grant from the Internet Society Foundation as part of efforts to decolonize the internet. He is partnering with colleagues from UDS, the Internet Society Ghana Chapter and Vrije Universiteit Amsterdam.

“I wouldn’t call myself a language activist, although I have in recent months learnt a lot about language issues through this project. I’d rather call myself an ICT4D champion for change,” says Francis, who speaks English, his mother tongue of Frafra, and a bit of Twi, Dagbani, and Dutch.

Francis’ first practical foray in ICT4D work began in 2019 through two Development of Information Access System projects. One called Mr. Meteo was a voice-based weather forecasting system. Users made Global System for Mobile Communication (GSM) voice calls to the system to receive information. Another, named Tibaηsim (originally called RadioNet), broadcast information from a low-cost Raspberry Pi computer over FM frequencies. It came third in the Internet Society’s international Chapterthon Competition in 2019, after teams from the USA and South Africa.

He acknowledges that he learnt many of his most valuable lessons about community engagement and information sharing from his father. Prof. Joseph Saa Dittoh, also at UDS, is widely recognized for his work in agricultural development.

Francis is a music lover who composes and performs gospel songs. Therefore, it’s no wonder that he would also use his IT prowess to further the work of Ghanaian authors and musicians through his involvement in developing two apps, Woom and Sefarim.

Woom is a customized mobile music app that will deliver Ghanaian music to music lovers. A Beta version of Sefarim is already available on the Google Play Store. It is a customized Android-based provider of eBooks by local Ghanaian authors.

“We are working on these apps because too often local content is drowned out by international voices. It makes it difficult to find Ghanaian authors and musicians on the web,” he explains.

He has previously provided IT services to help develop a website and databases in a United Nations Development Programme project, UNDP/GEF CLME+, to help manage the Caribbean and North Brazil Shelf Large Marine Ecosystems. On the business side, he helped to build a Ghanaian e-commerce site, Hypestore. Currently he is developing Python-based financial processing modules for a Dutch firm.

Francis, who also serves as a pastor in his local church, clearly likes keeping busy.

About the many projects that he has on the go, he just smiles: “My wife and mother may tell you there’s too many, but I’m always up for a challenge.”

It’s no wonder that Francis is looking forward to the upcoming 10th Heidelberg Laureate Forum – from meeting other researchers in his field and enjoying the social events to gaining insights about his field from this year’s laureates.

“There is always an eye-opening dimension to hearing their insights rather than just reading about them,” Francis notes. “I’d like to pick Raj Reddy’s brain about the potential use of artificial intelligence in resource-constrained areas of the world and gain insights from Vinton Gray Cerf on how the (re)design of hardware and software protocols can influence the decolonization of the internet.”

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>Dilrukshi Gamage, a postdoctoral fellow from Sri Lanka who studies how society is influenced by technology, is already an HLF 'veteran'. She attended the 5th HLF in 2017, midway through her PhD in Computer Science. Since then, she has carried with her the following advice offered by ACM A.M. Turing Award recipient and cryptologist, Martin Hellman: “The interesting thing in research is to be curious rather than furious.” Read more

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>*The Heidelberg Laureate Forum has a single purpose: To provide some of the brightest minds in mathematics and computer science with the space and time to make connections and find inspiration. The HLFF Spotlight series shines a light on some of the brilliant young researchers attending the event, their background and research, as well as their expectations for the HLF.*

Dilrukshi Gamage, a postdoctoral fellow from Sri Lanka who studies how society is influenced by technology, is already an HLF ‘veteran’. She attended the 5th HLF in 2017, midway through her PhD in Computer Science. Since then, she has carried with her the following advice offered by ACM A.M. Turing Award recipient and cryptologist, Martin Hellman: “The interesting thing in research is to be curious rather than furious.”

His words have since focused her while she weathered the logistical and administrative struggles that are typical of the career of an up-and-coming researcher, and one working in a field fraught with controversy and ethical dilemmas to boot. This includes studies not going to plan, and bracing oneself for the storm of reviewers who do not fully appreciate the value of such work.

“This mantra emphasizes to me the value of approaching challenges with an open, inquisitive mindset instead of becoming frustrated. It really encourages me to channel my energy more effectively into exploring, adapting, and collaborating. This has ultimately led to more meaningful discoveries and a positive research experience,” explains Dilrukshi, who in 2021 received her PhD from the University of Moratuwa in Sri Lanka.

As a postdoctoral fellow at the Tokyo Institute of Technology in Japan, she is now part of the research team of Kazutoshi Sasahara, an expert in computational social science and data analysis. Her current research is supported by the Japanese Science and Technology Agency (JST-CREST).

“Ethical considerations must be taken into account when adopting new technologies, in an effort to mitigate possible harm to society,” says Dilrukshi.

She studies how to better detect AI-generated images, soundbites, and deepfakes, as well as how to counter misinformation and disinformation. Her work includes finding ways to improve the discourse and landscape analysis around these topics through digital traces by incorporating tools and methodologies inherited from computational social science and human-computer interaction. This is done to track how society and its processes react to digital shenanigans. Dilrukshi also conducts quantitative and qualitative ethnographic and computational research on digital communities such as Reddit and Facebook, as well as on large-scale open learning communities such as Massive Open Online Courses (MOOCs).

For a recent presentation at the 8th International Conference on Computational Social Science (IC2S2 2022) in the USA, for example, Dilrukshi and a colleague analysed how people discuss deepfakes on social media sites like Twitter and Reddit. The word “deepfake” (which refers to the use of deep learning programmes to create fake images or sound) was first used on a Reddit forum in 2017, along with pornographic videos altered using machine learning algorithms. Deepfakes were also the topic of a chapter Dilrukshi was the lead author of in the book *Frontiers of Fake Media Generation and Detection*, published by Springer in 2021.

She worries that people too readily forward so-called “too good to be true” images or audio clips without checking their validity.

“It’s not that people don’t know how to find the truth. There are tools which can easily do so. It’s that people are not cognitively trained to think whether some content may be true or not, or do not have time to evaluate whether there is truth in it,” Dilrukshi explains.

Dilrukshi and others working on the project leverage human-computer interactions and computational social sciences to derive insights and design tools and technologies. They hope their analysis of data and trends will help the world stay one step ahead of criminals and pranksters. She is increasingly worried about how sophisticated AI-generated voice technology is becoming.

“AI-generated voice technology can make you say things that you never did, without a person listening on the other side of a phone even realising that it’s not you,” she explains, ever worried that especially elderly people might easily be fooled when for instance receiving fake voice notes asking for help from fake “family and friends”.

She foresees that people will increasingly have to become “old school” in their interactions with others again, and in important cases may gradually have to swap screen-based meetings for in-person interactions to counter the possibility of being conned.

Dilrukshi estimates that she had memorable chats with roughly 60% of young researchers attending the 5th HLF in 2017. She hopes to improve on her record in 2023.

She still fondly remembers swapping stories about Sri Lanka with British laureate Tony Hoare (the 1980 Turing Award recipient), who was born in Sri Lanka.

Dilrukshi says that going into her first HLF, she did not fully grasp how many opportunities to informally engage with laureates there would be on the programme.

“There were far fewer formalities than I expected,” she remembers. “It was really such a privilege to connect with the laureates of prizes in our field that are on par with the Nobel Prizes. It was fascinating to listen to their private stories, and to receive real-life advice about more than just academic matters.”

“What I learnt from their life stories was that none of them set out to become laureates. They had a vision and they had passion. And then something like prizes might come to you.”

The people-centred Dilrukshi hopes to have similar inspirational interactions when visiting Heidelberg this year, and to reconnect with laureates she met in 2017, such as Martin Hellman, Leslie Lamport, and Vinton Cerf. As a woman in science, she is eager to meet Yael Tauman Kalai, who in 2022 received the ACM Prize in Computing for her work in the field of cryptology.

“As a postdoc, I hope to receive advice from experienced researchers on career transitions and life hacks that will help me grow as a scientist and to conduct impactful research. Finally, as a Sri Lankan from the Global South, I hope to share my experiences and knowledge about the rich culture of my part of the island,” she adds.

Dilrukshi believes in working in pursuit of a specific vision, as it provides purpose to one’s life. To this end, she is glad that she can through her scientific endeavours work towards a meaningful purpose, even if she only adds small bits of change to the greater whole.

She describes herself as “an advocate of open research, open science, open access, open data, and open education.” She is thankful that the availability of open resources, as well as collaborative, data driven, and web-based research efforts – not to mention online training opportunities – have allowed her to fulfil the multiple roles of an involved mother, an active, internationally engaged researcher, and an activist for women in science – all this mostly from her home base in Sri Lanka.

She lives in the town of Piliyandala with her two teenage daughters and husband, engineer Manodha Gamage, with whom she partners in a technology consultancy company. Because of COVID-19 regulations that were in place until recently, she was able to spend the greater part of her postdoctoral fellowship working from home.

“Open science practices empower me to be a hardworking, helpful and quality researcher who can make a meaningful contribution to the world,” says Dilrukshi, who is prepared to share all her data and methods based on the principles of open science.

During the first years of her PhD, she was, for instance, part of a crowdsourcing research project led by Michael Bernstein of Stanford University’s Human-Computer Interaction (HCI) Group. The team worked towards prototyping, designing, and experimenting with a scalable community-led open platform that gig worker communities could manage themselves.

She is currently an affiliate at the Berkman Klein Center for Internet & Society (BKC) at Harvard University, having been a participant in its Research Sprint since 2020. She is also involved in the Credibility Coalition, a research community that fosters collaborative approaches to understanding the veracity, quality, and credibility of online information to further civil society.

These online endeavours have subsequently opened doors to conferences and training opportunities elsewhere in the world, and she has therefore been able to visit countries such as Sweden, South Korea, the USA, the Netherlands, the UK, Italy, Germany, India, and Japan.

In 2021, Dilrukshi won the award for the best paper presented at an Asian CHI symposium that formed part of the endeavours of the annual ACM Conference on Human Factors in Computing Systems (CHI). It highlighted best practices on how to leverage a sense of community when following online MOOC training sessions. In this regard, Dilrukshi has plenty of personal experience, having already completed more than 50 MOOC topics on seven training platforms.

Dilrukshi only started working towards her PhD after the birth of her second daughter. She admits that the responsibilities of studying and being the mother of two children are challenging, even at the best of times. However, she sees it as an opportunity to inspire not only her own daughters, but also other young women who might be considering careers in the sciences or elsewhere. To this end, she has, for instance, served on committees in the Society of Learning Analytics Research (SoLAR) and the ACM Special Interest Group for Computer-Human Interaction (SIGCHI) focussed on matters of diversity and inclusion. She served on programme committees for the IEEE Women in Engineering (WIE) International Leadership Conferences from 2019 to 2021 and is involved in IEEE activities in Sri Lanka.

Through her interactions with the global SIGCHI, she also founded and chaired a chapter in Sri Lanka in 2019 to bring together researchers working in the field of human-computer interaction.

Dilrukshi shows that a woman in academia – and especially in the STEM fields – does not have to choose between motherhood or a research career. She believes one can be a researcher who is curious about life and work from the home front, yet still be taken seriously on the world stage.

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>To Matias Palumbo, mathematics is beautiful, and even artistic; rigorous and complex, yet also simple – and often-times all these descriptions at once. He is one of the 200 young mathematicians and computer scientists from across the world who will have the opportunity at the 10th Heidelberg Laureate Forum to learn from recipients of some of the most prestigious prizes in their respective fields.

Matias hails from Argentina – the same country as Luis A. Caffarelli, who in 2023 became South America’s first ever Abel Prize winner.

Read more

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>To Matias Palumbo, mathematics is beautiful, and even artistic; rigorous and complex, yet also simple – and often-times all these descriptions at once. He is one of the 200 young mathematicians and computer scientists from across the world who will have the opportunity at the 10th Heidelberg Laureate Forum to learn from recipients of some of the most prestigious prizes in their respective fields.

Matias hails from Argentina – the same country as Luis A. Caffarelli, who in 2023 became South America’s first ever Abel Prize winner.

“It’s an incredible inspiration to know that someone from a city just four hours away from mine has received this highest possible honour,” the 22-year-old enthuses about the achievements of Caffarelli, who was born in Buenos Aires and is associated with the University of Texas at Austin in the USA.

For Palumbo, the city of Rosario in the province of Sante Fe is home. There he is working towards a Bachelor’s degree in the Department of Mathematics at the local National University of Rosario, after initially enrolling to study computer science.

In his own way, Matias has already made his mark on Argentinian mathematics. In 2022, he received the first prize in the National Monograph Contest of the Mathematical Union of Argentina. The monograph, titled “Transcendental Number Theory”, opened doors for him so that he could present his work at the Union’s annual national congress.

Also in 2022, he received a Friends of Fulbright 2022 scholarship from the Fulbright Association. It allowed him to participate in a seven-week cultural and academic exchange program at the University of Arkansas in the USA – an opportunity that not only proved to be valuable in terms of networking with peers from across the world and developing extracurricular skills, but which also allowed him to delve into historic or cross-disciplinary aspects of mathematics as part of weekly seminars presented by fellow students.

His undergraduate thesis work in functional analysis is part of a research project carried out in the French-Argentine International Center of Systems and Information Sciences (CIFASIS) inside Argentina’s National Science and Technical Research Council (CONICET). It focuses on the dynamics of linear operators and Hardy spaces of Dirichlet series.

Matias sees a future in mathematics for himself and would one day like to pursue a PhD at a European university. This might even allow him to walk in the footsteps of another of his mathematical heroes, the 1700s Swiss mathematician Leonhard Euler.

“He was a very prolific, inspirational teacher, and I like the picture that has been painted of him in writings,” Matias says.

He explains the decision he took in his earlier undergraduate years to change course from computer science to mathematics as such: “It wasn’t that I didn’t like computer science any more; I just realised that I enjoyed mathematics more. I realised that what I wanted to do in the future had everything to do with mathematics. Like it was my calling.

“Maths is very exact and very rigorous, but it’s also very artistic in a way and very complex, yet often incredibly simple. It’s something I deeply enjoy. I can just sit for hours on end thinking about doing that. That’s something I truly enjoy doing,” he adds.

That said, Matias reckons others would be mistaken to box him into being a “pure mathematician”. That’s because so much of his evolving research interests, part-time work and outreach activities are linked to computer science, as well as his interest in programming and developing new ways of communicating science.

These broader interests and skills stand him in good stead as part of the Von Neumann Group, an initiative at his university under the leadership of HLF alumnus Dr. Demian Goos, who will also be attending this year’s HLF. The Group’s endeavours combine art and video games within the framework of different “rooms” to deconstruct different areas of mathematics, and to show how close it is to most aspects of everyday life. He hopes to one day have enough time to complete a “room” based on Mandelbrot fractal shapes.

As a teaching assistant at his home university, Matias tries his best to make aspects of Mathematical Analysis as understandable as possible to the first-year students whom he supports. Some of these ideas have found a home in his latest endeavour, a YouTube channel called Mathium. Matias describes it as part of “a college student’s efforts to make mathematics as interesting and visually compelling” as possible. The board game Monopoly, for instance, inspired a video that explains probabilistic models, and how to calculate a player’s chances of winning given the properties that they own or that they aspire to buy. To this end, he was inspired by Grant Sanderson’s mathematical YouTube channel, 3Blue1Brown.

Matias says he is ambitious and curious by nature. He is therefore keeping an open mind on what his future research field could be. He is – among other things – intrigued by the combined interdisciplinary practice of maths and computer science, and how the two fields support each other. To this end, he has worked as a support engineer for the payment software company Paddle since 2022, in charge of analyzing reports of discrepancies in the platform’s data and debugging potential errors in the codebase.

He is however by no means an “all work and no play” kind of guy. When Matias is not studying or working on maths related projects, he likes to read, play volleyball, and sing popular songs.

“And a fun fact is that I am one of a set of triplets,” Matias divulges.

And no, don’t even ask when you meet him at the 10th HLF: The triplets are by no means identical, as Matias has two sisters. One is studying to become a nurse, and the other is a fashion designer for a streetwear brand in Rosario.

The South American student is looking forward to what awaits him in Heidelberg: “From what I’ve gathered, I expect the 10th HLF to be an incredible experience. l believe that it will be eye-opening and very enriching to meet students and researchers from all around the world that are as passionate about their endeavours as I am.

“This year’s laureates are some of the brightest minds there are, and I know meeting them and engaging in conversation with them is going to be enlightening in many ways. I expect that not only will it be enriching to talk to them because of their trajectory, but it will also be inspiring to actually meet the people behind such achievements.”

There are three laureates with whom he’d especially like to rub shoulders.

“They are Vinton Cerf, who made incredible contributions towards building the foundations of the modern Internet; Sanjeev Arora, with whom I share the enthusiasm for developing the mathematics behind deep learning models and rethinking the traditional frameworks behind machine learning as a whole; and Leslie Valiant, who first introduced the Probably Approximately Correct (PAC) learning model which laid the theoretical groundwork for machine learning, and whose ideas I find deeply interesting.”

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>Interested in community outreach projects? Then make sure to brainstorm a few ideas at the 10th HLF with upcoming attendee Victoria Sánchez Muñoz over a round of tic-tac-toe. Community engagement is one of the (many) aspects about her PhD studies on quantum games at the University of Galway (UoGalway) in Ireland that the 29-year-old Spanish physicist enjoys the most. Read more

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>Interested in community outreach projects? Then make sure to brainstorm a few ideas at the 10th HLF with upcoming attendee Victoria Sánchez Muñoz over a round of tic-tac-toe. Community engagement is one of the (many) aspects about her PhD studies on quantum games at the University of Galway (UoGalway) in Ireland that the 29-year-old Spanish physicist enjoys the most.

She uses an adapted version of the traditional pen-and-paper game of tic-tac-toe (also known as noughts and crosses, or Xs and Os) to workshop the basics of quantum physics with Grade 4 to 6 learners. They are given a quick tour of devices that were developed with input from the quantum physics field, such as LED lights and the lasers in supermarket scanners. A brief history lesson on computers and the quest to develop faster, more secure, and ultimately more superior quantum computers forms part of the workshop programme.

Victoria enjoys watching how willingly learners tackle unfamiliar problems and seeing how the experience of free play – within the realm of quantum physics – ignites the inner scientist in children. She hopes that some of them will one day consider a career in the sciences.

“It’s fantastic to see how children start playing without really knowing the rules. Their faces light up when they begin to work out how the game works when quantum principles are involved,” reflects Victoria, in whose vocabulary the word “fun” regularly crops up. “I’ve done the workshop with adults too, seldom with that same reaction. Adults are just super boring!”

Her workshop’s roots are to be found in a team challenge that Victoria participated in during an online quantum game jam in September 2021. She helped to develop a new educational quantum computer game called Heisenberg’s Bicycle.

After the event, she remembered a paper that proposed that it would be possible to play tic-tac-toe using quantum rules. She subsequently challenged herself to tackle it as her “Christmas 2021 Project”, while on vacation at home in the village of El Pinós in Alicante, Spain. By then Victoria had already investigated her fair share of simple quantum games as part of her PhD thesis that combines quantum physics and game theory under supervision of Dr. Michael Mc Gettrick of the School of Mathematical and Statistical Sciences at UoGalway.

She readily admits that working on a quantum-based version of tic-tac-toe while holidaying was more time-consuming than she had anticipated. Using what available spare time she had after the December break, she only “cracked the code”, so to speak, four months later.

When the opportunity to develop school-based workshops came along, however, it proved to have been time well spent. As member and president for two years of the UoGalway Student Chapter of the Society for Industrial and Applied Mathematics (SIAM) she has since presented it at nine primary schools in Galway.

Victoria’s first ventures into outreach work started in 2013, when she and five university friends founded the Spanish educational YouTube channel “Tippe Top Physics” (now known as “Gastrofisica“).

“It was fun to do, and people liked it,” she remembers. “I realised that outreach was fun.”

During the COVID-19 pandemic, she again reached for this medium, when as part of her School’s Outreach Committee she helped produce an explanatory video about Möbius strips for the 2020 Galway Science and Technology Festival. Afterwards, she participated in the online version of FameLab Ireland.

“Perhaps my interest in science communication comes from the fact that I’ve always had to explain my work,” suggests Victoria.

Victoria, who is a teaching assistant to undergraduates in mathematics and statistics at UoGalway, also tutors modules in linear algebra and engineering mechanics. Previously, she had also tutored a student who lives with disabilities and helped older students with their courses as part of the University’s Access Centre.

She admits that she generally says yes when asked whether she’d like to participate in an event or be involved in a project.

“I know I should say ‘no’ more often, because I’m in the last stages of my PhD, but …”

Her handful of years in Galway, a city of 80 000 people in West Ireland with a strong student contingent, have taught her the value of opening the proverbial door when opportunity knocks.

“Anyone can get a PhD, but not everyone is for instance able to do outreach, go to schools and interact with children. I am grateful that I am allowed to do so.”

She makes a point of pursuing personal development goals and has for instance recently followed basic courses or workshops on CPR skills, the handling of sexual violence, and ways to turn research results into podcasts. She is also learning German and hopes to practise it more during the 10th HLF.

“Take advantage of opportunities for free learning that your university might offer. Once you are in the workplace, you’ll have to pay for such courses,” she motivates others to venture beyond the borders of their academic programmes. “And in any case, you cannot use your brain all the time just for your studies!”

Victoria comes from a family of teachers and nurses and enjoys tennis, reading, visiting the gym, and the outdoors.

She describes the story behind how she decided on a career in physics as “very cheesy, but all true.”

A career in the sciences was always a natural choice, because of her knack for maths and biology. By her last year in high school, when the subject matter became more interesting and challenging, physics began to enthrall her.

“One evening I was so busy solving a question about gravitational laws and its influence on a planet and its satellites that I completely forgot to have supper. I realised that if something could be so interesting that I’d skip a meal for it, it was worth pursuing as a career!”

She subsequently completed a Bachelor’s degree in Physics at the University of Valencia in 2016, followed by a Master’s degree in Theoretical Physics about entanglement and geometry at the Autonomous University of Madrid in 2017.

In 2018, Victoria was appointed as a junior programmer for a consultancy firm in Barcelona. She was soon underwhelmed by the experience of working in the private sector.

“I loved my colleagues and the city, but it just felt as if I was making rich people even richer,” she explains her subsequent decision to further her PhD studies. Now in its final stages, she hopes to submit her thesis by the end of 2023.

She loves the small “eureka” moments that are part and parcel of academic life and doing research. However, she admits that her greatest highlight yet – and also her biggest disappointment – has nothing to do with her PhD work.

It saw Victoria test her mettle against an insurance company unwilling to pay out when wind severely damaged her outdoor venetian blinds and her mother’s car was scratched in the process. The insurance company held fast: They only covered damages caused at wind speeds of 75 km/h or more, but it had only blown 73km/h on the night of the incident, the company claimed.

Their verdict was like the proverbial red muleta of a Spanish bullfighter, and Victoria went into “full research mode” to try and prove the company wrong. She downloaded weather and wind data from official sources. She then went to great lengths to use physics and geometry principles to prove that the wind had indeed blown stronger.

Emails went back and forth. Months later, a lawyer eventually informed her that her report could not be fully considered because she was not a qualified expert.

“The system wore me down, so I decided to finally let it go,” sighs Victoria, still not convinced that the decision was fair nor based on sound scientific principles.

As a form of academic therapy, she wrote an article about her experiences, calculations, and findings, eventually submitting it for the Graham Hoare Prize 2022, by which the Institute of Mathematics and Its Applications (IMA) in the UK supports the work of early career mathematicians. Even though the insurers did not value her work, her academic superiors did.

Victoria won first prize, and her article “The estimation of wind speed: Challenging the insurance company’s decision” was published in the December 2022 issue of *Mathematics Today*.

“And that paper will remain online forever,” she states proudly.

“I must confess, though: As is usual in life, I enjoyed the (research) journey even though the result was a bit disappointing.

“Science can be fun. Science should be fun. Life is about more than doing research or completing your PhD as fast as you can. The world is so big. Explore it,” enthuses Victoria.

It’s not only for the opportunity to practise her German vocabulary that Victoria is looking forward to the 10th HLF.

She’s looking forward to learning from as many laureates as possible about their expertise and experiences. There are two in particular who work on topics that interest her: Fields Medallist 2022 Hugo Duminil-Copin, and ACM Prize in Computing 2022 recipient, Yael Tauman Kalai.

“I’m very excited to learn new things in terms of maths and computer science from the laureates themselves and the young researchers. But mostly I hope to meet, learn from, and connect with everyone there; and to enjoy Heidelberg. In a word, I just expect to have fun.”

The post HLFF Spotlight: 10th HLF originally appeared on the HLFF SciLogs blog.

]]>The post Circling the Square originally appeared on the HLFF SciLogs blog.

]]>It has a hole down the middle, and instead of having one open edge around the outside, it now has two separate open edges, one at the top and one at the bottom.

Continuing to modify the square in this way leads to several more interesting mathematical shapes.

If we want to make things more interesting, we could consider what happens if we join the top and bottom of the square too – sticking the left and right edges first gives a cylinder, so then proceeding to stick the top and bottom edges is equivalent to glueing the two ends of the cylinder together. As I recently discussed in my post on Gaussian curvature, this would not necessarily be easy to do with a paper cylinder, since rolling it one way stops it curving the other way – but if it were a rubber square, we could bend it round and stick the ends of the tube together.

This would give us a new kind of surface – a donut shape, called a **torus** – which has no open edges at all, but has a different hole through its middle. Toruses are hugely important shapes in the mathematical field of **topology**, and any simple shape with one hole in it can be considered equivalent to the torus: this is the origin of the hilarious joke about how a mathematician cannot tell the difference between a donut and a coffee cup. Since explaining a joke always makes it funnier, I can clarify that this is because both have one hole: the hole in the torus, and the handle on the cup (the part you put the coffee in is not technically a hole – it is more of a large dent).

Since they are effectively a circle stretched out around a second circle, toruses can also be used to model two-dimensional spaces of possibilities: for example, a robot arm with two hinged elbows, moving in a 2D plane, has a range of possible positions it can be in, corresponding to the angles each of the two joints are set to. Each possibility corresponds to one point on the surface of the torus, given by one angle determining the distance around the torus, and the other the distance towards the centre hole.

The mathematical way to represent these shapes is using an identification diagram – we say that the edges which have been glued are identified with each other, and we denote it using an arrow, or a pair of arrows (if we have a second pair of edges, to determine which pairs go together) to indicate the edges are stuck that way round. Below are the diagrams for a cylinder and a torus.

Presenting the shapes in this way presents an intriguing additional possibility: what if the arrows were not pointing in the same direction? Applying this to our cylinder gives us a shape with some very interesting topological properties: the **Möbius band**. This can be made from a simple square piece of paper (although in practice, you will need to stretch your square so it is much wider than it is tall) by joining the ends with a twist.

Named after the mathematician August Möbius, this shape has some curious properties, which can be easily explored if you have a paper one in front of you. For example, if you draw a line down the middle of the strip, it becomes apparent that it is possible to draw a line all the way along both sides of the paper without lifting off the pen. This makes sense if you consider that in adding the twist, we have joined the front of the paper to the back – so this shape has only one side.

The next twist (pun intended) comes when you try to cut the band in half along this line. Since most well-behaved objects, on being cut in half, obligingly become two separate pieces, it can be surprising to discover that the twist has also made the Möbius band have only one edge, meaning when cut in half it uncurls into one long loop, rather than splitting in two as you would expect. The resulting shape (which is actually a two-twist Möbius band) has two edges, one of which used to be the outside edge of your band, and the other of which was created by your cut.

There are plenty of other fun things to do with a Möbius band – some of which can be seen in this classic video by Stand-up Mathematician Matt Parker – but topologically it is a fascinating space. Mathematicians like to compare the **local properties** of a shape with the **global** ones; locally – looking at a small section of the strip – it is identical to a cylinder. It is only when considered as a whole that the twist becomes apparent.

Another nice property of the band is that it is **non-orientable**: if we tried to define a coordinate system on it, it would be inconsistent. Imagine drawing a pair of axes at right-angles to each other, then following one of the axes along the length of the strip. When you return to the point you started, your axes will be upside down, due to the twist – so it is impossible to define a consistent orientation.

You might well be wondering what happens if we try to combine these two approaches: what would a Möbius torus look like? If we join two edges of the square, then try to join the other two edges with a twist, we find a slight problem: it is only possible to do this if we allow the shape to intersect itself and join on from the inside. Assuming this is permitted, we can create a shape called a **Klein bottle**: a tube which curves round, passes through itself and joins back on to the other end but from the inside.

Named after mathematician Felix Klein, this shape has similarly interesting properties to the Möbius band: it is non-orientable, and has only one surface. An ant standing on the outside of the shape could walk to the opening of the tube, crawl into the hole and follow the tube back out through the side, so they are walking on the inside of the part they were previously on the outside of. If you cut a Klein bottle in half, you get two Möbius bands, but if you cut one in half in a different direction, you get one Möbius band.

The self-intersection needed to make this shape work feels like cheating, but that is only true in 3 dimensions – if we are allowed to extend up into 4D the shape behaves itself perfectly, and connects up by just ‘going around’. In the same way that it is impossible to draw a 2D diagram of a knotted piece of string without a line crossing itself somewhere, it is impossible to perceive a 4D shape like a Klein bottle in 3D space without it intersecting itself.

There is one final combination we have not considered: putting a twist in both directions gives a shape called the **Real Projective Plane**. Also a 4D shape, it has to self-intersect to exist, and is incredibly hard to visualise. My favourite way of picturing it is to imagine a sphere with a hole in, and then glue a Möbius band’s edge (which it has only one of) along the edge of the hole.

If you’d like to try to picture this, here is an animation based on a video by Jos Leys, which shows the edge of a hole in a sphere being joined up Möbius-wise, intersecting itself in order to do so. This gives a shape known as a **cross-cap**, which is equivalent to the real projective plane and one possible way to embed it in 3D.

In fact, since this is topology and we are allowed to stretch things as long as we do not change their shape, we can consider even more possible presentations of the space. For example, if you were to take each point on the surface of a sphere and glue it to the point directly opposite, doing this for all the points simultaneously, the shape would need to intersect itself, and it would result in another object that is equivalent to the real projective plane. Notionally, this would be a space of points containing one point for each pair of opposite points on a sphere – we can think of it as having a point for each possible direction in 3D space, which is why it is useful for modelling rotations in 3D.

Starting from just a simple square, we can produce many interesting shapes with interesting and useful properties. Topologists love playing around with squares, and hopefully you can see why!

The post Circling the Square originally appeared on the HLFF SciLogs blog.

]]>The post Potenzreihen und Konvergenzkreise originally appeared on the HLFF SciLogs blog.

]]>Diesmal geht es gewissermaßen tief in den Maschinenraum der Mathematik, da, wo man sich manchmal an den öligen Zahnrädern die Finger schmutzig macht – zum Beispiel an dem Begriff der Differenzierbarkeit, den Sie vielleicht noch aus der Schule kennen und sich nicht so recht mit ihm anfreunden konnten; er ist ja wirklich etwas sperrig. Wenn Sie sich mit Potenzreihen auskennen, werden Sie am Ende alte Bekannte wiedertreffen.

Wie war das? Differenzierbar ist, wenn die Kurve eine Tangente hat. Tangente ist, wenn sie sich an die Kurve so eng anschmiegt wie überhaupt möglich. Das ist schon richtig. Aber was heißt das genau?

Hier Kurve, da Gerade. Kurve \(f(x)=x^2 \) und Gerade \(g(x)=8x-12\) treffen sich in einem Punkt, und zwar (2, 4). Wenn man mit \(x\) langsam auf den Punkt \(x=2\) zuwandert, geht die Differenz \(f(x)-g(x)\) gegen null. Kein Wunder, wenn die beiden sich in \(x=2\) treffen. Aber eine Tangente ist \(g\) offensichtlich nicht.

So sieht eine richtige Tangente aus:

Die Idee ist, dass die Differenz \(f(x)-g(x)\) „schneller“ gegen null geht als die Differenz \(x-2\). Das wiederum wäre zu präzisieren zu der Aussage, dass \(f(x)-g(x)\) selbst dann noch gegen null geht, wenn man es durch \(x-2\) (was ja auch gegen null geht) dividiert. Mit ein bisschen Umrechnen kommt dann die übliche Definition heraus, nämlich dass \( (f(x)-f(2))/(x-2) \) einem endlichen Grenzwert zustrebt, den man dann die Ableitung von \(f\) an der Stelle 2 oder kurz \(f’(2)\) nennt.

Hier will ich auf einen anderen Aspekt hinaus: Die Tangente ist die beste Möglichkeit, die Kurve anzunähern, wenn man nur Geraden, sprich lineare Funktionen zur Auswahl hat. Die Annäherung ist richtig gut in der Nähe des speziell dafür ausgewählten Punktes, in unserem Beispiel \(x=2\), und wird mit zunehmender Entfernung schlechter – beliebig schlecht, um genau zu sein, denn die Tangente in dem speziellen Punkt hindert die Kurve nicht, weiter draußen die wildesten Sprünge zu vollführen.

Was heißt „in der Nähe“, und was heißt „weit draußen“? Das weiß man nie so genau. Die ganze Definition bleibt ja abstrakt und sagt erst einmal nichts darüber, wie viele Zentimeter lang die Näherung noch gut bleibt. (Wir wissen noch nicht einmal, ob die \(x\)-Werte überhaupt in Zentimeter gemessen werden!) Um dieses Problem präzise zu fassen, verwendet man die (gewöhnungsbedürftige) Epsilon-Delta-Definition, was hier nicht weiter vertieft werden soll.

Wie kann man die Näherung an die Funktion verbessern? Immer in der Nähe eines speziellen Punktes, der ab jetzt \(x_0\) heißen soll? Indem man das Sortiment der zur Näherung verwendbaren Funktionen vergrößert. Zum Beispiel indem man außer linearen auch quadratische Näherungsfunktionen zulässt. Die sind immerhin krumm und deswegen bereit, sich noch enger an eine krumme Funktion anzuschmiegen.

Oder man genehmigt sich nicht nur die erste und die zweite Potenz von \(x\), sondern die dritte; damit kommt man Funktionen bei, die sich mal linksherum und mal rechtsherum krümmen, die vierte, die noch größeres Gehampel mitmacht, die fünfte …

Mit jeder Potenz, die man hinzunimmt, wird die Näherung besser – kein Wunder, schlechter kann es ja nicht werden. Wenn die neu hinzugekommene Möglichkeit unter allen Umständen die Sache schlechter machen würde, könnte man sie ja mit dem Faktor null versehen und wäre wenigstens so gut wie zuvor.

Das wirklich Schöne an der ganzen Annäherei ist: Es gibt ein einfaches Kochrezept, um die jeweils optimale Menge an Zutaten zu bestimmen. Man nehme \(a_0\) Gramm von der konstanten Funktion \(f_0(x)=1\), \(a_1\) Gramm von der linearen Funktion \(f_1(x)=x-x_0\), \(a_2\) Gramm von der quadratischen Funktion \(f_2(x)=(x-x0)^2\) und so weiter, addiere die alle zusammen, und fertig ist die Näherung. Denn die Zutatenmengen („Koeffizienten“) \(a_0, a_1, a_2, \dots\) sind leicht auszurechnen: \(a_n = f^{(n)}(x_0) / n!\) Die \(n\)-te Zutat (die natürlich nicht in Gramm gemessen wird) ist gleich der \(n\)-ten Ableitung unserer Funktion \(f\) in dem Punkt \(x_0\), geteilt durch \(n!\) (\(n\)-Fakultät), das Produkt aller natürlichen Zahlen von 1 bis \(n\).

Diese Fakultätsfunktion wächst rasant an, schneller als jede Potenz von* *\(n\). Das heißt, die Zutatenmengen werden mit zunehmendem \(n\) sehr schnell immer kleiner – vorausgesetzt, die Werte der \(n\)-ten Ableitungen halten sich einigermaßen in Grenzen. Und natürlich vorausgesetzt, es gibt sie überhaupt. Schon die Differenzierbarkeit ist eine ziemlich spezielle Eigenschaft einer Funktion, und ob deren Ableitung ebenfalls differenzierbar ist, die Funktion also überhaupt eine zweite Ableitung hat, ist nicht von vornherein klar. Und so weiter. Zu jeder Bedingung, die man gerne erfüllt sehen würde, findet sich ein Gegenbeispiel.

Eigentlich sind also die Funktionen, deren erste, zweite, dritte … Ableitungen sämtlich existieren, geradezu exotische Ausnahmeerscheinungen. Aber in der mathematischen Praxis laufen sie einem die ganze Zeit über den Weg. Und zwar diejenigen, die nicht nur unendlich oft differenzierbar sind, sondern bei denen die nach obigem Rezept zubereitete Näherungsfunktion (sie heißt übrigens „Taylorpolynom“) auch tatsächlich funktioniert. Selbst dafür gibt es Gegenbeispiele.

Was genau heißt es, dass die Näherung durch ein Taylorpolynom optimal ist? Dass der Unterschied zwischen Funktion und Näherung schneller gegen null geht als die höchste Potenz von \(x-x_0\), die unter den Zutaten des Taylorpolynoms vorkommt. Und wenn das für jedes Taylorpolynom der Fall ist, dann darf man „unendlich viele Zutaten zusammenschütten“ und erhält nicht nur eine Näherung an die Funktion, sondern eine exakte Darstellung der ganzen Funktion.

Na ja. Wenn es unendlich viele Summanden zu addieren gibt, läuft das auf einen Grenzwert hinaus, dann will wieder bewiesen werden, dass er existiert, und um ihn auszurechnen, muss man manchmal erheblich um die Ecke denken. Aber zumindest der Existenzbeweis ist in diesem Fall nicht besonders problematisch. Unsere unendliche Summe (die „Taylorreihe“) lautet ausgeschrieben

\[ f(x)=\sum_{n=0}^\infty {f^{(n)}(x_0) (x-x_0)^n \over {n!} }\; ,\] und solche unendlichen Summen pflegen zu konvergieren – das heißt, es gibt einen endlichen Grenzwert, der als Wert der Summe gelten kann –, wenn die einzelnen Summanden hinreichend schnell klein werden. Zum Kleinwerden trägt einerseits die große Zahl \(n!\) im Nenner bei, andererseits der Faktor \( (x-x_0)^n\); denn wenn \(x\) nahe bei \(x_0\) liegt, also der Betrag von \(x-x_0\) klein ist, dann gehen seine Potenzen gegen null wie eine geometrische Folge. Das hilft bis zu einem gewissen Grad sogar, wenn die Ableitungswerte \(f^{(n)}(x_0)\) mit zunehmendem \(n\) gegen unendlich gehen.

Genauer: Es gibt für jede Taylorreihe einen Maximalabstand \(r\), den „Konvergenzradius“. Wenn der Betrag \(|x-x_0|\) kleiner als \(r\) ist, konvergiert sie; ist er größer als \(r\), konvergiert sie nicht, und wenn er gleich \(r\) ist, weiß man es nicht so genau.

Insbesondere darf der Konvergenzradius unendlich sein. Dann passiert etwas sehr Merkwürdiges: Das Verhalten der Funktion auf der ganzen reellen Achse ist bestimmt durch ihr Verhalten in einem einzigen Punkt \(x_0\)! In einer beliebig kleinen Umgebung dieses einzigen Punktes, um genau zu sein, denn es kommt auf den Wert der Funktion und aller ihrer Ableitungen in diesem Punkt an, und die Ableitungen sind ja Grenzwerte, in die Funktionswerte in der Nähe von \(x_0\) eingehen. Also muss man, um die Funktion vollständig zu kennen, das gedachte Fernrohr nicht nur auf den Punkt \(x_0\) richten, sondern muss sich eine beliebig kleine Umgebung von \(x_0\) mit anschauen.

Analytische Funktionen, das sind solche, deren Taylorreihe überall konvergiert, haben sozusagen einen starken inneren Zusammenhalt. Man kann nicht an einer Stelle ein bisschen wackeln, ohne die ganze restliche Funktion in Mitleidenschaft zu ziehen – wenn sie analytisch bleiben soll.

Auch wenn die Taylorreihe überall konvergiert, wird die Konvergenz typischerweise um so mühsamer, je weiter man sich von dem speziellen Punkt \(x_0\) entfernt. Das wird deutlich an Funktionen, die zwar analytisch sind wie die Sinusfunktion, aber sich vollkommen anders verhalten als jedes Polynom.

Die Sinusfunktion (braun im Bild) bewegt sich brav zwischen 1 und –1 und kreuzt dabei unendlich oft die \(x\)-Achse. Polynome dagegen haben das so an sich, dass sie für große Werte von \(x\) gegen unendlich gehen; und ein Polynom vom Grad \(n\) kann auch höchstens \(n\)-mal die \(x\)-Achse kreuzen. Entsprechend haben die ersten paar Taylorpolynome (blau im Bild) die größte Mühe, sich der Sinusfunktion anzupassen.

Und wenn eine Funktion an irgendeiner Stelle unendlich wird, kann es dort keine Konvergenz mehr geben. Nehmen wir \(f(x)=1/(1-x) \). An der Stelle \(x=1\) wird der Nenner null, der Funktionswert ist dort also nicht definiert, und in seiner Nähe geht es rasant gegen unendlich. Vielleicht kennen Sie die Potenzreihe dieser Funktion noch aus der Schule. Es gilt \(f(x) = 1 + x + x^2 + x^3 + \dots \), das folgt aus der Summenformel für die geometrische Reihe. Und die konvergiert bekanntlich nur, wenn \(|x|\), der Betrag von \(x\), kleiner als 1 ist.

Aber was ist mit einer Funktion wie \(f(x)=1/(1+x^2)\)?

Die ist nun wirklich das Paradebeispiel eines Schlaffis. Ewig lang passiert nichts; einmal im Leben, in der Nähe von \(x=0\), rafft sie sich zu einer gewissen Aktivität auf, um dann alsbald wieder nachzulassen und in alle Ewigkeit monoton auf die Null zuzuschleichen. Von Unendlichkeitsstelle keine Spur. Da sollte doch die Taylorreihe kein Problem haben, dieses friedliche Wesen wiederzugeben, weniger jedenfalls als mit der Sinusfunktion. Aber weit gefehlt! Der Konvergenzradius ist 1, wie bei ihrer wilden Schwester \(1/(1-x)\).

Wieso? Hier hilft ein Blick auf die komplexen Zahlen. Ich habe zwar die ganze Zeit nur von reellen Funktionen geredet; aber das lässt sich alles ohne nennenswerte Änderungen auf komplexe Werte von \(x\) und \(f(x)\) übertragen. Dann ergibt auf einmal der Begriff „Konvergenzradius“ einen Sinn, denn in der komplexen Ebene ist es tatsächlich ein Kreis, in dem die Taylorreihe konvergiert.

Im Komplexen läuft unsere Schlaffifunktion zu großer Form auf; denn für \(x=\pm i\) wird \(x^2=-1\), der Nenner wird null, und unsere Funktion hat an diesen beiden Stellen Pole; so nennt man diese milde Form von Unendlichkeitsstelle. Der Konvergenzkreis mit Mittelpunkt null stößt bei \(\pm i\) sozusagen auf eine natürliche Grenze, über die hinaus er nicht wachsen kann.

Philosophische Schlussbemerkung: Selbstverständlich ist es legitim, sich ausschließlich mit reellen Funktionen zu beschäftigen und von komplexen Zahlen nichts wissen zu wollen, vor allem für einen Menschen, der die Mathematik auf reale Gegenstände anwenden will. Seltsamerweise bleiben dabei Fragen offen („Wieso ist der Konvergenzradius von \(1/(1+x^2)\) so klein?“).

Diese Fragen finden eine elegante Antwort, wenn man sich auf komplexe Zahlen einlässt. Eigentlich ist das Grund genug, sich ihnen zuzuwenden.

The post Potenzreihen und Konvergenzkreise originally appeared on the HLFF SciLogs blog.

]]>The post What is an Aperiodic Monotile? originally appeared on the HLFF SciLogs blog.

]]>A team of mathematicians from all over the world had come together to write a research paper explaining an ingenious new family of shapes: one which solved a long-standing question that had stumped mathematicians for decades, while being pleasingly simple and elegant. The shapes, which were discovered by a retired print technician from Yorkshire, provide an answer to the question “Can there be an aperiodic monotile?”

When you mention tilings, most people will think of a regular repeating pattern of tiles on a wall. Mathematically, we consider tilings to be on an infinite plane, going on forever in all directions, and there is an important property of tilings that concerns whether a tiling repeats, called **periodicity**.

A **periodic tiling** is one which, if shifted across or down, gives the same pattern. For example, shifting a tiling by squares across or down by the side length of one square would give a tiling which matches up the outlines of the original set of tiles exactly.

If we wanted to break this, we could take an infinite tiling of squares and cut a random set of the tiles in half horizontally or vertically, giving two rectangles where there was previously a square. Assuming we picked our cuts randomly, this tiling could never be slid across or down to match itself again, and would be technically classed as a **non-periodic** tiling – it does not have a repeating pattern, or a translational symmetry.

It turns out that it is possible to design tiles, or sets of tiles, which are not just capable of non-periodic tilings, but in fact require them. Such a tile set is called **aperiodic** – it can only be used to produce a non-periodic tiling, and such tilings will contain as large a region as you like which has no repeats. Such tile sets produce beautiful non-repeating designs, but as you might imagine, they are a bit hard to find, and it is even harder to prove this aperiodic property.

Aperiodic tilings are of particular interest to physicists, as they are related to quasicrystals – structures found in certain minerals and chemical compounds, which exhibit this same non-periodic behaviour. Prior to this connection being found, work on aperiodic tilings was considered to be more recreational than proper research, and was often undertaken by amateur mathematicians such as Robert Ammann. But from the 1970s, physicists started to take an interest.

The most famous type of aperiodic tiling is the Penrose tiles. Designed by physicist Roger Penrose, there are various versions of it out there but the most famous is a tiling by two rhombuses, one wide and one narrower. The particular dimensions of the rhombuses are related to the Golden ratio, and the angles at the corners are all related to pentagons. These tiles can be used to produce beautiful non-periodic tilings of the plane – but only if you use them correctly.

The Penrose rhombs themselves are fundamentally not aperiodic: if I wanted to, I could arrange the two rhombuses into a periodic tiling (below left). But if we apply certain rules to the way the rhombuses are allowed to combine – for example, we are not allowed to put two together so they form a parallelogram – we can force them to be aperiodic. The easiest way to do this in practice is to mark the tiles so they must be matched up correctly, or add notches to the sides so they fit together like jigsaw pieces in the right way.

This tiling is periodic.

This seems dissatisfying – and up until fairly recently, all the attempts at aperiodic tile sets were disappointing in some aspect. Several variations on Penrose tile sets exist, all of which need some kind of matching rules to prevent them tiling periodically. It is also annoying that you need more than one type of tile to cover the plane: a single tile with this property would be called a **monotile**, and while examples of aperiodic monotiles were known, they have so far been made up of multiple disconnected pieces that form one ‘tile’, like the Socolar-Taylor tile.

Enter maths enthusiast David Smith, who enjoys playing with tilings and patterns, and uses his computer to generate tilings with different shapes. He spent a lot of his spare time playing with tiles and trying to fill the screen, and one day he discovered an intriguing shape. Smith had split the whole plane into hexagons, then cut each hexagon along lines perpendicular to the centres of its faces to make six kites. These kite pieces could be joined together into tiles, and he found one particular combination that seemed to tile in a strange way.

Suspecting this might be an interesting discovery, Smith contacted his friend in Canada, research mathematician Craig Kaplan – they shared an interest in tilings and shapes, and Kaplan was able to take a look at this new tile to see if it was anything exciting. And it turned out, it was – Smith had discovered a single tile which tiles the whole plane non-periodically, and the non-periodicity is forced merely by the shape of the tile: not requiring any additional matching rules.

Kaplan and Smith pulled in some other mathematical friends, mathematician Chaim Goodman-Strauss and software developer Joseph Myers. Between them, they worked out a way to prove that this tile had the aperiodic property they were looking for. (If you would like more detail on this, Christian Lawson-Perfect has written a great explainer over at The Aperiodical). They wrote up the discovery and published it online – and the response was huge.

The breakthrough was unexpected and definitely new mathematics – making progress in an area where nobody had discovered anything in over 50 years – but it was also fairly simple to understand and explain. Plus, it involved a friendly-looking shape that makes pretty tilings that you can share and look at. This meant everyone was soon talking about it, creating their own tiles using laser cutters and 3D printers, and even sharing maths memes. Everyone was talking about it, with excellent write-ups in Quanta and The Guardian.

In fact, the researchers had discovered a whole infinite family of solutions, made by varying the two different lengths of edges of the shape, all of which provide monotiles. There are two members of this family which make particularly pleasing tiles, corresponding to length parameters of \((1, \sqrt{3})\) and \((\sqrt{3},1)\) and they have become known as ‘the hat’ and ‘the turtle’.

Image CC BY-SA Cmglee on Wikipedia

As exciting as this discovery was, there was still one outstanding question, relating to one more slightly unsatisfying aspect of the discovery. The tilings produced by these shapes are non-periodic, but in order to fill the whole plane they need some of the pieces to be flipped over, or mirrored (the darkest pieces in the tiling above). Roughly one in six tiles – or less roughly if you prefer, one in ϕ^{4},^{ }where ϕ is the Golden ratio – needs to be facing the other way in order for it to work.

People immediately started wondering if this was as far as we could go, or whether someone would be able to find the elusive aperiodic monotile which does not need to be present in its flipped form in order to work. The team behind the first monotile discovery also wondered this, and continued working on the problem.

And less than two months later, they have announced that this problem is now also solved – and in fact, they had kind-of already solved it! One of the stages in the infinite family of solutions they had already seen – the half-way point in the video above, corresponding to edge lengths of \((1,1)\) – was previously discounted as an aperiodic tile, since the tiling it produces is periodic.

But on closer inspection it turns out that if you use this tile only in one orientation, this forces it to be a non-periodic tiling – if you are looking at examples including flipped versions, this must make it harder to see the aperiodicity. But it is there, and the authors have now produced a proof that this tile works, is aperiodic, and can tile the whole plane on its own when used without reflections.

Since using the tile in different mirror orientations can still lead to periodic tilings, we need to make one tweak to ensure it is aperiodic, and that is to change the straight lines making up the edges into some kind of curve – any curve will do, but just enough to force us to use the tile the right way round. The resulting slightly spooky-looking shape has caused this new tile to be christened ‘the spectre’ – and again, this means there is a whole infinite family of shapes we can create from it.

If you would like to hear more about these cool new shapes, we had the pleasure of chatting with Chaim Goodman-Strauss, one of the four paper authors, for my podcast Mathematical Objects. This chat took place between the first and second discoveries, so we mainly talk about the original monotile – but it comes across how excited the team were about this discovery, and I can understand why. This is a really cool, actually new piece of mathematics – and I can make myself a keyring from it.

The post What is an Aperiodic Monotile? originally appeared on the HLFF SciLogs blog.

]]>The post Komplexe Analysis und der harmonische Oszillator originally appeared on the HLFF SciLogs blog.

]]>Auf diese Frage, die in den Kommentaren dieses Beitrags intensiv gestellt wurde, gibt es viele Antworten. Die komplexe Analysis (so wird die Theorie der holomorphen Funktionen auch gerne genannt) sitzt so zentral im Beziehungsnetz der Mathematik, dass man ihre Wirkungen an zahlreichen Stellen wiederfindet. Da die ganze Fülle hier schwerlich auszubreiten ist, soll es stattdessen ein kleines Einzelbeispiel geben. Das Schöne an diesem Beispiel ist: Man kommt von einem physikalischen Problem bis zu dessen vollständiger Lösung – für Physiker wie Mathematiker ein eher seltenes Vergnügen.

Das physikalische System, um das es gehen soll, ist unter dem Namen „harmonischer Oszillator“ bekannt. Im Physikunterricht pflegt der Lehrer / die Lehrerin eine Schraubenfeder an einem Haken aufzuhängen, unten dran kommt ein Gewichtsstück („Massenstück“ ist die korrekte Bezeichnung), und dann lässt man das Ding zappeln.

Zur Beschreibung dieser Bewegung genügen fürs erste zwei einfache physikalische Gesetze: Kraft ist Masse mal Beschleunigung (Newton II), und die Rückstellkraft der Feder ist proportional ihrer Auslenkung (hookesches Gesetz). Eigentlich gibt es auch noch die Schwerkraft; aber die können wir uns merkwürdigerweise wegdenken. Im Ruhezustand zieht die Schwerkraft, die an der Masse angreift, die Feder ein Stück auseinander, bis deren Rückstellkraft das Gewicht gerade kompensiert. Wir legen unsere Längenskala so, dass an genau dieser Stelle der Nullpunkt sitzt, und legen den Nullpunkt der Kraftskala auf diese kompensierende Federkraft. Und wie in der Physik üblich, denken wir uns die Masse in einem einzigen Punkt konzentriert. Dann nimmt die Bewegungsgleichung für diesen Massenpunkt eine besonders einfache Form an:

Nennen wir \(u(t)\) die zeitabhängige Auslenkung aus der Ruhelage – positive \(u\) nach oben gerechnet –, dann ist die Kraft einerseits gleich \(m u’’(t)\); \(m\) ist die Masse, die Beschleunigung \(u’’(t)\) ist die zeitliche Ableitung der Geschwindigkeit \(u’(t)\), und das ist die zeitliche Ableitung der zeitlichen Ableitung des Ortes \(u(t)\). Andererseits ist die Kraft, die von der Feder ausgeübt wird, gleich \(-k u(t)\). Dabei ist \(k\) (die „Federkonstante“) ein Maß für die Härte der Feder, und das Minuszeichen muss sein, weil die Federkraft immer der Auslenkung entgegen gerichtet ist: abwärts, wenn die Masse zu hoch hängt, und umgekehrt. Damit ist unsere Gleichung \[m u’’(t) = -k u(t)\; ,\] und wir suchen eine Funktion \(u(t)\), die diese Differenzialgleichung erfüllt.

Der folgende Witz stammt aus einer Zeit, als Konservendosen noch nicht mit einem Ring und einer vorgeritzten Linie zum Öffnen versehen waren, sondern ein spezielles Werkzeug erforderten: einen Dosenöffner. Die Pointe des Witzes hat dagegen den Zeitablauf unbeschadet überstanden.

Ein Experimentalphysiker, ein theoretischer Physiker und ein Mathematiker werden hungrig in je eine Gefängniszelle gesteckt, mit nichts als einer verschlossenen Dose Hering in Tomatensoße. Am Ende des Tages schaut der sadistische Wärter nach, was seine Gefangenen angestellt haben.

Die Zelle des Experimentalphysikers ist ziemlich verwüstet, und rote Soße tropft von den Wänden. Der Mann hat die Dose so oft gegen die Wand geknallt, bis sie schließlich ihren Inhalt freigab. Nun leckt er sich die Reste von den Fingern.

Der theoretische Physiker hat irgendwo einen Bleistiftstummel gefunden und die Wände der Zelle mit umfangreichen Flugbahnberechnungen bedeckt. Daraufhin kam er mit einem einzigen gezielten Wurf zum Ziel.

Der Mathematiker sitzt vor der geschlossenen Dose, ist aber durchaus glücklich damit und sagt nur „Angenommen, diese Dose wäre offen …“

An dieser Stelle verfahren wir wie der Mathematiker mit der Fischdose. Wir nehmen an, wir hätten eine Lösung, und werden glücklich damit.

Unser virtueller Dosenöffner ist in diesem Fall die Exponentialfunktion. Man schreibt sie \(f(x) = \exp(x)\) oder auch \(e^x\), denn in der Tat ist sie gleich der Zahl \(e=2{,}718281828\dots\), der Basis des natürlichen Logarithmus, hoch der Zahl \(x\). Wie man Potenzen mit irrationalen Exponenten definiert, ist eine Sache für sich, aber das braucht uns hier nicht zu interessieren. Worauf es hier ankommt, ist die Tatsache, dass die Exponentialfunktion gleich ihrer eigenen Ableitung ist: Für \(f(x) = \exp(x)\) gilt \(f’(x) = \exp(x)\). Und wenn noch irgendein konstanter Faktor vor dem \(x\) steht, nennen wir ihn \(a\), dann ist die Ableitung von \(\exp(ax)\) gleich \(a \exp(ax)\). Dahinter steckt kein weiterer Tiefsinn, sondern die Kettenregel der Differenzialrechnung. Und die zweite Ableitung ist entsprechend \(a^2 \exp(ax)\).

Na gut. Wir nehmen an, unser gesuchtes \(u(t)\) wäre gleich \(\exp(at)\).

Ein kleiner Moment der Verwirrung: Unsere unabhängige Variable heißt jetzt nicht mehr *x*, sondern *t*, weil es sich um die Zeit handelt. So sind die Bräuche: Wenn die unabhängige Variablde nicht *x* heißt, ist man verwirrt, und wenn die Zeit nicht *t* heißt, ist man auch verwirrt. Also muss ich Ihnen wohl oder übel die kleine Verwirrung, die durch die Umbenennung der Variablen entsteht, zumuten.

Setzen wir das in die Gleichung ein, dann ergibt sich \[m a^2 \exp(at) = -k \exp(at) \; .\] Das sieht doch schonmal ganz nett aus. Den Faktor \(\exp(at)\) können wir auf beiden Seiten der Gleichung wegdividieren (die Exponentialfunktion nimmt niemals den Wert null an), und übrig bleibt \[m a^2 = -k\] mit der einzigen Unbekannten \(a\). Das nach \(a\) aufgelöst ergibt \[a=\pm \sqrt{-{k \over m}} \; .\] Das sieht schlecht aus. Massen sind positiv, Federkonstanten sind positiv, verrechnet haben wir uns auch nicht, also hilft nichts: \(a\) ist die Wurzel aus einer negativen Zahl, und schon stecken wir mit \(a = \pm i \sqrt{k/m}\) in den komplexen Zahlen. (Wie war das? \(i\) war die imaginäre Einheit, von der man zunächst nur weiß, dass \(i^2=-1\) ist. Daraus kann man komplexe Zahlen machen, mit Rechenregeln und einer Darstellung in der gaußschen Zahlenebene. Einzelheiten hier.) Die beiden Lösungen unserer Gleichung wären \(u = \exp(\pm i\sqrt{k/m} \; t)\), und die sind definitiv nicht reell, sollen aber eine reelle Auslenkung \(u\) beschreiben. Die Schraubenfeder mit der Masse dran driftet ja offensichtlich nicht ins Reich des Imaginären ab.

An dieser Stelle sind wir in derselben Lage wie der gute Herr Gerolamo Cardano (1501–1576). Der hatte eine Lösungsformel für kubische Gleichungen gefunden, also solche, in denen die Unbekannte in der dritten Potenz vorkommt. Bei deren Anwendung tauchten unterwegs auch diese merkwürdigen imaginären Größen auf – und verschwanden wieder, mit dem Effekt, dass am Ende reelle Zahlen als Lösungen herauskamen.

Das Vergnügen werden wir auch haben. Allerdings müssen wir dafür eine weitere Tatsache zu Hilfe nehmen. Unsere Differenzialgleichung ist linear, das heißt, die Summe zweier Lösungen ist wieder eine Lösung, und wir dürfen jeden der Summanden noch mit einem konstanten Faktor multiplizieren. Wieso? Die zweite Ableitung einer Summe von Funktionen ist die Summe der zweiten Ableitungen, und der Rest ist einfaches Ausrechnen.

Zwei Lösungen haben wir schon, also ist \[ u(t) = b_1 \exp(i\sqrt{k/m}\; t) + b_2 \exp(-i\sqrt{k/m} \;t) = b_1 \exp(i c t) + b_2 \exp(-i c t) \] auch eine Lösung. Diesem nervigen Wurzelausdruck \(\sqrt{k/m}\) habe ich den neuen Namen \(c\) gegeben, damit es übersichtlicher aussieht. Und über die beiden Konstanten \(b_1\) und \(b_2\) können wir noch verfügen.

Die kommen uns auch gerade zurecht. Denn im konkreten Fall will man zum Beispiel wissen, was passiert, wenn man die Masse nach unten zieht, dabei die Feder anspannt, und dann loslässt. Oder wenn man ihr beim Loslassen noch einen Schubs mitgibt. Oder allgemein ausgedrückt: Wir wollen die Lösung des „Anfangswertproblems“ bestimmen, das ist die obige Differenzialgleichung mit der Zusatzbedingung, dass zum Zeitpunkt 0 die Position und die Geschwindigkeit des Massenpunkts vorgeschriebene Werte \(u_0\) beziehungsweise \(v_0\) annehmen: \(u(0)=u_0\), \(u’(0)=v_0\).

Na gut, setzen wir das in unsere allgemeine Lösung ein: \[u(0) = u_0 = b_1 + b_2 \; ,\] denn \(\exp(0) = 1\). Für \(v_0\) müssen wir die allgemeine Lösung zunächst ableiten: \[u’(t) = b_1 i c \exp(ic t) – b_2 ic \exp(i c t) \] und dann einsetzen: \[u’(0) = v_0 = ic (b_1 – b_2)\] Das sind zwei lineare Gleichungen für die beiden Unbekannten \(b_1\) und \(b_2\), was man mit Schulmitteln lösen kann. Das Ergebnis ist nicht schwierig, aber ein bisschen unübersichtlich. Schauen wir uns lieber den Spezialfall „auslenken und loslassen“ an, also \(v_0 = 0\). Dann folgt aus der zweiten Gleichung \(b_1 = b_2\) und aus der ersten \(b_1 + b_2 = u_0\), also \(b_1 = b_2 = u_0/2\). Also sieht unsere Lösung so aus: \[u(t) = u_0 {\exp(ic t) + \exp(-ic t) \over 2}\] Und jetzt kommt uns eine Formel aus der komplexen Analysis zu Hilfe und verschafft uns das Cardano-Erlebnis: \( (\exp(iz)+\exp(-iz))/2 = \cos z \). Damit finden wir das Endergebnis \[u(t) = u_0 \cos(ct)\; ,\] die imaginären Größen haben sich hinweggehoben, und alles geht mit reellen Dingen zu!

Das ist ja alles ganz nett; aber kann das sein, dass wir mit Kanonen auf Spatzen geschossen haben? Am Ende kam, vielleicht etwas überraschend, die Kosinusfunktion heraus. Aber wirklich neu ist das nicht. Dass die Masse an der Feder periodisch schwingt, hatten die Physiker schon vorher gemerkt. Dass Sinus und Kosinus geeignete Funktionen zur Beschreibung periodischer Bewegungen sind, hatte sich auch schon herumgesprochen. Und wenn man eine Funktion sucht, deren zweite Ableitung gleich dem Negativen ihrer selbst ist, ist es nicht so schwer, sie zu finden: Sinus und Kosinus erfüllen diese Bedingung. Man kommt, wenn man sich ausreichend Mühe gibt, auf die Lösung, ohne an komplexe Zahlen oder holomorphe Funktionen überhaupt zu denken.

Stattdessen habe ich die Exponentialfunktion, die Sie vielleicht noch kennen, aber eben nur für reelle Zahlen, stillschweigend auf komplexe Werte angewandt und dann noch diese merkwürdige Beziehung zwischen ihr und den Winkelfunktionen Sinus und Kosinus, die man üblicherweise als \(e^{ix} = \cos x + i \sin x \) liest, aus dem Hut gezaubert, nicht zu vergessen die Rechenregeln für komplexe Zahlen und die ganze zugehörige Theorie. Insgesamt ist das ziemlich viel Aufwand.

Aber es lohnt! Ja, man kommt auch im Dunkeln die Kellertreppe heile hinunter und wieder herauf, wenn man sich auskennt und vorsichtig ist. Aber warum nicht einfach das Licht anmachen? Ja, eine Leitung zu legen und eine Lampe zu montieren ist mühsam. Aber wenn man sie einmal hat, entschädigt einen die totale Erleuchtung für alle Mühen.

Wir haben uns ja bisher nur um den einfachsten Spezialfall gekümmert. Aber mit Erleuchtung erledigt man den allgemeinen Fall gleich mit. Die Feder nicht auslenken, sondern nur aus der Ruhelage anschubsen? Die Rechnung geht genau so, bloß dass am Ende die Sinusfunktion herauskommt. Und beide Spezialfälle kann man zum allgemeinen Fall zusammenmischen – die Differenzialgleichung ist ja linear.

Um die Kraft, die in der Realität auf die Dauer die dominierende Rolle spielt, indem sie den Oszillator über kurz oder lang zum Stillstand bringt, habe ich bisher sorgfältig herumgeredet: die Reibung. Die theoretischen Physiker mögen sie nicht so richtig, weil sie nicht durch eine einfache Formel beschreibbar ist wie „Kraft gleich Masse mal Beschleunigung“, sondern sich aus vielen kleinen Molekülinteraktionen zusammensetzt, über die man wenig Allgemeines sagen kann. Die übliche Annahme lautet: Die Reibungskraft ist der Bewegung entgegengesetzt und proportional der Geschwindigkeit, also gleich \(-r u’\), und \(r\) ist ein Proportionalitätsfaktor, der die Stärke der Reibung beschreibt. Das stimmt wenigstens ungefähr.

Damit sieht unsere Differenzialgleichung jetzt so aus: \[m u’’(t) = -k u(t) – r u’(t)\] Das ist immer noch eine lineare Differenzialgleichung mit konstanten Koeffizienten; so ist die Schublade beschriftet, in die die Fachleute diesen Gleichungstyp einsortieren. Auf alle Gleichungen aus dieser Schublade kann man den Ansatz mit der Exponentialfunktion anwenden. Für den unbekannten Faktor \(a\) ergibt sich in dem Fall mit Reibung eine quadratische Gleichung. Da findet endlich mal die gefürchtete Mitternachtsformel aus dem Schulunterricht eine sinnvolle Anwendung, und am Ende steht wieder eine Mischung aus Sinus- und Kosinusfunktionen, aber diesmal gedämpft mit einem Faktor der Form \(e^{-\lambda t}\).

Damit hat das Fischdosenverfahren („Angenommen, die Lösung wäre eine Exponentialfunktion …“) seine Möglichkeiten noch längst nicht erschöpft. Lineare Differenzialgleichungen mit konstanten Koeffizienten gibt es reichlich, und was nicht passt, wird passend gemacht – wenn es geht. Die Physiker sind ganz groß darin, Dinge, die eigentlich nicht linear sind, für linear anzusehen, und solange es sich um kleine Abweichungen von einem Referenzzustand handelt, stimmt das sogar so ungefähr. Das Verfahren ist in der Tat ein Hammer – da kommt man schon mal auf die Idee, überall Nägel zu sehen oder dies oder jenes zum Nagel zurechtzubiegen.

Dass eine einzige Funktion in der Lage ist, sowohl Schwingungs- als auch Dämpfungsvorgänge zu beschreiben (und das exponentielle Wachstum obendrein): Zu dieser Erkenntnis kommt man erst, wenn man das Licht im Keller anmacht. Das allein rechtfertigt schon die Mühe, sich mit den komplexen Zahlen und komplex-analytischen Funktionen zu beschäftigen. Ganz zu schweigen von der Eigenschaft, die diesen Funktionen das Etikett „komplex-analytisch“ eingebracht hat: dass sie alle durch eine unendliche Potenzreihe darstellbar sind. Aber davon erzähle ich ein andermal.

The post Komplexe Analysis und der harmonische Oszillator originally appeared on the HLFF SciLogs blog.

]]>The post A Question of Curvature originally appeared on the HLFF SciLogs blog.

]]>This occurred to me recently when I was thinking about Gaussian curvature. This is a geometrical property of objects and surfaces which describes how the surface curves in three-dimensional space. Gaussian curvature is the answer to a very specific definition: if an object curves in one direction, what does it do in the other direction?

To find the Gaussian curvature at a point on a given shape, you need to measure two perpendicular axes of curvature; for those of you holding a sharpie pen and/or who would like a simpler way of phrasing that, you need to draw a cross on the shape, made from two lines which intersect at right angles.

Once you’ve picked a point and drawn a cross, you need to compare how those two lines curve in 3D. Imagine your object is a shape like a sphere – a football, a bowling ball or an orange – and that you’ve drawn your cross on the very top of it. The two lines will both be curving downwards as they move away from the centre of the cross.

How much are they curving by? Well, we measure the curvature of a circle as 1/r, where r is the radius of the circle. So each line is curving at a rate of 1/r in the same direction. The Gaussian curvature is calculated as the product of these principal curvatures: for two identical curves, 1/r × 1/r = 1/r², which for any sensible value of r will be a small but definitely positive number.

This is because a sphere is an example of a shape which has **positive Gaussian curvature**, and hopefully with a little thought you can convince yourself that it will remain true whichever point on the sphere you choose: if it curves away from you in one line, the perpendicular line curves away from you too. This means the sphere has positive Gaussian curvature everywhere.

If instead we wanted a shape with **zero Gaussian curvature**, we could picture something flat – a piece of paper, a section of floor (in most sensible places), or a pizza. If we measure the curvature in two directions at a chosen point, it will be zero, and the product of zero with itself is still zero. But you didn’t need me to tell you that something which does not curve has no curvature.

Imagine instead a shape like a cylinder: a rolled-up tube of paper, a pencil or a fancy cylindrical pillow. If we pick a point on the flat end of the cylinder, that will also have zero curvature, just like on our flat objects. But if we pick a point on the curved side, and draw a cross there, something interesting happens.

To find the principal curvatures, we need to look for a maximum and a minimum: this forces the orientation of our cross to have one line running horizontally parallel to the cylinder’s length, and the other one wrapping around the cylinder’s circular face (On the sphere, the curvature is the same in every direction, so the orientation of the cross is unimportant).

Since we have a curve in one direction, the curvature is 1/r (where r is the radius of the cylinder). But in the other direction, it will be zero – and 1/r times zero is just zero. So even though a cylinder has a curved face, when considering Gaussian curvature, it isn’t curved at all. And this is where things start to get interesting – because Gaussian curvature is an intrinsic property of an object and cannot be changed.

So if I start with a piece of paper, and want to curve it into a cylinder, I can do that pretty easily: a cylinder has the same curvature as a flat object, so it will do this with no problems. However, if I tried to curve it further – into a donut shape or a sphere, things would start to get messy – the paper would need to be crumpled, or cut, or torn and stuck together, in order to create the shape.

As anyone who has tried to peel an orange will tell you, you don’t get a whole piece of flat peel; similarly, anyone who has attempted to map the globe will quickly have found that in order to draw it all on a flat page, you need to stretch and squash parts of the surface, or leave gaps along the edge (á la the magnificent Goode homolosine projection). An object with positive curvature can’t map onto an object with zero curvature, and vice versa.

So if you have an object with zero curvature, it will always have zero curvature: if you bend it in one direction, it will remain steadfastly straight in the perpendicular direction. And if this sounds like a familiar idea, that will be because it is something you have very likely used regularly without even thinking about it as a ‘life hack’, namely when eating a pizza.

I deliberately included pizza on my list of zero-curvature objects, because a pizza, and indeed a single slice of pizza, is a flat object with the zero curvature property. And if you pick up a piece of pizza by its crust, you might find it starts to unhelpfully droop, threatening to discharge all of its delicious toppings onto the floor. In this situation, the solution is simple: introduce curvature in a perpendicular direction.

Again, translating that into a more sensible instruction: bend the crust of the pizza upwards at the sides. This will make the pizza curve in that direction, meaning that according to the rules of geometry, it will not be able to curve in the direction at right-angles to that: along the length of the slice.

Of course, this depends slightly on the sogginess of your pizza dough: a soft, stretchy base will rashly ignore the universe’s rules of geometry and probably stretch and curve in whichever directions it wants. But assuming a reasonably structurally intact pizza base, this ‘hack’ has been used by millions of people around the world to prevent their slice from drooping. But how many of them, I wonder, credit Carl Friedrich Gauss with its success?

The post A Question of Curvature originally appeared on the HLFF SciLogs blog.

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