Department of Mathematics · 2017-06-21 · Julian Palmore, Professor – Spaceflight...

2 0 1 3Depar tment of Mathematics

College of Liberal Arts and Sciences • University of Illinois at Urbana-Champaign

Robert Bauer, Associate Professor – Stochastic Processes in Microbiology and MedicineA research project of Robert Bauer’s is modeling the dynamics of the cytoskeleton—the skeleton of the cell. An important phenomenon is the alternating of growing phases with rapidly shrinking phases for microtubules, known as “dynamic instability.” Bauer has proposed a queueing theory model to capture and reproduce the basic aspects of this phenomenon. In the living cell, microtubule instability is tightly regulated. Dysregulation can lead to diseases such as Alzheimer’s. On the other hand, modulation of microtubule instability is also being exploited for the treatment of cancer by interrupting the dynamic role played by microtubules in cell division. The picture shows the mitotic spindle of microtubules (green) attached to the kinetochores (red) of chromosomes (blue) of a dividing cell.

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Department of Mathematics • University of Illinois at Urbana-Champaign • 1409 W. Green, Urbana, IL 61801 • Tel: (217) 333–3350 • e-mail: [email protected] • www.math.illinois.edu

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Marius Junge, Professor – Operator Spaces and Quantum Information TheoryWho would have thought that abstract topics such as tensor norms in Banach spaces and the relatively new research area of operator spaces have anything to do with real applications, in particular in Quantum Information Theory? For the research team of Professor Marius Junge and Professor Zhong-Jin Ruan, one of the founding fathers of operator spaces, this comes as a pleasant surprise. This connection was discovered because Junge participated in a Ph.D. defense in Spain and met David Garcia-Perez and Nacho Villaneuva while talking about an exotic question on tensor norms. Two years later, after Garcia-Perez made a career change towards Quantum Information Theory, Junge learned that the results discussed above could solve a longstanding open problem about violation of Bell inequalities with three parties. Since then the connection between Operator Spaces and Quantum Information theory has become more deeply worked out, and Junge and Ruan are now working with a group of UI experimental physicists, including Paul Kwiat. The picture here is from a ‘loop hole free’ experiment.

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Kay Kirkpatrick, Assistant Professor – Statistical Mechanics of Condensed Matter Bose-Einstein condensation is a cool and unusual kind of matter near absolute zero, where many quantum particles can condense into the same state and behave like a giant quantum particle. We have been analyzing probability distributions for these quantum particles, proving limit theorems that make the rigorous connection between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schroedinger equation. This illustration depicts the distributions of the particles' velocities: at warmer temperatures (warmer than 200 nanokelvin, anyway), they spread out; but at cold enough temperatures, they move in a coherent beam.

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Julian Palmore, Professor – Spaceflight Interplanetary flight requires mathematics for the design of spaceflight trajectories and other facets of a mission. A trip to Mars can take six to eight months each way, with a waiting period on Mars of one to eighteen months to allow the earth and Mars to be in relative position for a return flight. Just as the international space station is a global enterprise, an international expedition to Mars may be possible as soon as 2035 if undertaken by a global effort.

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Rinat Kedem, Professor – Combinatorics and Statistical ModelsDiscrete statistical mechanical models sometimes have particularly nice properties, such as integrability, which allow us to solve them, or identify them as bijectively equivalent to different models for which a solution is available. One can use such systems to model physical systems, but it is also interesting to use them to solve problems from representation theory, enumerative geometry and combinatorics. This illustration is part of the proof of the positivity conjecture from cluster algebras for a particularly ubiquitous example, solving the problem by mapping it to domino tilings of regions of the punctured plane, and generalized non-intersecting paths.

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Lee Deville, Assistant Professor – Multiscale Methods for Simulation of AerosolsAerosols—small airborne particles suspended in the atmosphere—are a significant driver of both anthropocentric climate change and local/regional weather patterns. These particles have a direct effect through absorption and scattering of solar radiation, but also drive the formation of clouds, which themselves play a significant role through radiation, absorption, and the water cycle. It is thus natural to seek to simulate the effects of aerosols on weather and climate. However, there is a significant challenge in that these populations are extremely numerous and diverse: the air over a typical city can contain up to one quadrillion aerosol particles, and these can each be made up of about twenty different chemical constituents. It is currently intractable to simulate directly such a large and diverse population —and will be so for the foreseeable future—and thus sophisticated simulation and data-processing methods of large-scale random processes are required.

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Sheldon Katz, Professor – Initiative for Mathematical Sciences and Engineering (IMSE)The Initiative for Mathematical Sciences and Engineering (IMSE) is a campus-wide initiative whose long term goal is to become a national Center for the interaction of mathematics and engineering. IMSE will accelerate advances and innovations in mathematics and engineering through cross-fertilization, with the dual objectives of fostering cutting-edge mathematics as indispensable in addressing significant engineering applications and of advancing the next generation of mathematics through the infusion of new classes of problems. Created in March, 2012, IMSE has already hosted numerous lectures including a two-day symposium, and has provided seed funding for seven collaborative research projects involving mathematicians, engineers, and graduate students.

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Zoi Rapti, Assistant Professor – Mathematics and BiologyThe freshwater plankton Daphnia dentifera (competent host) gets infected by the fungal spores of the parasite Metschnikowia bicuspidata. In many freshwater lakes the presence of Daphnia pulicaria (incompetent host) has been linked to decreased disease prevalence. Both Daphnia species consume spores as they feed on algae. We have been investigating the interactions between the two hosts, their food resources, and the parasite. The resulting dynamics are very rich and we have observed period doubling and torus bifurcations as the model parameters vary.

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Jared Bronski, Professor – Nonlinear Waves and Coherent StructuresThe Kuramoto model is a system of differential equations modeling coupled oscillators. This model is thought to govern the blinking of fireflies, the firing of cells in the cardiac pacemaker, unstable oscillations in London’s Millennium Bridge, and other phenomena involving the synchronization of different oscillators. As the strength of the coupling between oscillators is increased there is a phase transition between an incoherent state and a phase-locked state, where all of the oscillators are in phase. The proof of the existence of a phase transition requires one to understand the geometry of a high dimensional polytope, shown here (inset) in the three dimensional case.

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Rick Gorvett, Associate Professor – Risk ModelingModeling a complex system such as a hurricane (or an earthquake, or a terrorist event, or any other risk) involves the use of sophisticated mathematics from a number of areas. Statistical analysis of historical data can provide insights into the future probabilities of different events. Regression analysis can help to explain the causes of events, and the interrelationships between the system’s underlying risk factors. Stochastic processes and differential equations permit mathematical descriptions of how systems can change over time. Taken together, these and other mathematical techniques provide a foundation for modeling the frequency and severity of potential risk events, and thus the possible impact on our life, health, and economic well-being.

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George Francis, Professor – Mathemematical Visualization in the ArtsDeBruijn’s methods realize quasicrystals as projections of 3D sublattices of the 6D cubical lattice to pack 3-space. These 3D Penrose tilings are quasicrystalline because they display icosahedral symmetry locally, but are aperiodic globally. My lifelong avocation to visualize mathematics turned to real-time interactive computer animation when I could no longer hand-illustrate sphere eversions and Thurston’s non-Euclidean space geometries. With the advent of the CAVE and Cube, I expanded my collaboration beyond scientific visualization to the arts. New York mathematical artist Tony Robbin, my students and I here explore quasicrystals in the Beckman Cube immersive virtual environment. The ability to change latitude, season, and hour of day, saves Tony much guesswork in designing physical quasicrystal installations. Since the tools resulting from this synergy are familiar to the artist, there is no learning curve to impede creativity.

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Yuliy Baryshnikov, Professor – Quantum Random WalksQuantum random walks were born as a mathematical construct underlying Grover’s quantum search algorithm. Quantum random walks on lattices in Euclidean space do not resemble their classical counterparts at all: where the latter keep close to the starting point (deviating, after N steps, by the distance of order √N and spread in the familiar and universal Gaussian shape, the quantum random walks spread to the distance of order N in moire-like sharp patterns. The first steps in understanding these patterns were made by Baryshnikov, Pemantle and their students. The image here shows amplitudes of a planar quantum random walk with unitary 4-coin.

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Department of MathematicsUniversity of Illinois at Urbana-Champaign

1409 W. Green StreetUrbana, IL 61801

www.math.illinois.edu

Ahlgren, Scott Albin, Pierre Ando, Matthew Athreya, Jayadev Balogh, Jozsef Baryshnikov, Yuliy Bauer, Robert Bergvelt, Maarten Berndt, Bruce Boca, Florin Bradlow, Steven Bronski, Jared D’Angelo, John DeVille, Lee Di Francesco, Philippe van den Dries, Lou Dutta, Sankar Dunfield, Nathan Duursma, Iwan Erdogan, Burak Feng, Runhuan Fernandes, Rui Loja

Ford, Kevin Francis, George Gorvett, Rick Haboush, William Hieronymi, Philipp Hinkkanen, Aimo Hur, Vera Mikyoung Ivanov, Sergei Johnson, Paul Junge, Marius Kapovich, Ilya Katz, Sheldon Kedem, Rinat Kerman, Ely Kirkpatrick, Kay Kirr, Eduard-Wilhelm Kostochka, Alexandr Laugesen, Richard Leininger, Christopher Lerman, Eugene Li, Xiaochun McCarthy, Randy Merenkov, Sergiy

Mineyev, Igor Muncaster, Robert Nevins, Thomas Nikolaev, Igor Palmore, Julian Rapti, Zoi Rezk, Charles Reznick, Bruce Rosenblatt, Joseph Ruan, Zhong-Jin Schenck, Hal Solecki, Slawomir Song, Renming Sowers, Richard Tolman, Susan Tumanov, Alexander Tyson, Jeremy Tzirakis, Nikolaos Wu, Jang-Mei Yong, Alexander Zaharescu, Alexandru Zharnitsky, Vadim

Current Faculty

this calendar was designed by tori Corkery for the Department of Mathematics at the University of Illinois at Urbana-Champaign © 2013.

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Department of Mathematics · 2017-06-21 · Julian Palmore, Professor – Spaceflight...

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