Category: Talks

A special panel discussion featuring:

Maddie Brown `18 – NSF REU in mathematical analysis and applications at University of Michigan – Dearborn.

Caroline Edelman `18 – REU in dynamical systems at Boston College

Nate Mattis `19 – TEU program at Brown University

Alexander Murph `18 – Nielsen Professional Summer Analytics Experience (PSAE)

Leo Orozco `18 – Summer Security Intensive, IT Lab Fellow at Carnegie Mellon University

Pizza and sodas for everyone!

Title: What Do You Mean, It’s Hard?

Abstract: Suppose someone gives you a computer and asks you to perform one of the following tasks:

1. solve a 17 × 17 × 17 Rubik’s cube, or
2. decide if a given list of 100 integers can be broken into two parts having equal sums.

If your life depended on it, which task would you choose? Which is harder, computationally?

In 1971, Stephen Cook proposed a strong measure of efficiency – polynomial time, or simply P – as a desirable standard to which we should hold solutions to computational problems. Task A is an instance of a problem with such a solution. He also identified a seemingly less stringent measure – nondeterministic polynomial time, or simply NP – in which one merely has to check, efficiently, that a given solution is correct. Tasks A and B are both instances of problems satisfying this condition. The big question raised by Cook is whether these two measures of computational efficiency are actually distinct.

One can argue that the “P ≠ NP Problem,” as it is now known, is the most important open problem in all of mathematics and computer science. Certainly, it has far-reaching implications both within these two fields and beyond. Cook showed in his 1971 paper that there are NP problems that seem really hard: remarkably, if you can solve any one of these problems efficiently, then you can solve every NP problem efficiently. Problem B is an instance of one of these “NP-complete” problems. To solve P ≠ NP, one could look for NP problems that are unlikely to be hard in this sense and try to show that they’re also not easy. We have yet to find such a problem, but in this talk I will try to persuade you that there are some candidates worthy of further scrutiny!

 

Title:  Fluid Flow Around Slender Bodies in Viscous Fluids: From Swimming Worms to Bacterial Carpets

Abstract:   There are many biologically relevant situations which involve long slender bodies (e.g. worms, flagella, bacterial bodies, etc.) where it is important to understand the dynamic interactions of the body and the low Reynolds number fluid in which it moves. In this presentation, I will be discussing applications of the method of regularized stokeslets to periodically moving bodies in fluids. These models have applications to the study of locomotion as well as fluid mixing.

Title:  Proofs (not) from the Book

Abstract:  The eminent mathematician of the 20th century, Paul Erdos, often mention “The Book” in which God keeps the most elegant proof of every mathematical theorem. So, attending a mathematical talk, he would say: “This is a proof from The Book”, or “This is a correct proof, but not from The Book”. M. Aigner and G. Ziegler authored the highly successful “Proofs from THE BOOK” (translated into 13 languages). In this talk, I shall present several proofs that are not included in the Aigner-Ziegler book but that, in my opinion, could belong to “The Book”

Title:  What can topology tell us about the neural code?

Abstract:  Cracking the neural code is one of the central challenges of neuroscience. Neural codes allow the brain to represent, process, and store information about the outside world. Unlike other types of codes, they must also reflect relationships between stimuli, such as proximity between locations in an environment. In this talk, I will explain why algebraic topology provides natural tools for understanding the structure and function of neural codes.

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:  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.