Research Experience for Undergraduates Program Research Reports Indiana University, Bloomington Summer 2019 Contents Unusual Topographical Features of Laplacian Eigenfunctions .. 5 Kelly Chen and Olti Myrtaj Extremal Length of Winding Curves .................. 15 Max Newman Mixing of Random Walks on Punctured Graphs .......... 32 Chung Kyong Nguen Cauchy-Fueter Formulas for Universal Clifford Algebras of In- definite Signature ............................ 43 Ely Sandine On the Large Time Diffusion of Isotropic and Anisotropic Particles 72 Mikhail Sweeney Bernoulli Percolation on Hyperbolic Graphs ............. 92 Christine Sullivan Spectral Properties of Reversible Indel Models ........... 105 Linden Yuan Preface During the summer of 2019 eight students participated in the Research Experi- ence for Undergraduates program in Mathematics at Indiana University. This program was sponsored by the National Science Foundation through the Re- search Experience for Undergraduates grant DMS-1461061 and the Department of Mathematics at Indiana University, Bloomington. The program ran for eight weeks, from June 3 through July 26, 2019. Eight faculty served as research advisers to the students from Indiana University: Kelly Chen and Olti Myrtaj were advised by Chris Judge. • Max Newman was advised by Dylan Thurston. • Chung Kyong Nguen was advised by Graham White. • Ely Sandine was advised by Matvei Libine. • Mikhail Sweeney and Linden Yuan was advised by Louis Fan. • Christine Sullivan was advised by Chris Connell. • Following the introductory pizza party, students began meeting with their faculty mentors and continued to do so throughout the next eight weeks. The students also participated in a number of social events and educational oppor- tunities and field trips. Individual faculty gave talks throughout the program on their research, about two a week. Students also received LaTeX training in a series of work- shops. Other opportunities included the option to participate in a GRE and subject test preparation seminar. Additional educational activities included tours of the library, the Low Energy Neutron Source facility, the Slocum puz- zle collection at the Lilly Library, and self guided tours of the art museum. Students presented their work to faculty mentors and their peers at various times. This culminated in their presentations both in poster form and in talks at the statewide Indiana Summer Undergraduate Research conference which was hosted at Indiana University - Purdue University at Indianapolis. On the lighter side, students were treated to weekly board game nights as well as the opportunity to do some local hiking. They also enjoyed a night of \laser tag" courtesy of the Department of Mathematics. Contents 3 The summer REU program required the help and support of many different groups and individuals to make it a success. We foremost thank the National Science Foundation and the Indiana University Bloomington Department of Mathematics without whose support this program could not exist. We espe- cially thank our staff member Mandie McCarty for coordinating the complex logistical arrangments (housing, paychecks, information packets, meal plans, frequent shopping for snacks). Additional logistical support was provided by the Department of Mathematics and our chair, Elizabeth Housworth. We are in particular thankful to Jeff Taylor for the computer support he provided. Thanks also go to those faculty who served as mentors and those who gave lectures. We thank David Baxter of the Center for Exploration of Energy and Matter (nee IU cyclotron facility) for past personal tours of the LENS facility and his infor- mative lectures. Thanks to Andrew Rhoda for his tour of the Slocum Puzzle Collection. Chris Connell September, 2019 Contents 4 Figure 1: REU Participants, from left to right: Chris Connell, Kelly Chen, Mikhail Sweeney, Ely Sandine, Max Newman, Linden Yuan, Olti Myrtaj, Chris- tine Sullivan and Chung Kyong Nguen (not pictured). Unusual Topographical Features of Laplacian Eigenfunctions Kelly Chen∗ Olti Myrtaj† [email protected] [email protected] Summer 2019 Abstract Solutions to the Laplace eigenvalue problem satisfying Dirichlet boundary conditions on a rectangular region of the plane are well known. In this work, we study interesting features of the level sets of such eigenfunctions. We consider the possibility of degenerate critical points with Poincar´e-Hopfindex 0, which correspond to cusp-like points on the level sets. In particular, we construct a one-parameter family of eigenfunctions and prove the existence of an index-0 critical point. 1 1 Introduction Let Ω be a domain with piecewise smooth boundary. A smooth function ψ :Ω R is called a Dirichlet eigenfunction of the Laplacian if and only if there exists a real number λ such that → ∆ψ(x, y) = λ ψ(x, y) for (x, y) Ω, (1) ψ−(x, y) = 0 · for (x, y) ∈ ∂Ω. ∈ The number λ is called the eigenvalue associated with the eigenfunction ψ. It is known [Ulb76] that for a generic choice of domain, the critical points of each eigenfunction are nondegenerate. In particular, there are no zeros of the gradient vector field ψ that have Poincar´e-Hopfindex equal to zero. (Lemma 2.5 describes the connection between the index and the nondegeneracy∇ of a critical point.) We restrict our attention to the case where Ω is the square [0, π] [0, π], which is not a generic domain. We show that in this case, there are in fact critical points of index 0: × Theorem 1.1. There exists a Dirichlet function of the Laplacian on [0, π] [0, π] with an index-0 critical point in (0, π) (0, π). × × On the square, one may verify that if m and n are integers, then the function ψm,n(x, y) = sin (mx) sin (ny) 2 2 is a solution to (1) with λ = λm,n = m + n . In fact, it is is well-known [Crn-Hlb, pp. 300-301] that each eigenvalue has the form m2 + n2, where m and n are integers, and a function ψ is a Dirichlet eigenfunction of the Laplacian with eigenvalue λ if and only if ψ = am,nψm,n, 2 2 m +Xn =λ where am,n are real numbers. In this paper, we consider λ = 65, the smallest λ that generates a four-dimensional eigenspace. We observe that 12 + 82 = 42 + 72 = 65, so we construct the linear combination f(x, y) = a ψ (x, y) + a ψ (x, y) + a ψ (x, y) + a ψ (x, y). 1,8 · 1,8 4,7 · 4,7 8,1 · 8,1 7,4 · 7,4 ∗Massachusetts Institute of Technology †Virginia Tech 1This material is based upon work supported by the National Science Foundation under Grant No. DMS-1461061. 1 We study the one-parameter family of eigenfunctions f t(x, y) = cos(t) sin(x) sin(8y) + sin(t) sin(4x) sin(7y), t [0, π], ∈ which contains all possible f with a8,1 = a7,4 = 0, up to scalar multiplication. To prove Theorem 1.1, we show that there exists a t such that f t has an index-0 critical point. 2 Preliminaries Recall the following definitions: 2 Definition 2.1. A function g : R R has a critical point at (x0, y0) if and only if the gradient g(x, y) = (gx(x, y), → ∇ gy(x, y)) is the zero vector at (x0, y0). Definition 2.2. A critical point (x0, y0) of g is called degenerate if the Hessian gxx(x, y) gxy(x, y) g (x, y) g (x, y) yx yy has determinant equal to 0 at (x0, y0). 2 2 2 Definition 2.3. Let v: R R , where v = (v1, v2), be a smooth vector field, and let γ be a simple closed curve in → R2. If v does not vanish on γ, then the Poincar`e-Hopfindex (or simply index) of v along γ is given by 1 v dv v dv ind(v, γ) = 1 2 − 2 1 . (2) 2π v2 + v2 Zγ 1 2 2 2 Note that in order for ind(v, γ) to be well-defined, v1 + v2 must be nonzero on γ. If v is the gradient vector field of a function g : R2 R, then this condition is equivalent to g having no critical points on γ. Next, we consider→ some properties of the index and some relevant examples. Proposition 2.4. Let v be a smooth vector field and let γ be a smooth simple closed curve. Then ind(v, γ) is an integer. Proof. Abusing notation slightly, we let γ :[a, b] R2 be a parameterization of γ. If we define → x dy y dx ω := − x2 + y2 then v1dv2 v2dv1 v∗(ω) = 2 − 2 . v1 + v2 Therefore, by a standard fact concerning differential forms we find that v1dv2 v2dv1 − = v∗(ω) = ω. v2 + v2 Zγ 1 2 Zγ Zv(γ) The curve v(γ) is a closed curve—not necessarily simple—that does not contain the origin since v = 0 on γ. Define 6 α :[a, b] R2 by → v γ(t) α(t) = ◦ . v γ(t) | ◦ | Then the image of α lies in the unit circle. Moreover, a straightforward argument shows that (v γ)∗ω = α∗ω, and so ◦ ω = ω. Zv(γ) Zα 2 1 2 2 To be more precise, γ is an oriented curve—that is, an equivalence class of C paths where two paths p:[a, b] R and p0 :[a0, b0] R 1 → → are equivalent if and only if there exists a strictly increasing C map φ:[a, b] [a0, b0] such that p = p0 φ. For simplicity, we will not explicitly state this again in the rest of the paper. → ◦ 2 Define the map p from R to the unit circle by p(t) = (cos(t), sin(t)). A straightforward calculation shows that p∗(ω) = dt. Since [a, b] is contractible, the path lifting lemma implies that there exists a path α :[a, b] R so that α(t) = p α(t).