Title: The Mathematics of Soap Films
Abstract: Every child knows that when you blow bubbles the surface that results is a sphere. But why? We’ll talk about this, more exotic examples, and a little bit about the group of Princeton mathematicians that became obsessed with soap films in the 1960s.
Title: Weighted shifts on directed trees
Abstract: The main goal of this research is to implement some methods of graph theory into operator theory. We do it by introducing a new class of Hilbert space operators, which we propose to call weighted shifts on directed trees. We have studied the structure of these operators since 2012 and obtained some remarkable results about those operators. In this talk we discuss some of the established properties for such operators, for examples, normality, hyponormality, and subnormality, etc. It is well-known that the subnormality of Hilbert space operators is closely related to Stieltjes moment sequences. We consider directed trees with one branching vertex and establish a connection between the subnormality and Stieltjes moment sequences by using the k-step backward extendability of bounded subnormal weighted shifts. In addition, some exotic examples of weighted shifts on directed trees related to Stieltjes moment sequences are discussed.
Title: Beyond Standard Scores: Using Item-level Responses From Clinical Measures to Detect Atypical Developmental Patterns
Abstract: Developmental deviance (DDEV) refers to the non-sequential attainment of milestones within a developmental domain. This observation is in contrast to developmental delay (DD), where milestones are reached in the typical sequence, but the timeline of attainments is delayed. There is evidence that DDEV is associated with certain neurobehavioral diagnoses, such as autism spectrum disorder (ASD). Clinically, the attainment of developmental milestones is assessed through standardized measures of developmental domains. Many psychometric tests are arranged hierarchically, and on the surface, two individuals with the same overall score on a clinical measure may appear to be impaired to a similar extent. However, at the itemized level, individuals with DDEV exhibit a more scattered pattern of incorrect answers. Differentiating between DDEV and DD may inform prognosis and predict long-term outcomes. We developed a measure of scatter, called deviance index (DI), to differentiate between DD and DDEV using standardized measures of language ability. We tested the accuracy of DI to predict ASD diagnosis, and by extension DDEV, among individuals from the New Jersey Language and Autism Genetics Study (NJLAGS) cohort. Using our DI metric, we found that individuals with ASD and a language impairment (LI) exhibit more DDEV across measures of expressive, pragmatic, and metalinguistic language compared to individuals with LI alone. By distinguishing between DDEV and DD, DI was able to predict ASD diagnosis among LI/LI+ASD probands. DI can be applied to measures across multiple developmental domains in order to characterize developmental profiles of individuals with DDEV/DD.
Title: Modeling lamprey swimming with mathematics and computation
Abstract: Locomotion — swimming, running, flying — is one of the most basic animal behaviors. In order for an animal (like a lamprey) to swim, a lot has to happen, including neural signaling, muscle contractions, interactions between the body and the water, and adjustments from sensory feedback. How do we use mathematics to better understand the underlying mechanisms that drive locomotion? We will explore how we can use mathematical models to describe these individual systems and then coordinate the systems to produce a naturally emerging behavior in computer simulations of swimming lampreys.
Title: The multiplicity one theorem for paramodular forms.
Abstract: Classical modular forms are known to have the “strong multiplicity one” property: A cuspidal eigenform is determined by almost all of its Hecke eigenvalues. Siegel modular forms, on the other hand, do not have this nice property. The main theorem presented in this talk states that strong multiplicity one still holds for an important class of Siegel modular forms known as “paramodular forms”. The latter have gained prominence because of their appearance in the “paramodular conjecture”, a degree-2 version of Shimura-Taniyama-Weil.
Title: What is Number Theory?
Abstract: In its original meaning, Number Theory is concerned with the properties of the “natural” numbers 1, 2, 3, … In this talk we will attempt to explain how the consequent study of “elementary” properties of numbers leads naturally to the theory of automorphic forms and the vast web of conjectures known as the “Langlands program”.
Title: 1+2+3+4+… = -1/12
Abstract: Infinite series. Every Calculus II student hates them; every mathematician loves them. So, why do mathematicians love infinite series? Well, ask one, and you’ll never hear the end of it… (just sum math humor).
We’ll see how two standard examples of series, the geometric series and the zeta function (p-series), lead to beautiful applications, such as in economics and signal processing. We’ll also see how even divergent series are fascinating.
Title: Wavelet-based Probit Functional Models
Abstract: Recent advances in functional regression include the development of a wide range of function-on-scalar regression models where outcome is a function of time regressed onto a set of scalar covariates. Most of these methodologies require the outcome to be normally distributed. However, very little work has been done on the case where the outcome is categorical. In this talk, we will discuss a flexible Bayesian procedure for function-on-scalar regression for categorical functional outcomes of varying levels. Simulations and data examples will be presented to illustrate the methodology.
Title: Losing Altitude: A story of airplanes, heart rate, and one “controversial” dataset
Abstract: When you ride in an airplane, the lowered pressure in the cabin causes your blood oxygen levels to decrease. If oxygen saturation levels go low enough, you may experience some interesting side effects (like temporary color blindness, for example), that much we know. What we don’t know is if there is also an impact on the functioning of the heart. Partially motivated by a rash of flight related medical case studies, a study was conducted in 2007 to more formally assess the effects of exposure to altitude on heart health while in flight. We will explore this study, the complex dataset it generated, and the “controversy” of conducting research a certain airline manufacturer would rather we not.
Title: What Did You Do Last Summer?
Moderator: Alexander Murph ’18
Presenters:
Trevor Adriaanse ’17 – Cryptanalysis & Exploitation Services Summer Program, at the NSA.
Alexander Murph ’18 – Research Apprentice for the Bucknell Geisinger Research Initiative (BGRI)
Laura Papili ’17 – Actuarial Internship at Genworth in Richmond, VA. Genworth’s Actuarial development program.
Ryan Buzzell ’17 – AEW Capital Management, L.P. (Boston Office). Real Estate Investment Firm.
Katie Lunceford ’17 – Susquehanna International Group LLP (SIG)/Statistical Options Trading/Intern – Bala Cynwyd, PA
Tung Phan ’17 – Susquehanna International Group LLP (SIG)/Statistical Options Trading/Intern – Bala Cynwyd, PA
Naba Mukhtar ’18 – REU: Partial Differential Equations and Dynamical Systems – Florida Institute of Technology
Abstract: There are many exciting summer opportunities for students in the mathematical sciences! These range from internships in financial companies to research experiences at other universities to leadership development programs. In this week’s colloquium, a panel of your peers will tell you their experiences. What did they enjoy about their experiences? When did they apply? There will also be ample time for questions and answers. These varied opportunities, as well as being terrific fun, are also immensely valuable as you begin to think about your careers after Bucknell.