Studio Applied Math

"Studio Applied Math: Topics in Emergent Behavior" is a student-made experimental applied math class in Spring 2015 at Brown.

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And it’s done!

We had our final presentations on May 15th. Despite a (tiny) audience, the presentations were great—fully-formed, clear, engaging, and informative. People took different venues to articulate their final projects: Mara focused on a narrative over the entire semester, for instance, while Henry and Uthsav illustrated the results of their final push.

We are still considering best ways to give this project some long-lasting institutional memory. The only person graduating from the class is Mara—all others are here in Providence at least one more semester.

Thanks again to Björn Sandstede for being a communicative, supportive, helpful, and energizing advisor.

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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.

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Wrapping up Studio Applied Math: Topics in Emergent Behavior. Big thanks to the Brown CRC, our advisor Björn Sandstede, and the team of students that has made it awesome!

Had a poster fair today to share our project with friends and fellow GISP-ers this semester.

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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.

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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.

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Working on our "poster."

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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!

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Studio Week 1

Today was our first true studio week in the class, and our guest critic was Dr Martin Maxey, whose research highlights multi-phase fluid dynamics and computational schemes.

Three projects presented:

Uthsav & Henry reviewed their work on CA and L-system computation in python before seguing into the focus for their final project, which is to develop a neural network to classify 1D cellular automata. We focused on discussions of machine learning and how to transition to higher-dimensional CAs.

Mara, after a review of phytoplankon blooms based on nutrients and competing populations, moved to discussions on two plankton populations and methods of computation (PDE2 package on MATLAB) at the surface of ocean flows. The first goal is to replicate a bateria model found in a paper by Schmitz et al., which is a similar problem to phytoplankon blooms, but without movement. Maxey helped us discuss concerns with various numerical techniques.

Shawn spent time summarizing his Hodgkin-Huxley model from the midterm and the theta model for neuron modeling. He has implemented modeling of networked two neurons and discussed various ways to extend the model for the end of the semester. His focus is on introducing Synaptic Time-Dependent Plasticity (synapse synchronicity dependence for capacitance) and the crit revolved on public communication of the work.

We moved into discussing, as a group, the end-of-semester GISP presentation and the possibility of a poster, deciding that for a general audience it may be more advantageous to focus on the pedagogical impetus for the class rather than our particular research projects. We also discussed some guidelines for future facilitators and presenters to be more directive and specific in their questions to get more out of guest critics.

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Some Studio Reminders

Some key points as we enter studio weeks:

Guest critics & presenting students should both receive information about the other, such that we can go smoothly and not spend much time on trivial background information.

Direct your presentations with questions! While they are presentations, they are primarily opportunities to learn from your peers and the guest critic.

Visual and non-visual representations generate different questions; try prompting us with both or multiple descriptions.

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We had our first CRIT today (for the Midterm projects) at 180 George (the Brown CCV).

Visiting was Brown professor Brad Marston from the Physics department.

Crit went smoothly, and as we progress into final research topics we will refine our process of getting our guest critics up to speed with the variety of topics we will discuss in a single day.

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Final Projects Brainstorm

Last week, we discussed final project ideas with Bjorn. Projects are still in the planning stage, so ideas are subject to change.

Mara: Looking at systems of phytoplanktons in the ocean. Striped population patterns show up in equilibrium, but what happens when the turbulence in the water is factored in?

Shawn: Look at results of paper about a system consisting of an excitatory and inhibitory neuron. Feeding different inputs into the excitatory neuron gives different patterns. Can we investigate more?

Uthsav: Machine learning for different patterns of cellular automata. How well can we learn? Will we have to make a training set?

Dan: Looking at arithmetic dynamics over prime order fields. Perhaps over these fields we can tighten some bounds?

Afterwards, we had a small studio session where we worked on our midterm projects.

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Lesson 5: Daniel

Daniel's lesson focused on the subject of Arithmetic Dynamics—that is, looking at iterated functions and how the groups and fields they are performed on can affect stability.

We saw the functions and algebraic structures as frameworks, out of which steady-state solutions emerged. Interesting connections to directed graph theory were possible with functions considered in fields (i.e. with modular arithmetic).

Elements of a walk could be split into three types:

Periodic (a fixed point is a periodic point with period 1)

Pre-Periodic (those that lead into a periodic loop)

Wandering (non-periodic; requires an infinite parameter space)

Natural questions arise: how long is the typical walk until a periodic point? What proportion of a domain is wandering? How do you predict if walking "backwards" in a graph results in necessary jumping outside the target field?

We sat and discussed various questions like this while Henry quickly created a walk visualizer in Mathematica.

Tomorrow is a discussion on the shapes and forms for final research proposals. Our midterm crit (with guest Prof. Brad Marston of Physics) is next week Tuesday.

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CRC Check-In

We met with Wayne (one of the [G]ISP coordinators) this morning to check-in with our GISP's progress. He probed us with some good, reflective questions. By and large things are going well—we are feeling more comfortable with each other as peers and Bjorn is an amazing addition to our classroom.

Some ideas on things to do:

Collect anonymous feedback for the class at Midterms and review it collectively

Establish & codify a group-level learning goal beyond the soft ones in our syllabus.

Make a document at the end of the semester that captures the pedagogical goals and reflection on our success and failures

We are almost at the transition point into studio weeks!

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Planning Meeting

Due to a student drop, a lesson planned to cover Statistical Mechanics was cancelled. We found it worth our time to have a week to discuss a variety of things in the class:

scheduling our midterm critique & dates around spring break, to accomodate a few shifting schedules.

further ideas into faculty/guests/graduate students to invite to our weekly studio "crits"

how to structure the time & flow of the studio crit days

In addition, each class member pitched a proposal for their midterm projects. Our syllabus indicates this should be a "visualisation, simulation, document, or demo that can be used as a stand-alone pedagogical tool." We are all excited for GUI possibilities our topics can take.

We discussed the role of the weekly guest critics and roles that "crit" versus "studio work" time will play in our weeks to come and are striking a balance between structure and flexibility, to allow serendipitous conversations and collaboration to occur. Mara proposed having a "conversation facilitator" who takes charge of priming the conversation with leading questions and nurturing the group dynamics as we present and talk.

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Lesson 4: Shawn

For lesson 4, Shawn presented material on neural dynamics. We began with a discussion about small networks of neurons. In particular, the properties of a network consisting of a connected excitatory neuron and inhibitory neuron, and a model of how they fire with different levels of connectivity. Part of this discussion on firing neurons, and a MATLAB demonstration on varying parameters for the network, brought us back to Lukas’s lesson on synchronicity.

We then examined the Hodgkin-Huxley model, which describes action potential in neurons. After an introduction to the biology of action potentials, we broke down the equations of the model into their constituent components, comparing them to a more complicated (essentially extended HH) model from a paper on neuron firing and spreading depression by Schiff et al. We considered the consequences of the model under a Hopf Bifurcation (Mara’s lesson coming in handy here) and increasing injected current.

After some brief words of wisdom from Bjorn* on ways to approach reading scientific and mathematical papers, we concluded our discussion by examining a much simpler model proposed by Eugene Izhikevich, bifurcations on these models, and the advantages and disadvantages of the simpler model in the Izhikevich paper.

*A good way to go about reading papers is to first read the abstract, introduction, discussion, etc. to try and understand what the paper is trying to conclude. Then move on to diagrams and technical discussion. Repeat.

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Lesson 3: Lukas

This week’s topic was on Synchronicity and Small World Dynamics. As an intro, to better understand synchronicity, we discussed our previous notions of emergence. After some looks at disappointing Wikipedia definitions, we agreed that these topics described a disconnect between behavior at the different levels of a system.

On the subject of small-world dynamics, we looked at Strogatz method of rewiring random networks: connecting each node to its closest k neighbors, and then with probability p, reconnecting each node in a clockwise fashion to a randomly chosen node in the graph. We then examined the local clustering coefficients as p increased, trying to intuit at every step. Strogatz furthered that this method worked well to model disease transmission, and collaboration between artists on imdb.

Later on we discussed random graphs, particularly Erdős graphs where edges are created with probability p. At around p > 1/N, for N nodes in a graph, it is almost guaranteed to have unconnected parts with one large connected component. These graphs have a rigid degree distribution that does not model real-world networks well. To better understand those real networks, we then looked at a method by Barábasi to describe such ‘scale-free networks.’ This method starts with a strongly connected component, such as the Erdős graph, and adds new nodes that are connected a pre-existing node i with probability proportional to the degree of i. It can be shown that the resulting degree distribution is a negative exponential. The intuiting of the method is that new friends are more likely to connect with the most popular nodes.

After we had become more familiar with creating graphs, we looked at another feature of graphs: connectedness. To this light, we tried to create and describe different varieties of connected graphs: those that require more and more node deletions to become disconnected. After playing around with these graphs, we realized that much of our intuitions of the graphs could not be immediately described mathematically. Eventually, however, we discovered that higher eigenvalues corresponded to higher graph density.