Abstract: Large scale geometry is almost the dual of topology, and a large scale structure is almost dual to a topology (large scale structures are dual to uniform structures, and every uniform structure induces a topology). Topology is concerned with properties holding for smaller and smaller scales, while large scale geometry is concerned with properties holding for larger and larger scales. Persistent homology uses algebra to measure topological properties. Topological properties which persist through multiple scales are considered important features of the space, while those appearing in few scales are considered noise.
This talk will discuss the relationship between large scale geometry and persistent homology. In particular, I will show how large scale geometry induces a filtration of complexes for a space; this filtration is then used to compute the persistent homology. Both theory and examples will be discussed.
“Why would they change math!?”
Free movie night sponsored by Women in Math!
Snacks and lots of BYU Math t-shirt giveaways!
Everyone welcome. Come to 275 TMCB anytime this week for your tickets.
Title: Convolutions, Singular Integrals, and Several Complex Variables
Abstract: The Dirichlet problem for the unit disk is to find a harmonic function with certain prescribed boundary values on the unit circle. The solution is given by taking the convolution of the boundary value function with the Poisson kernel. The theory of such convolution operators is quite well understood. The theory is more subtle when we generalize to consider integral operators that have non-integrable kernels or are not of convolution type. In this talk we give an overview of this theory and discuss its connections with current research in several complex variables.
Abstract: The relationship between modular forms and quadratic fields is exceedingly rich. For instance, the Hilbert class field of an imaginary quadratic field may be generated by adjoining to the quadratic field a special value of the modular j-invariant. This is the underlying reason why “Ramanujan’s constant” eπ√ 163 is close to an integer, and is related to the famous prime generating polynomial n2 +n + 41.
The connection between class groups of real quadratic fields and invariants of the modular group is much less understood. In my talk I will discuss some of what is known in this direction and present some new results (joint with W. Duke) about the asymptotic distribution of integrals of the j-invariant that are associated to ideal classes in a real quadratic field. The proof brings together ideas from hyperbolic geometry, harmonic analysis, and analytic number theory.
Problems to Be Solved: Applied Math in South America
By Lilian Whitney
In Spring of 2018, students took their study of math equations from the orderly classrooms of BYU campus to a much larger classroom: South America.
Dr. Michael Dorff, chair of BYU’s Department of Mathematics, led the three-week study abroad tour through four South American countries: Brazil, Chile, Peru, and Argentina. The trip allowed sixteen STEM students to study math in a real-world, culturally diverse setting. Students toured various companies including Itaú Bank, Walmart, General Motors, and Laureate.
In addition to learning about applied math, students rafted below the thundering waters of Iguazu Falls, explored the vibrant streets of Rio de Janeiro, and hiked the lush mountains to Machu Picchu. Though the stunning sites of South America were memorable, what students remember most is seeing applied math combat poverty, strengthen businesses, and change lives.
Making a Difference
Seeing the growth students experience in new cultures inspired Dorff to lead the first applied math study abroad in Europe in 2017. Dorff decided to lead the 2018 program in South America, allowing students to connect with alumni who live and work there. Dorff believes enabling students to see math applied in unfamiliar cultures increases their capacity to work and serve others both at home and internationally. Each visit was made possible through connecting with BYU alumni, a unique aspect of the program which reflects the diverse opportunities available to BYU graduates.
“The goal is for students to see how math and data analytics are used in different companies that you probably wouldn’t expect,” Dr. Dorff said. “When people think of tech stuff they probably think of companies like Google and Microsoft, but they might not think of Walmart or General Motors. For students to see that these types of companies use data to improve their services and production is really important. When students graduate and are looking for a job, they don’t have to restrict themselves to the standard tech companies. I also want them to see opportunities outside of the United States. It’s part of BYU’s famous catchphrase: Enter to Learn, Go Forth to Serve.”
Visiting the Walmart headquarter in São Paulo, Brazil helped many students realize how much of a difference math can make for families in poor economic circumstances. Paulo, a BYU alum who is a native Brazilian, taught the students about Walmart’s innovative automated-inventory services. Walmart employees use data to create automated inventory services, which ensure the correct number of goods are stocked in stores. Increasing production efficiency enables Walmart to sell goods at lower prices. This allows many Brazilians and individuals around the world to afford to feed their families.
After teaching students about automation services, Paulo divided the students into two groups and gave each group 120 reais, the equivalent of around $32 USD. Each group then had to use that money to buy groceries to feed a Brazilian family of four for one week—a difficult task to accomplish.
“Trying to work out how we could get enough food to feed four people for a week with just 120 reais was really hard and really humbling,” student Tyler Mansfield said of the experience.
The student teams picked out food basics like rice, beans, and fruit. The food students picked was then donated to Brazilian families in need. Learning how applied math individuals afford to feed their families was a priceless experience for many students.
“On the study abroad we would walk through the streets and see these people in need that had so little, which was really difficult. But then we got to go into Walmart and saw how employees use their math abilities to make it possible for others families to afford food. [Applied math] helps people put food on their tables at the end of the day because it allows businesses to increase productivity which helps stabilize the economy on a large scale, which is a really powerful concept,” Mansfield said.
While many STEM students look forward to lucrative careers in STEM, Dorff believes it’s individuals who combine math savvy with a sense of gratitude for the circumstances of their lives who will find the most fulfillment in their careers.
“When you visit places and see people who are less fortunate than yourselves it makes you appreciate what you do have, but also makes you think of ways to help others who are less fortunate than you are,” said Dorff. “There are many big problems that people with STEM backgrounds will be able to solve. You can use math and data to solve pollution problems, to solve hunger problems. You can make energy use more effective and more efficient.”
Problems to be solved
Visiting foreign countries allows students to see just how vast and exciting the opportunities are to apply math in their future careers.
“When I tell people that I’m studying math, they’re like, ‘what will you do with that?’” said student Tyler Mansfield. “What I’ve realized is that so many people have real-world problems that they need math to help them solve. Together, professionals and mathematicians can solve problems in the economy and business that they never would have alone.”
Students also visited the Instituto de Matemática Puras e Aplicadas in Rio de Janeiro, where math and culture intersected as BYU and Brazilian students spent time together.
“We got to sit down with the students there and talk about their lives and how they experience Brazil and their schooling,” said student Gentry Carter. “Learning math changes how everyone views the world, but there is an intersection between culture and math that we shouldn’t ignore. Talking to them about their culture and how they learn math was so cool. They are learning the exact same things we are, but in a different language and applied in different ways.”
At GM, students saw how cars are automated and manufactured. They also met with corporate employees to learn how data analytics help improve production and safety of GM vehicles. Data analytics allow employees to identify production errors in the early stages and to perform cost-performance analysis. Data from consumers also helps employees create future car designs that meet customer needs.
“There are a lot of jobs right now analyzing data,” said Dorff. “If you can get people with different skills in mathematics, computer science, and statistics to work together, their backgrounds will complement each other to help create a better solution to a problem. This is a great opportunity for students to see the interdisciplinary aspects of math. Not everyone that we visited was a math alum—they were engineering, they were computer science, they were finance. Math majors are interacting and solving problems with people of other backgrounds.”
Culture and people
From ancient Inca sites to boisterous soccer stadiums, students enjoyed the food, business tours, and hikes the trip included. According to Dorff, it was by forming relationships with people of different backgrounds that students’ global perspectives changed the most.
“I travel a lot, and people ask me: What’s the best thing you see or the best thing you do when you travel? And it’s meeting the people,” Dorff said. “Whether its Uber drivers or people at restaurants or buses, that’s what’s most important. It’s not about how much money we earn or our belongings, it’s about the people we can learn from and we can help. That happens when you interact with people—you don’t get that in a classroom.”
Friendships spanning cultures flourished throughout the trip, from Uber drivers who shared stories of struggles and diverse circumstances with students to LDS members who reached out to Dr. Dorff about careers in math.
For student McKenna Pitts, experiencing the beauty of new countries granted a new perspective on God’s love for his children and our unique roles in using our talents—whether they be math, music, or science—for good.
“I remember one of the coolest moments for me was at the Christ the Redeemer statue in Rio de Janeiro,” Pitts said. “The statue is ginormous, and you’re at the Savior’s feet and you just feel so, so small, so insignificant when you look out and see all of Rio and all of the ocean. But we know how much Heavenly Father and Jesus Christ really love us. Even though in the grand scheme of things we are absolutely nothing, we mean the world to them. It’s just another testament that God built this earth for us. We are in this huge world and there’s so much you can do because you can make so many important changes, even when you feel small and insignificant.”
Pitts and her fellow students brought home countless memories and photos of hikes, colorful streets, passionate soccer games, and luscious jungles. But what they remember most is the knowledge that wherever they may go in the world, their ability to solve problems can change the lives of others for the better.
As student Tyler Mansfield puts it, “There are so many problems out there that I can help be the solution to.”
Abstract: Data science represents some of the biggest opportunities and challenges to science today. However, many of the algorithms underlying machine learning are not well understood. In this talk I will discuss a number of ways that mathematical analysis can help in understanding these algorithms. In particular, I will discuss how a broad range of tools from modern analysis (including differential equations, variational analysis and probability) can be used to understand crucial questions in machine learning relating to optimization routines, overfitting and reinforcement learning. This represents joint work with a number of statisticians, engineers, and mathematicians. Most of this talk is designed to be accessible to undergraduates.