#studio applied math

[studioapma-blog]

Studio Week 5

We experimented with a different structure for our time this week. Visiting critic was Dr Jeff Hoffstein from the Math Department, whose research is largely in number theory.

Each research project gave a 5-minute pitch of their topic and current questions, and what they saught to work on during the rest of class. Following this, both Hoffstein and Bjorn were able to jump between groups as they worked and offer advice or just to talk.

Mara continues to refine her methods for numerical PDE solving. Henry and Uthsav have abandoned their originally-intended neural network in favor of one with hand-selected features for the machine learning algorithm to pick up on when comparing CAs. Lukas and Daniel have completed a classification of their linear polynomials and have refined their code so that they can generate some statistical results on higher-degree polynomials.

After some work time, we turned our attention to a GISP evaluation form that gave us a moment to reflect on the semester.

We think this type of open work day could be really effective if more of our projects overlapped or with a guest critic with more encompassing research interests.

This was our last day of class! The next time we meet will be for a final presentation, on Friday, May 15th.

[studioapma-blog]

Studio Week 4

We were fortunate to have Professor Richard Kenyon from the Mathematics Department as our guest critic. With a background in statistical mechanics and discrete geometry, he had a lot of great insight into the projects by Henry & Uthsav and Daniel & Lukas, which deal more with number theory and discrete math. Mara also presented, though Professor Kenyon noted he had less familiarity with the topic of ecological models. Some interesting suggestions he proposed: for Henry’s project, potentially examining a cylindrical, continuous domain on which to compute cellular automata - since there are a finite number of possible patterns on the cylinder of fixed radius, all rules become periodic - a metric that could be used to classify rules. For Daniel & Lukas, Professor Kenyon suggested looking at special cases of polynomials with no translational component, and to rework their algorithm to calculate number of components by running an iterated labeling scheme.

[studioapma-blog]

Studio Week 3

We were joined by Dr John Gemmer of Brown's Division of Applied Mathematics this week, with Henry as our facilitator. He first spoke briefly about his own research in stretching on elastic sheets and physical PDEs.

Shawn was first to present, and has abandoned implementation of STDP in lieu of focusing on implementing multiple timescales in the forcing voltage, and analyzing the regimes of the resultant two-neuron activity. He has been perfecting his pitch of the science to a general audience.

Mara more fully illustrated her research's connection to her thesis work and continues to seek turing bifurcations in her two-species nutrient-competition model. She is working on technical details of PDE solving packages in MATLAB, which Gemmer contributed a lot of thoughts to.

Lukas and Daniel revisited the structural questions they have on graphs and showed some output from recent simulations. They are narrowing in on a specific goal for their research. Gemmer proposed examining only polynomials constructed by multiplication of linear terms, and that this will work well with extensions of fields into the complex numbers.

In addition to our research projects, we are preparing for our poster fair next week, where we will join all other Group Independent Study Projects at Brown in a presentation of our work. We have chosen to focus on our pedagogical framework rather than research subjects to better capture public attention.

[studioapma-blog]

Studio Week 2

Unfortunately Professor Geman was unable to make it to our crit today. However, we still had a crit among ourselves, which yielded very fruitful discussions and insights.

Daniel and Lukas- lots of results are looking at large finite fields (as n goes to infinity), but what about small graphs? Those are a lot more intuitive and easy to graph. Consider looking at adjacency matrices, maybe try using spectral theory results (for instance, the Laplacian has nice properties related to clustered components). There are many questions one can ask: what’s the average number of components? Which graphs in a finite field have structurally isomorphic graph? Got into a discussion on efficient ways to compare graphs and check for isomorphisms.

Uthsav and Henry- Implemented neural net for classifying MNIST digits. Going to apply to CA classification next week.

Mara- no updates for this week, but she’s presenting her thesis next week on very related material!

Shawn- Small updates to model (adding different types of currents for synapse delay), mostly still reading papers. Going to try and implement the papers’ suggestions into the algorithm for next week.

We reflected on the course as well what we can add to our poster. We figured that this structure would serve well for senior classes as well as some introductory classes, although care must be taken re: size/scope of class, strength of students, and other factors. But it’s certainly doable!