Research StatementDaniel Visscher

IntroductionI study dynamical systems and ergodic theory. Dynamical systems has its origins in the di�erential

equations of classical mechanics, while ergodic theory is rooted in the statistical analysis of complexsystems. Work of Poincaré and Boltzmann in the late 19th century popularized the use of qualitativemethods that characterize these fields, and which have been fruitful in many contexts: from modelingthe physical movement of planets and molecules, to animal populations and infectious diseases, to socialand economic networks, all rooted in the assumption that the present state of a system determines allfuture states. In studying the global statistical properties of gas molecules, Boltzmann developed anergodic hypothesis that the amount of time a system spends in a given state can be measured by the sizeof that state in a phase space. This is a basic tool in computational sciences, and it has also received a lot ofa�ention theoretically. In a foundational paper from the 1960s, Anosov showed that a su�icient conditionfor a dynamical system to satisfy the ergodic hypothesis is that it is uniformly hyperbolic—characterizedby the local expansion and contraction of the system under the dynamics [1].

My research focuses on two questions: Which dynamical systems exhibit hyperbolicity? And how muchhyperbolicity is needed to generate interesting statistical properties such as ergodicity? I am especially inter-ested in geometric dynamical systems. For example, in billiard dynamics, a ball bounces o� boundariesof a table specularly, and its orbit is determined by the shape of the table. Or, in a geodesic flow, a beamof light travels along a straight path, and its orbit is determined by the shape of the space. The tools Iuse come from analysis, di�erential geometry, and topology. Computations are an important facet of mywork, and computer programs, such as Mathematica, help me make computations as well as simulateexamples for further study. I am enthusiastic about mentoring undergraduate students in mathematicalresearch, and my scholarship is well-suited for undergraduate research projects.

Additionally, I have research interests in the teaching and learning of undergraduate mathematics,especially in alternative formats for supporting and assessing student learning.

Geometric systems and hyperbolicity—Which dynamical systems exhibit hyperbolicity?

Hyperbolicity is a local dynamical property of a system. When hyperbolicity is uniform over theentire phase space, a dynamical system is called Anosov. The standard examples of Anosov geodesicflows, coinciding with those that appeared in Anosov’s original paper, come from manifolds with strictlynegative curvature. While negative curvature of either a manifold or billiard table boundary is a su�icientcondition for producing Anosov dynamics in the corresponding flow, it is not a necessary condition. I aminterested in nonstandard examples of Anosov geodesic and billiard flows.

Hyperbolic geodesic flows. A surface that is embedded in R3 necessarily has some areas of positivecurvature, so such surfaces cannot be everywhere negatively curved. It is known that there exist surfacesof very high (but unspecified) genus that can be embedded in R3 such that they have an Anosov geodesicflow [2]. V. Donnay and I have explicitly constructed such a surface with specified genus, thus providingexamples of these surfaces [3]. Our construction is based on quantifying the Anosovity of a non-embeddedmodel metric, and then quantifying the dynamical e�ects of a perturbation to the metric. This tells ushow far we can safely wander from the model metric while looking for an embedded metric. We plan tocontinue work on finding smaller genus embedded surfaces with Anosov geodesic flow, as well as studyingother dynamical and geometric properties of the examples we have produced.

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Figure 1: Pictures from REU work of I. Garfinkle. The middle figure is part of the phase portrait for a biangle billiard table onthe surface of revolution with profile curve f(z) = (1− z2)5. One of the regular orbits is shown on the le� (table boundary inblack, orbit in blue); an orbit that does not appear to be part of the integrable part of the billiard system is on the right.

Hyperbolic billiard dynamics. Hyperbolicity is also important in billiard dynamics. Some exam-ples of hyperbolic billiards are well-known: Sinai billiards, whose boundaries are negatively curved andcause families of parallel trajectories to spray out upon impact, or billiards with positively curved focusingcomponents that cause such families of trajectories to focus and then spray out before hi�ing the nextboundary. One such billiard table is the Bunimovich stadium, which provides a good example of howsmall changes in the parameters of a dynamical system can cause drastic changes in the behavior of thesystem ([4] gives an application of this idea to climate change).

One can also consider billiard tables on surfaces other than the flat plane, in e�ect making the in-terior surface of the billiard table curved. Separately, geodesic flows on curved surfaces and flat billiardtables with curved boundaries are well-studied. I am interested in how these two types of curvature in-teract in producing billiard dynamics. Billiard tables on the round sphere and on the hyperbolic planeare commonly studied examples of billiards on curved surfaces; billiard tables on surfaces with variablecurvature are not well-studied. This past summer, I supervised research by Isaac Garfinkle, an under-graduate student from Carleton College, considering billiard tables on surfaces of revolution, which havevariable curvature. He programed geodesic equations for surfaces of revolution and specular reflectionsfor billiard tables into Mathematica and plo�ed phase portraits of the resulting dynamical systems (seeFigure 1). He found, in contrast to tables formed by two intersecting geodesics on the round sphere thatare always completely integrable, that there are tables formed by two intersecting geodesics on surfacesof revolution that appear to not be integrable and in fact exhibit hyperbolicity along at least some or-bits [5]. This work provides some further advanced research directions. For instance, one way to provenon-integrability of a dynamical system is by studying the behavior of local stable and unstable mani-folds, which can in addition say something about the topological entropy of the system. I also envisionusing this Mathematica code for other undergraduate research projects investigating other sets of tableson surfaces of revolution (e.g., circular tables, or perturbations of tables known to produce hyperbolicityon the sphere).

Global statistical properties and partially hyperbolic systems—How much hyperbolicity is needed to generate interesting statistical properties such as ergodicity?

Uniformly hyperbolic systems exhibit a range of desirable dynamical properties. Such systems arestably ergodic, and they have positive Lyapunov exponents and positive entropy, unique maximizing mea-sures of various kinds, and good statistical properties concerning closed orbits and how they distribute.Weaker forms of hyperbolicity, in some cases, also produce these properties.

Partial hyperbolicity is a natural way to relax the conditions of hyperbolicity. A flow is hyperbolic if thetangent space perpendicular to the flow can be decomposed into sub-spaces that exponentially expandor contract under the flow, and it is partially hyperbolic if the tangent space has directions of hyperbolic

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behavior whose expansion and contraction dominate the behavior of the remaining central directions.

Stable ergodicity. An ergodic dynamical system whose close-by systems are all ergodic is calledstably ergodic. In many contexts, almost all partially hyperbolic dynamical systems are stably ergodic.Conversely, it is known in some se�ings that stable ergodicity implies partial hyperbolicity (e.g., [6]).For geometric systems, this stability is with respect to geometric perturbations of the manifold or billiardtable. In order to study such behavior, one needs to know how perturbing the system geometrically a�ectsthe local and global dynamics. In my thesis, I describe the connection between geometric perturbations ofa manifold or billiard table and algebraic perturbations of the dynamics along geodesics [7, 8]. This is animportant step towards showing that stably ergodic dynamics necessarily have some kind of hyperbolicity,which is still an open question for geodesic flows. I plan to continue developing perturbation tools forgeodesic flows, as well as investigating tools in similar se�ings such as billiard flows or contact flows.

Maximizingmeasures. I also study the global statistical properties generated by partially hyperbolicdynamics. A special case of partially hyperbolicity occurs when the central directions of the dynamics areisometric. This is the case for frame flows on negatively curved surfaces, where orthonormal bases areparallel transported along geodesics. The geodesic flow is hyperbolic, but the frames get rotated as theytravel along a geodesic and so generate isometric directions in the flow.

One way to study the global statistical properties of a dynamical system is by asking for measuresthat maximize a certain quantity. For instance, the pressure of a dynamical system is a number that eval-uates the complexity of a dynamical system weighted according to a “potential” function. The measure-theoretic pressure assesses the amount of entropy and the amount of function value that the measuresees. Measures that maximize this quantity are called equilibrium measures. In thermodynamics, multipleequilibrium measures indicate the presence of a phase transition in the system.

R. Spatzier and I show that for certain frame flows and other partially hyperbolic systems that alsohave isometric behavior in the central direction, there exists a unique equilibrium measure, with respectto which the frame flow is ergodic [9]. As a corollary, we reprove a classic result of Brin and Gromov thatframe flows are ergodic with respect to the natural volume measure. We are working on generalizing ourmethods to a larger class of potential functions.

Undergraduate mathematics educationI also have research interests in the teaching and learning of undergraduate mathematics. I am es-

pecially interested in studying how di�erent forms of instruction and assessment influence and gaugestudent learning. One such project grew out of looking for ways to improve calculus students’ concep-tual understanding. M. Bode, M. Khorami and I conducted a study concerning student use of interactivefeatures in a calculus e-book [10]. We found that students reported these resources improved their con-ceptual understanding, but also that they did not o�en incorporate the interactive features into theirpersonal studying and probably need support for utilizing such resources e�ectively.

N. White and I are currently working on a comparative study of oral and traditional wri�en assess-ments given to pre-service elementary teachers. We investigate why certain students perform be�er (com-paratively to their peers) on one form of assessment over the other, and we find that the type of oralassessment given (“oral assessment with tutor”) benefits students with problem solving anxiety [11]. Thisfinding has further led to a refinement of our instrument for measuring types of math anxieties. Whilethere are many surveys that measure math anxiety, their ideas of what math is do not look at all like themathematics of our active learning classrooms. We developed 15 new items to test for problem solvinganxiety and math explanation anxiety, and we ran a study validating the instrument [12]. We believe thistool will be useful to other researchers and practitioners beyond our own study.

Additionally, we are interested in the instructional value of these oral assessments. We are analyzing

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video data and coding observed learning gains in order to be�er understand how students make newconnections in their knowledge. This past academic year, I co-mentored an undergraduate researcher,Cooper Agar, as part of this project. He looked at video recordings and categorized the types of learninggains over the course of the oral assessment, which were evident from the students’ verbal and visualpresentations, and he compared these with the types of words and phrases that the students used toanswer questions [13]. We have received support for this research from a Center for Research on Learningand Teaching grant at Michigan, as well as through an NSF-funded REBUILD grant, and the project isongoing.

References

[1] D.V. Anosov. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Stek. Inst.Math. 90 (1967).

[2] V. Donnay, C. Pugh. Anosov geodesic flows for embedded surfaces. Geometric methods in dynamicsII, Astérisque 287(xviii), 61–69 (2003).

[3] V. Donnay, D. Visscher. Constructing embedded surfaces with Anosov geodesic flow. In progress.[4] V. Donnay. Is our climate headed for a mathematical tipping point? TED-Ed original

video, http://ed.ted.com/lessons/is-our-climate-headed-for-mathematical-chaos-victor-j-donnay

[5] I. Garfinkle. Billiard dynamics on surfaces of revolution. University of Michigan Summer REU report(2016).

[6] R. Saghin, Z. Xia. Partial hyperbolicity or dense elliptic periodic points for C1-generic symplecticdi�eomorphisms. Trans. Amer. Math. Soc. 358, no. 11, 5119–5138 (2006).

[7] D. Visscher. A new proof of Franks’ lemma for geodesic flows. Discrete & Continuous DynamicalSystems - Series A 34(11), 4875–95 (2014).

[8] D. Visscher. A Franks’ lemma for convex planar billiards. Dynamical Systems 30, 333–340 (2015).[9] R. Spatzier, D. Visscher. Equilibrium measures for certain isometric extensions of Anosov systems.

Ergodic Theory & Dynamical Systems. To appear. doi: 10.1017/etds.2016.62.[10] M. Bode, M. Khorami, D. Visscher. A Case Study of Student and Instructor Reactions to a Calculus

E-Book. PRIMUS 24(2), 160–174 (2014).[11] D. Visscher, N. White. Using oral assessments in mathematics content courses for pre-service ele-

mentary teachers: expanding our measurement of student learning. In progress.[12] D. Visscher, N. White. Measuring mathematics engagement anxiety. In progress.[13] C. Agar. Oral assessments and learning opportunities. University of Michigan UROP poster (2016).

Updated October 2016 4 / 4