Monday

Mini-course 1. Mathematical Modeling in Ecology: Population Dynamics

Lidia Mrad

Abstract: The aim of this mini-course is to explore some models which have been central to the development of mathematical modeling, and which provide examples of how mathematics can be used in understanding ecological interactions. We focus on dynamical systems, which historically have been bound up with the calculus. However, we assume no knowledge of calculus and build up ideas using computational tools. Models we will study include the logistic growth model, competitive and symbiotic species interaction models, and predator-prey models.

Mini-course 2. What is dimension?

Sean McCurdy

Abstract: We all have the intuition that lines are one-dimensional, planes are two-dimensional, and cubes are three-dimensional. But why? What does that mean? And how do we apply that intuition to sets which are not so “nice”? It turns out, there are many different ways to make our intuitions of “dimension” rigorous. Surprisingly, these different ways don’t always agree! This mini-course invites participants into mathematics as a creative process through the question “What is dimension?” Attempts to answer this question will take us through 5 definitions of “dimension,” with plenty of opportunities for participants to come up with their own definitions along the way. We will see some strange things (fractional dimensions, infinite dimensions, even negative dimensions!) as well as some big ideas in Linear Algebra, Topology, and Analysis. The course will be structured loosely as a dialectic. Each class will develop a new definition of “dimension” and end with an exploration of its strengths and limitations. Problem Sessions for this mini-course will give participants the opportunity to A) further explore the theory in which a given definition of “dimension” is framed, or B) invent new definitions of dimension which fix the limitations. The next class will focus upon seeing where these the new definitions lead.

Informal conversation

Robert Bridges

Talk. In an era of data, can we permit meaningful analytics and protect our privacy?

Robert Bridges

Abstract: As collection and storage of data on entities and individuals becomes more frequent, methods for ensuring privacy while permitting data analytics are needed. Quintessential examples include releasing analytics from US Census data for demographic studies or using Machine Learning (ML) on electronic healthcare records to better predict future treatments and diagnoses. Differential privacy (DP), a field of math and computer science, has become the accepted standard for releasing analytics from (esp. releasing ML models trained on) a private dataset. The fundamental idea is to allow outsiders to query or “work on” the data but protect privacy with clever use of, well, lying a little bit… or maybe fibbing, … fudging it!! By injecting randomness into any algorithm used on the data, DP releases information as a sample of a probability distribution and in doing so provides a formal mathematical guarantee of privacy of the entities represented in the data. The tradeoff is a loss of accuracy. (Contrast this concept with encryption which seeks to 100% conceal private data from outsiders but furnishes it exactly—no accuracy problems—to those authorized).

When approaching DP initially, one may find the subject is not accessible; it is presented in terse, esoteric mathematical constructions, requiring expertise and ample time to distill it into intuitive concepts. The goal of this talk is to give an introduction to DP that focuses on the concepts behind the mathematics of DP up to a basic understanding of the state of the art (SOTA) for training and releasing ML on private datasets. Finally, I plan to conclude with a brief overview of our current research that is beginning to push the state of the art in differentially-private ML.

Worktime

Talk. Circles, sets, neighborhoods, and killing monsters

Katherine Brubaker

Abstract: When I say that I’m a mathematician who doesn’t like numbers, I get funny looks. But it’s true! I like shapes, functions, order and abstract curvy things; I love the sense of logical adventure in “Math Land”, where there always is an answer. (Though, whether you can prove it is another matter.)

We’ll explore some math-y things with no numbers in them. You can use circles to organize information and settle arguments. Marking off neighborhoods is a good way to measure accuracy in killing monsters. And killing monsters is a fun way to explore the epsilon-delta definition of a limit, which is fundamental to any analysis class and understanding how to work with “abstract curvy things”. If you like, we can talk about my research, which involves an infinite-dimensional abstract curvy thing made up of different ways to measure distance. Fun!