Title: Proofs Without Witnesses: Zero Knowledge Proofs
Abstract: Peggy wants to convince Victor that she possesses a secret. Victor doesn’t believe that Peggy is telling the truth while Peggy doesn’t trust Victor enough to show him the secret. Zero knowledge proofs provide a method by which Peggy can convince Victor that she has the secret without giving Victor any information about the secret itself. In this talk we’ll construct and analyze several zero knowledge proofs, and discuss how they can be used in a wide array of areas including computer security and nuclear disarmament.
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: Quasinormality and weak quasinormality of operators
Abstract: There are two notions to define the quasinormality of unbounded operators by Kaufman and Stochel-Szafraniec respectively. Our results show that Kaufman’s definition of an unbounded quasinormal operator coincides with that given by Stochel-Szafraniec. In this talk we discuss various characterizations of unbounded quasinormal operators. Examples demonstrating the sharpness of our results are constructed. An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. This approach establishes a new definition to be called weakly quasinormal operators. Some characterizations concerning to the weakly quasinormal operators are discussed. In addition, various examples and counterexamples illustrating the concepts of this work are constructed by using weighted shifts on directed trees.
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.