## Fall 2016 – Summer 2017 – Abstracts

Wednesday, September 7, 2016. Mathematics Colloquium.

Prof. Peter Winkler, Dartmouth College Mathematics and Computer Science.
“Probability and Controversy”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Mathematics boasts perhaps the fewest controversies of any academic subject—if you make a mathematical claim you have to prove it, otherwise it’s just a conjecture.

Nonetheless there are a few contentious areas and several are connected to the notion of probability. We’ll explore some probability puzzles and paradoxes, and perhaps put some dents in the idea that in math there’s always a right answer.

Wednesday, September 21, 2016. Mathematics Colloquium.
Prof. Mary Lou Zeeman, Bowdoin College Mathematics.
“A dynamical systems framework for exploring resilience.”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Resilience is a slippery concept that has different meanings in different contexts. It is often described as the ability of a system to absorb change and disturbance while maintaining its basic structure and function. The ambiguity of the description provides an anchor for rapid intellectual connection across disciplines. The struggle for precision in the context of a particular application deepens the interdisciplinary conversation. From a management point of view, the goal may be to increase resilience (e.g. of biodiversity or of crop yield to weather extremes), or to decrease resilience (e.g. of invasive species, or of the collapsed state of a fishery).

From a dynamical systems point of view, the different meanings of resilience are often about interactions between transient dynamics of a system and disturbance to the system. In this talk, we subject the flow of an autonomous system of ODEs to regular shocks (“kicks”) of constant size and direction. The resulting flow-kick systems occupy a surprisingly under-explored area between deterministic and stochastic dynamics. Natural questions to ask include: Does the resulting “flow-kick” system equilibrate? If so, where? What are the dynamics near the flow-kick equilibrium? And does that represent resilience?

Wednesday, September 28, 2016. Mathematics Colloquium.
Prof. Thomas Enkosky (B.A. ’03 & M.A. ’04 UMaine), Boston University Mathematics and Statistics.
“The Algebra of Rigid Structures”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: A graph is a collection of vertices and edges and is often depicted visually with points and straight line segments. Suppose we draw a graph in the plane and fix the lengths of the line segments but allow the edges to pivot about the vertices. The drawing is rigid if the graph keeps its shape when the vertices are moved. In this talk we will explore some of the algebra of graph rigidity.

Wednesday, October 5, 2016. Mathematics Colloquium.
Prof. Rob Niemeyer, University of Maine Mathematics and Statistics.
“The T-fractal billiard and fractal flat surface.”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: The T-fractal billiard is a rather interesting example of a fractal billiard table. Much is known on the existence of periodic orbits of the T-fractal billiard. We will discuss a candidate for the T-fractal billiard phase space. The second half of the talk will then focus on giving the construction of the corresponding fractal flat surface and an infinite interval exchange transformation derived from a particular trajectory on the surface.

Thursday, October 13, 2016. Mathematics Colloquium.
Prof. Chris Johnson, Wake Forest University Mathematics and Statistics.
“Non-singular ergodic theory, affine interval exchanges, and the T-fractal”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: The motion of an ideal point-mass in a self-similar polygonal region in the plane, called the T-fractal, naturally leads to the study of an infinite interval exchange. The self-similarity of this infinite interval exchange yields a finite affine interval exchange. Though such maps do not preserve Lebesgue measure, many ergodic theoretic questions can still be asked about these maps.

In this talk I will describe a family of maps generalizing the affine interval exchange associated with the T-fractal, and describe a construction which associates a measure-preserving transformation on a space of infinite measure to each such map.

Wednesday, October 26, 2016. Mathematics Colloquium.
Prof. Lauren Ross, University of Maine Civil & Environmental Engineering.
“Three-dimensional tidal flow in a frictional fjord-like basin with converging width: an analytical model”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: A three-dimensional analytical model was used to understand tidal wave dynamics in fjord-like basins. This model allows the width of the basin to decay exponentially with along-channel distance from the mouth. Both the length scale of exponential convergence, Lb, and the friction parameter, Av (vertical eddy viscosity), were the free parameters. Model results show amplification of the tidal amplitude toward the head of the basin. Amplification depends on the narrowing rate of the funnel-like width of the channel and on friction. Cross-channel variations in along-channel tidal flow are also sensitive to the friction parameter. A typical along-channel tidal flow distribution was found across the channel when the vertical eddy viscosity was characteristic of a basin with strong friction. Maximum along-channel tidal velocities (ranging from 0.25 to 0.5 m s-1 depending on width convergence strength) were located in the center of the basin and at the surface. Decreasing friction resulted in along-channel velocity maxima located near the lateral boundaries and subsurface in the middle of the channel. These tidal flow distributions were verified with observations from Reloncavi Fjord, Chilean Patagonia.

Wednesday, November 2, 2016. Mathematics Colloquium Video Lecture.
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB).

Ali Özlük was a Professor of Mathematics at the University of Maine for 25 years until his untimely death in 2012. Later that year, the Quebec-Maine number theory conference was held in his memory. The plenary speaker was Prof. Peter Sarnak who spoke about Ali’s work and its relation to a larger program in number theory. This is a video presentation of Sarnak’s 2012 lecture.

Prof. Peter Sarnak, Institute for Advanced Study, Princeton.
“Symmetry types for families of L-functions”

Abstract: We review some of A.E. Özlük’s work concerning correlations of zeros of Dirichlet L-functions and in particular his work (with C. Snyder) on low lying zeros of quadratic L-functions. His results can be explained by a symmetry associated with general families of automorphic L-functions, a theory that has been actively pursued over the last 15 years. We will describe some of the basics of this theory as well as some recent advances.

Wednesday, November 9, 2016. Mathematics Colloquium.
Prof. Thomas Hulse, Colby College Mathematics & Statistics.
“Counting Lattice Points in Spheres”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: In number theory, there is an old problem due to Gauss which asks how precisely we can estimate the number of integer lattice points inside the interior of a circle of a variable radius. In higher dimensions, there are analogous questions about the number of integer lattice points in n-Spheres. In this talk, we will explore these questions, related questions, progress that has been made, and progress that we are trying to make now. This is based on joint work with Chan Ieong Kuan, David Lowry-Duda and Alexander Walker and should be accessible to a general math audience.

Wednesday, November 16, 2016. Mathematics MA Thesis Defense.
Ayesha Maliwal, University of Maine Mathematics & Statistics. (Advisor: Franzosa)
“Sperner’s Lemma, The Brouwer Fixed Point Theorem, The Kakutani Fixed Point Theorem and their applications in social sciences”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB).

Abstract: Does an envy-free division of a cake exist such that each individual is content with their share? In a game, does an equilibrium strategy exist where no player is motivated to change their approach? Does an equilibrium exist in a goods market where the demand for goods is entirely met by the supply and there is no tendency for prices to change? In this talk, we will explore the answers to these questions using three important results in mathematics: Sperner’s Lemma, the Brouwer Fixed Point Theorem and the Kakutani Fixed Point Theorem. We will also examine the interconnection between these theorems, treating Sperner’s lemma as a combinatorial analog of The Brouwer Fixed Point Theorem, and the Kakutani Fixed Point Theorem as a generalization of the latter.

Friday, December 2, 2016. Graduate Seminar.
Zach Connerty-Marin, Mathematics MA student.
“Applications of the 2D Wavelet Transform Modulus Maxima Method to Off-Axis Holograms”
3:00 – 3:50 pm, 421 Neville Hall.

Abstract: Off-axis holography provides high resolution images of small structures, but image processing is difficult due to noise and interference fringes inherent to the methodology. The 2D wavelet transform modulus maxima (WTMM) method is a noise-resilient mathematical microscope, which allows for spatially-localized examination of image gradients across a continuous range of size scales. This talk covers the concept of wavelets, the 2D WTMM methodology, and an example application to bacterial tracking in holograms. The 2D WTMM method is applied to off-axis holograms to distinguish signal from noise, characterize and model noise, and segment image structures. This coupled with basic properties of Airy disks may be used to automatically track particles in off-axis holograms.

Wednesday, December 7, 2016. Mathematics Colloquium.
Prof. David Krumm, Colby College Mathematics & Statistics.
“The global dynamics of quadratic maps on the projective line”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB).

Abstract: An important open problem in the field of arithmetic dynamics is the Uniform Boundedness Conjecture of Morton and Silverman. Despite many efforts over the last two decades, even the simplest case of the conjecture has not been proved. In this talk we will discuss an ongoing research program that involves refining the conjecture in the setting of quadratic maps on the projective line, and developing techniques to tackle certain simplified cases of the conjecture in this setting.

Friday, March 3, 2017. Graduate Seminar.
Caroline Reno, UMaine Mathematics MA student.
“Sequences on the Circle: Gaps, Density, and Uniformity”
3:00 – 3:50 pm, 421 Neville Hall.

Abstract: If two bodies are orbiting in the same plane at different velocities, where do they align? We will discuss the history of this problem and consider the sequence of points where bodies align. In particular, we will look at the density of this sequence on the circle with unit circumference, the size of the gaps between successive points, and the uniformity of these gaps. This will lead to a discussion of what it means for a sequence to be uniformly distributed modulo 1, a concept which is stronger than density.

Wednesday, March 22, 2017. Mathematics Colloquium.
Prof. Naomi Tanabe, Dartmouth College, Department of Mathematics.
“Central Values of L-functions”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB).

Abstract: Analyzing the special values of L-funcions has been a significant target of research ever since the Riemann zeta function was introduced in the eighteenth century, as they play an important role in many fields including number theory. For example, it is known that the zeros of the Riemann zeta function are deeply connected to the distribution of prime numbers. In this talk, I will survey various L-functions and results particularly concerning the vanishing or nonvanishing of their central values.

Wednesday, March 29, 2017. Mathematics Colloquium.
Prof. Rob Benedetto, Amherst College, Department of Mathematics and Statistics.
“The arboreal Galois group of a PCF cubic polynomial”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB).

Abstract: Let $K$ be a number field (such as the field $Q$ of rational numbers), let $f$ contained in $K(x)$ be a rational function of degree $d$, and let $a$ be contained in $K$. The roots of $f^n(z)-a$ are the $n$-th preimages of a under f, and they have the natural structure of a $d$-ary rooted tree $T$. There is a natural Galois action on the tree, inducing a representation of the absolute Galois group of $K$ in the automorphism group of $T$. In many cases, it is expected that the image of this arboreal Galois representation has finite index in the automorphism group, but in some cases, such as when f is postcritically finite (PCF), the image is known to have infinite index. In this talk, we will present the first complete calculation of the arboreal Galois group of a PCF map, namely $f(z)=3z^2-2z^3$, that is not induced by an endomorphism of an algebraic group.

Wednesday, April 5, 2017. Statistics Colloquium.
Prof. Liam O’Brien, Colby College Mathematics & Statistics.
“Marginal Pairwise Associations Arising from Multiple Source Data”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB).

Abstract: We consider the situation when prediction of a univariate outcome is of primary interest in a regression setting. In this scenario we assume that data have been collected from more than one predictor. Standard regression methods allow one to estimate the effect of each predictor on the outcome while controlling for the remaining predictors. The regression coefficient for each predictor has an interpretation that is conditional not only on the predictor being considered, but is conditional on all other predictors as well. In settings in which interest is in comparison of the marginal pairwise relationships between each predictor and the outcome separately, standard regression methods can no longer be used. Instead, we consider maximum likelihood (ML) estimation of these marginal relationships allowing comparisons to be made among the separate marginal associations for each predictor.

Thursday, April 27, 2017. MA Thesis Defense. (Advisor: Khalil)
Zachary Connerty-Marin, Mathematics MA candidate.
“Wavelet-Based Particle Tracking in Unreconstructed, Off-axis Holograms”
12:00 – 12:50 pm, 421 Neville Hall.

Abstract: Standard numerical focusing and reconstruction of hologram time series for detailed 3D particle tracking is slow and computationally expensive. Motion detection and particle tracking is an unsolved problem in unreconstructed, off-axis holograms. This thesis proposes an automated wavelet-based method of tracking particles in unreconstructed, off-axis holograms, with the purpose of providing rough estimates of the presence of motion and particle trajectories in hologram time series.

The wavelet transform modulus maxima (WTMM) multifractal method is used to estimate background noise as a Hurst exponent 𝐻~ − 0.04 in a real hologram time series. This is used to create hologram time series simulations for calibration purposes. A WTMM segmentation method is developed to extract Airy disks, which represent particles, from hologram time series. This method is run on simulations and extended to a real hologram time series. The method accurately tracks particle positions in the 𝑥𝑦-plane. Depth estimation works on some simulations, but breaks down in noisy real data.

Depth estimation may be improved by coupling the method in this thesis with phase demodulation methods. Particle density is limiting in this method as the desired closed chains are not found when the central lobes of Airy patterns overlap. Particle centroids can be stitched together using algorithms more robust to missed detections. The method proposed in this thesis shows potential for motion detection, and estimating particle tracks in low-particle- density time series or in time series where all tracks behave roughly the same.

Friday, April 28, 2017. MA Thesis Defense. (Advisor: Knightly)
Caroline Reno, Mathematics MA candidate.
“Uniform Distribution of Hecke Eigenvalues”
3:00 – 3:50 pm, 421 Neville Hall.

Abstract: We will give three results about uniform distribution of Hecke eigenvalues of modular forms. The first two are known results and the last is a new result. We will start with some background about modular forms, including two examples: the Eisenstein series and the Poincaré series. We will then define a specific set of orthonormal polynomials, the Chebyshev polynomials of the second kind. These polynomials will be useful to prove the main results, which are about the distribution of certain sequences relative to a measure on [-2,2] called the Sato-Tate measure. The sequence we will look at consists of normalized Hecke eigenvalues, obtained from Hecke operators of modular forms.

Friday, May 5, 2017. Graduate Seminar.
Kenneth Bundy, Mathematics MA candidate.
“Signal Processing, the Hilbert-Huang Transform, and Leak Spectrum Identification”
3:30 – 4:20 pm, 421 Neville Hall.

Abstract: Air leaks in spacecraft pose a danger to the crew and the success of the mission.  In work at the WiSe-Net Lab at UMaine, we are using simulated leaks in various materials to address this issue.  The technique uses spectral decomposition and other frequency based analysis techniques to identify leaks in aluminum, steel, plastic and rubber.

However, extracting information from noisy signals can be difficult; common techniques include Fourier Analysis and the Hilbert-Huang Transform. This talk will focus on the formulation of the Fourier transform, and then the Hilbert-Huang Transform (HHT), building parallels to Fourier Analysis as we go. Once the HHT has been introduced, we will discuss this application to identification of leaks based on their frequency spectrum.