Juan C. Meza Division of Mathematical Sciences

1. The Quite Reasonable Effectiveness of Mathematical Sciences Juan C. Meza Division of Mathematical Sciences

2. https://commons.wikimedia.org/wiki/File:Siemens_Magnetom_Aera_MRI_scanner.jpghttps://commons.wikimedia.org/wiki/File:Siemens_Magnetom_Aera_MRI_scanner.jpg

3. Compressed Sensing Beginnings • In 2001, DMS funded a group of researchers (Donoho, Candes, Huo, Jones) in geometric analysis, statistics, computational mathematics, and astronomy • Led to the idea now known as compressed sensing • Essential idea – For objects that have sparse representations in some fixed, known basis it is possible to gather FAR fewer sample than traditionally information theory suggests • Need to implement the principle of `random projections’ and the reconstruction is nonlinear, based on L1 minimization Division of Mathematical Sciences

4. Fast Forward Fifteen Years • In 2017 FDA approved 2 new MRI devices that dramatically speed up scanning, between 8 and 16 times faster than conventional methods. • One machine (CS Cardiac Cine) allows movies of the beating heart; the second (HyperSense) allows rapid 3D imaging, for example of the brain. • The speedup will allow more patients to be served at a lower cost per patient, giving US taxpayers a better return on the tens of billions of dollars in annual MRI charges. A Focused Research Group on Multiscale Geometric Analysis – Theory, Tools, Applications. DMS – 0140698/Donoho (Lead), DMS – 0140540/Candes, DMS – 0140587/Huo, DMS – 0140623/Jones Division of Mathematical Sciences

5. Mathematics of Complex Biological Systems • Biology is becoming much more quantitative due to advances in technologies • complex causal relationships leading to emergent properties of molecular, cellular and organismal systems • emergent properties resulting from the complex integration across different levels of organization at different time scales Structural complexity emerges during the development of a fruit fly’s eye. (Northwestern University) Division of Mathematical Sciences

6. Studying Interface Between Mathematics and Biology • Researchers at Harvard are studying how the shapes of various insect wings form • Dragonfly wing at right, one is computer generated, one is real • VoronoiTesselation Used to Develop Computational Model for Wing Structures Division of Mathematical Sciences

7. Understanding DNA Reconnections • Using Knot Theory, Low-Dimensional Topology, and Computer Simulations can help understand DNA Reconnection and local crossing changes • Vazquez et al. developed software to model the locations where reconnection enzymes could cut and reconnect DNA chains. • Modeled millions of configurations representing 881 different topologies, and identified hundreds of minimal pathways to get two DNA circles linked in up to nine places down to two separate circles. Vazquez, DMS-1716987 Division of Mathematical Sciences

8. Topological and Geometric Data Analysis • Engaging experimental and mathematical neuroscientists to better understand the brain • In this fMRI image of the human brain, the regions of the brain that are active when a person is focused on rhythm appear in red, the regions active when thinking about grammar are in green, and the overlap of both appears in blue. • Develop online course on topological and geometric data analysis, for undergraduate students to better prepare them for graduate education and/or entry into the workforce. Division of Mathematical Sciences

9. Sustainable Transportation Solutions • providing efficient, cost-effective, and sustainable transportation solutions is a challenging problem • On-demand transit modes - bikesharing, ridesharing, and micro-transit - have the potential to solve many of the challenges faced by cities. • Using Stochastic Network Models to design an efficient and sustainable integrated transit ecosystem, including mass transit, ridesharing, bikesharing Division of Mathematical Sciences

10. Automatic Recovery of Road-Networks • Goal is to recover the underlying road network given a collection of GPS trajectories or a satellite image • This is modeled as reconstructing a geometric graph embedded in the plane • In mathematical terms, the method recovers hidden graph-like structures from noisy data using discrete Morse-based graph reconstruction algorithms A topological data analysis technique invented by Dey, Wang, and their students at TGDA@OSU TRIPODS Center recovers the road network from GPS trajectory data. Graph Reconstruction by Discrete Morse Theory, T. Dey, J. Wang, Y. Wang, SoCG 2018, 31:1-31:15 Division of Mathematical Sciences

11. Identifying Emergency Events • improve the ability to use publicly-available human-generated geospatial data sources for identification of emergency events • Need to Identify location, magnitude, and impact of the Event. • Machine Learning Techniques coupled with generalized gaussian graphical models that include non-Gaussian multiway tensor data Dobra and Williams, Geospatial Graphical Models of Human Response to Emergencies, DMS – 1737746 Division of Mathematical Sciences

12. Emerging Frontiers Division of Mathematical Sciences

13. Towards a Quantum Computer • Concepts of topology came from considering properties of objects under deformations • These concepts were applied by physicists, specifically the 2016 Nobel Prize winners Thouless, Kosterlitz and Haldane to describe topological matter in a way applicable to crystalline solids and systems with periodic potential • The field of topological systems is now enjoying an explosive growth – One such concept is called symmetry protected topological order, where a state cannot change unless a specific symmetry of this state is broken. Division of Mathematical Sciences

14. Topological Quantum Computing 5. Readout • the building block of a quantum computer will be a topological qubit – a quantum two-level state protected by specific symmetry. • Such states are proposed to be realized e.g. with anyons, or exotic quasiparticles living in 2-dimensional space, and forming an entity in-between fermions and bosons. • Calculations on such a computer are performed using mathematical theory of braiding. 4. Fuse 3. Braid 2. Create anyons 1. Initialize Division of Mathematical Sciences

15. Genotype to Phenotype • Phenotype is the set of observable characteristics resulting from the interaction of a genotype with the environment. • Decades taking cells and organisms apart to describe them at the molecular level. Going the other direction—from molecule to cell to organism—is the harder and more important goal of biology. • Tools of classical biology alone will not help us put those cells and organisms back together. Experimental, statistical and mathematical approaches to model and test hypotheses about causal relationships among environment, genome, epigenome, and phenotypic characteristics (morphology, physiology, behavior). Division of Mathematical Sciences

16. Windows On the Universe • supermassive black hole M87*, 55 million light years from Earth, at the heart of galaxy Messier 87. • 6.5 billion times the mass of our sun, it distorts spacetime like few objects in the universe • Maximum Likelihood, Bayesian Optimization, … Using the Event Horizon Telescope, scientists obtained an image of the black hole at the center of galaxy M87, outlined by emission from hot gas swirling around it under the influence of strong gravity near its event horizon. Credit: Event Horizon Telescope collaboration et al. Division of Mathematical Sciences

17. Summary • Mathematical Sciences research has a big impact in science, engineering, and Society • Opportunities in Mathematical sciences? • Every Field and Sub-Field • Academia, Industry, Government • Emerging Trends point to even more challenges and opportunities Division of Mathematical Sciences

18. Thank You! Division of Mathematical Sciences

19. Extra Slides Division of Mathematical Sciences

20. Training the Next Generation Division of Mathematical Sciences

21. DMS Programs Address a Wide Range of Stages in the Pipeline Institutes Conferences RTG Scope MSGI EDT REU GRFP CAREER MSPRF IIA Grad Student UG Postdoc PI Division of Mathematical Sciences

22. Mathematical Sciences Graduate Internship • Provide an opportunity for mathematical sciences doctoral students to participate in internships at national laboratories, industry and other approved facilities • Aimed at students who are interested in understanding the application of advanced mathematical and statistical techniques to "real world" problems, regardless of whether the student plans to pursue an academic or nonacademic career • 40 graduate students from 38 universities worked in 10 National Labs in the Summer of 2017 SIAM News Article (12/01/2017) Managed by Oak Ridge Institute for Science and Education Division of Mathematical Sciences