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Research Seminars: Graduate Student Colloquium


Past Graduate Student Colloquium:

Spring 2017

Time & Location: All talks are on Tuesdays in Stanley Thomas 316 at 4:30 PM unless otherwise noted.
Organizer: Alexej Gossmann



September 5

A simplified human birth model - translation of a rigid cylinder through a passive elastic tube

Roseanna GossmannTulane University

Abstract:

In order to better understand the forces on an infant during birth, this work uses a simplified model to explore the effects of fetal velocity and viscosity of the surrounding fluid on the forces associated with human birth. The model represents the fetus moving through the birth canal using a rigid cylinder (fetus) that moves at a prescribed velocity through the center of an elastic tube (birth canal). The entire system is immersed in highly viscous fluid. Low Reynolds number allows for the use of the Stokes equations to govern the fluid flow. The discrete elastic tube through which the rigid cylinder passes has macroscopic elasticity that may be matched to tubes used in physical experiments. This framework is used to explore the force necessary to move the rigid inner cylinder through the tube, as well as the buckling behavior of the elastic tube. More complex geometries as well as peristaltic activation of the elastic tube can be added to the model to provide more insight into the relationship between force, velocity, and fluid dynamics during human birth.


September 12

Beyond Perfect Graphs -- Hypercycles and Perfect Hypergraphs

Jonathan O'RourkeTulane University

Abstract:

In attempting to extend the notion of perfect graphs to the class of hypergraphs, my research partner and I studied a class of hypergraphs which bear some resemblance to cyclic graphs. We studied the associated primes of the cover ideals associated to this class of hypergraphs, as well as their index of stability. This study resulted in an easy-to-describe class of hypergraphs which answer a question of Francisco, Van Tuyl, and Ha regarding the relationship between the index of stability and the chromatic number of a family of hypergraphs, and in fact proving a stronger result. I will explain the preliminaries necessary to understand the problem and some of the techniques used to solve it.



September 19

Private Set-Union Cardinality: a cryptographic protocol for privacy-preserving distributed measurement

Ellis FenskeTulane University

Abstract:

There are many contexts where we wish to collect data about use of a system (e.g. a computer network, medical system), but simultaneously wish to respect the privacy of these users, and it is not obvious how to do this. The Tor network is our motivating example: users connect through Tor to protect their privacy, and system operators are generally volunteers who believe in this mission and will not compromise the privacy of their users. Yet data about the network is crucial to improve it and for research and funding opportunities for network operators. While it is a solved problem to aggregate all measurements from each relay in a privacy-preserving way, the case where the same measurement can be recorded by two distinct data collectors so that we must aggregate *unique* measurements is much more complex. I will present work from a paper I have published in collaboration with researchers at Georgetown University and the US Naval Research Laboratory that solves this problem.


September 26

A New Notion of Constructive Cardinality

Nathan BedellTulane University

Abstract:

Many mathematicians in the constructive tradition have some misgivings about Cantor's theorem and the existence of uncountable sets. In this talk, I will explain some of the basic principles of constructive mathematics, and why one might be skeptical of the ontological claim that uncountable sets exist. I then show that this view is not unreasonable in light of Cantor's theorem by seeing the constructive view of Cantor's theorem as analogous to the classical view of Russell's paradox. This argument then motivates a new conception of cardinality in terms of graded category theory, which is more in line with constructive intuitions. In particular, I will show that there are non-trivial graded categories in which all infinite sets have, in my terminology, the same absolute cardinality.


October 3

Accurate Integration of High Dimensional Functions using Polynomial Detrending

Lin LiTulane University

Abstract: The accuracy of numerical integration of high dimensional functions is an important problem in many industrial applications. Numerical quadrature built on lattice grid can quickly suffer from the curse of dimensionality. Monte Carlo and Quasi Monte Carlo method have provided a convergence rate independent of dimensionality. Unfortunately, the errors of these Monte Carlo methods converge very slowly when there are large variations in the underlying high dimensional integrand. We proposed a new method, polynomial detrending as an efficient way of variance reduction, which can provide a desired accuracy for high dimensional integration problem even with a small number of sample points.

 




October 17

A Probabilist's Perspective

Cooper Boniece Tulane University

Abstract:

Probability Theory and Analysis are closely related disciplines.  As such, there are many results that lie squarely at the intersection of these two areas.  However, there are also some results in Analysis that are inherently non-probabilistic, for which a probabilist's perspective yields new understandings.  These perspectives also offer insight into the myriad connections between these two disciplines. In this expository talk, after introducing some facts about martingales and Brownian Motion, we'll explore some probabilistic approaches to a wide variety of topics and theorems, from Analysis and elsewhere, including: The Dirichlet Problem; Picard's Little Theorem; The Fundamental Theorem of Algebra and more!



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Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu