PERSPECTIVES, APPETIZERS, APPLICATIONS, AND BOOTSTRAPS 

ABSTRACTS

[09:30 – 10:15]

Kasper Olsen and Jakob Bohr, DTU Nanotech

Twist Neutrality and the Diameter of the Nucleosome Core Particle

The nucleosome core particle consists of DNA coiled around eight histone proteins. We suggest that its unique size is determined by two geometrical principles: Its DNA is both optimally packed, i.e. optimizes a volume fraction, and a zero-twist structure, i.e. has zero strain-twist coupling. Reference: J. Bohr and K. Olsen, Phys Rev Lett 108, 098101 (2012)

[10:30 – 10:50]

Magnus Ögren, DTU Mathematics, and Marcus Nyström, Humanities Laboratory, Lund University.

How do university students solve problems in vector calculus? Evidence from eye tracking

Today, learning outcomes in schools and universities are mainly measured with written exams, where calculations and answers to a number of problems are evaluated and graded. You can at best get a glance of how the students have identi?ed the problem, how the problems have been represented and the methods of solutions. However, too often it is impossible from the written information to obtain any detailed knowledge about the processes the students went through leading up to their answers. Several researchers have used eye tracking to trace the problem solving process in mathematics and related disciplines on a level corresponding to high school education. However, it seems to be a lack of studies in higher education, in particular within the mathematical domain. In this study, we investigate how students divide their attention across text, equations, and graphical illustrations of problems in vector calculus. Since vector calculus is a subject where there is a major need for ?gures explaining the physical interpretation of mathema­tical formulas, we are in particular interested in the dynamics between obtaining information from text, mathematical formulas, and ?gures. We collected eye movements and speech from 36 second year students from the engineering physics program two weeks into a vector calculus course. The students solved eight problems related to vector calculus presented with text and equations or text, equations, and a graphical illustration of the problem. They were asked to ’think aloud’ and verbalize their thoughts while solving the problems. The experiment was self-paced, with the restri­ction that each problem had a maximum allowed presentation time of two minutes. After each question, the students were asked to answer a true or false statement about the problem, and to state how con?dent they were about their answer. Overall, we found no evidence that illustrations increased the number of correct answers. However, there were large inter-problem differences; the illustrations signi?cantly increased the number of correct answers in two of the problems and decreased the number of correct answers in one problem. The illustrations also did not seem to change the relative amount of total dwell time on the text and equations compared to the non-illustrated problems. Instead, the time when the illustrations were inspected was taken equally much from other parts of the problem. These results suggest that care should be taken when designing an illustration to a problem in vector calculus, which, instead of helping the students, may have a detrimental effect on learning.

[11:00 – 11:30]

Magnus Ögren and Jakob Møller-Andersen, DTU Mathematics

General perturbative trace formula for U(n) → SO(n) symmetry breaking

We derive an approximate semiclassical trace formula for the density of states of an isotropic harmonic oscillator (HO) in an arbitrary spatial dimension n, perturbed by an arbitrary even term εr^α. This term breaks the U(n) symmetry of the HO, resulting in a hyper-spherical system with SO(n) symmetry. We treat the anharmonic terms in semiclassi­cal perturbation theory by integration of the action of the perturbed periodic HO orbits over a suitable manifold. The manifold can be a complex projective space CPⁿ⁻¹ or a higher dimensional sphere S²ⁿ⁻¹, dependent on the parame­terization chosen for the classical HO orbits. For the latter case i.e. using a S²ⁿ⁻¹ manifold, we here give the details of the derivation of the trace formula through the map to two lower dimensional Sⁿ⁻¹ spheres. In the limit of ε (or the energy) → 0, the trace formula is restored to the well known HO trace formula with U(n) symmetry, i.e. where each

1 _Πn−1

energy level N have a degeneracy dⁿ = _j=1 (N + j).

N (n−1)! A MORE POPULAR ABSTRACT: You learn during the ?rst course in quantum mechanics that a con?ned particle can have discrete energy levels that can be obtained by solving the Schrödinger equation from 1926. But already in 1913 Niels Bohr used a model with circular classical orbits for the electron in order to derive the energy spectra of the Hy­drogen atom. Although Bohr’s model contained several assumptions that contradict modern quantum mechanics, the derived spectra were correct in this case. The ?eld connecting classical physics and quantum mechanics is generally called semiclassical physics and is a vital theory and important toolbox in many sub?elds of theoretical physics today. In particular the energy levels of a quantum system can be obtained by starting from the solutions of the particle orbits in the corresponding classical systems, this is called periodic orbit theory (POT) and writing up a series for the density of energy states have some formal analogies to Fourier series.

[12:30 – 13:15]

David Brander, DTU Mathematics

Singularities in surface theory

A singularity of a map between two differentiable manifolds is a point at which the derivative does not have full rank. The word can also be used for a point in the domain where the map blows up, but this is not the meaning con­sidered in this talk. Singularities arise in varied contexts. An example is the the "outline"of the surface of some object in the real world as perceived by the eye. This is a "fold"singularity of the map from R² to R² given by the image of the object impinging on the retina. In surface theory, which we may take here to mean the theory of special surfaces in R³ or in some other 3-dimensional homogeneous space, singularities often arise naturally. For example because the surfaces have a very natural representation, such as the Weierstrass representation for minimal surfaces, in terms of some simple data, and this data usually includes surfaces with singularities. Surfaces with no singularities at all are generally very special solutions to the problem in question, and there are examples, such as constant negative curvature surfaces , which have no (complete) singularity-free solutions. Generic, or stable, singularities are those which persist under deformations of the surface within the same class. These can be thought of as the "typical"singularities. For the class of smooth surfaces in R³ there is only one kind of stable singularity, but for other classes of surfaces, there are more types, and it is a natural question to try to ?nd the stable singularities in a given surface class. For surfaces associated to integrable systems, which can be constructed using loop group methods, there are ma­ny singularities which are hard to understand, and also hard to avoid. This is because they arise in the process of constructing the surface through a loop group decomposition and cannot be seen in the simple data from which the surfaces are constructed. We will discuss some of the background of this, and say something about recent work on the singularities of constant mean curvature surfaces in Lorentz 2 + 1 space.

[13:30 – 14:15]

Ana Hurtado, Dept. of Geometry and Topology, University of Granada, Spain

The isoperimetric problem in the sub-Riemannian 3-sphere

We consider the sub-Riemannian metric gh on S³ given by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector ?eld. In this talk we study stable area-stationary C² surfaces with a volume constraint in (S³ ,gh) and we solve the isoperimetric problem assuming C² regularity of the isoperimetric solution.

[14:30 – 15:15]

Vicent Gimeno and Vicente Palmer, Dept. of Math. Universitat Jaume I -INIT, Castelló, Spain.

Volume growth of submanifolds and number of ends, Cheeger isoperimetric constants, and Laplacian eigenvalues

The ?niteness of the volume growth of certain submanifolds with controlled radial mean curvature in an appropri­ate ambient manifold which possess a pole determines the Cheeger isoperimetric constant and the number of ends of the submanifold. During the talk we will explore these relations for minimal submanifolds of the Hyperbolic space Hⁿ(b), and ?nally we will characterize the Dirichlet spectrum of the Laplacian for these submanifolds.

[15:30 – 16:15]

Antonio Alarcón, Dept. of Geometry and Topology, University of Granada, Spain

Minimal surfaces and approximation theorems

We will discuss about the basic theory of minimal surfaces and pay particular attention on how this theory has be­en strongly in?uenced by approximation theorems in complex analysis, such as Runge’s and Mergelyan’s theorems.