**What is the format of the workshop?**

Students work in groups through the worksheets (see files below). Informal presentations and discussions on the most important problems occur throughout the day. The style is informal, with students working in a collaborative environment.

**How are the worksheets structured?**

Each day’s worksheet contains a list of definitions, theorems, and exercises (about 40 each day). Some of the problems were modified from those in the references, while others are problems written for this working group.

**Day 1 Worksheet**(tex file, tex label index, pdf file)

Projections and the Gram-Schmidt Process

QR Factorization

Least-squares

Linear Models: Regression**Day 2 Worksheet**(tex file, tex label index, pdf file)

Diagonalization

Symmetric matrices

Spectral Theorem

Quadratic Forms

Singular Value Decomposition**Day 3 Worksheet**(tex file, tex label index, pdf file)

Principal Component Analysis and Dimensional Reduction

Brief look at Markov Chains

LU Factorizations

Duals and annihilators

Some multilinear algebra

The “tex label index” files are useful when editing the tex files, as they list the labels used in the tex code.

**Acknowledgment**

These Linear Algebra Workshop materials may be freely used by others. We ask that when materials are re-used, the following statement be included:

These materials were created by Stefan Klajbor Goderich at the University of Illinois, with support from National Science Foundation grant DMS 1345032 “MCTP: PI4: Program for Interdisciplinary and Industrial Internships at Illinois.”

]]>The program has four components. Students early in the graduate program usually do the Linear Algebra Working Group, Computational Bootcamp, and Prepare/Train Group. Students later in the graduate program usually do the Computational Bootcamp and an Internship.

1. **Linear Algebra Working Group**

**May 16-18** (239 Altgeld Hall), led by Stefan Klajbor Goderich

2. **Computational Mathematics Bootcamp**

Part I: **May 21-26** (239 Altgeld Hall) with focus on Data Science, led by David LeBauer.

Part II: **May 30-June 1** (239 Altgeld Hall) with focus on Mathematica Fundamentals, led by A. J. Hildebrand

3. **Prepare and Train Group – Algorithms for Analytic Combinatorics**

Dates: **June 4-July 13** (with holiday on July 4)

Location: 159 Altgeld Hall (room booked in mornings)

Instructor: Stephen Melczer (U. of Pennsylvania)

The program is a “Research Experience for Graduate Students” style endeavor: after a series of introductory lectures, the students form small groups (2-5 people) to work on open-ended interconnected problems.

*Overview.* The goal is to guide students through the transition from working on “canned” problems to tackling open-ended problems and formulating the problems themselves. We expect the group work to involve a mixture of computational experiments (to generate conjectures) and theory (to prove them).

One of the draws of combinatorics is its ability to draw on, motivate, and even push forward diverse areas of mathematics and computer science. The focus of this program will be to study the methods of analytic combinatorics – a field drawing inspiration from complex analysis, differential geometry, and algebraic geometry – from the perspective of computer algebra.

Students will begin by learning the underlying theory and implementing algorithms which have been previously described at a theoretical level. Later, students will engage with open problems of both a theoretical and computational nature, and examine new applications of this fast-growing theory. A wide range of problems of varying difficulty will be available for students with different backgrounds.

Topics include the new theory of analytic combinatorics in several variables, effective enumeration results for power series coefficients of algebraic functions, and decompositions of multivariate rational functions. Potential applications touch on areas of queuing theory, representation theory, theoretical computer science, transcendence theory, probability theory, and (of course) combinatorics.

A more detailed statement of problems can be downloaded here.

4. **Internships**

Various dates. Hosts to be arranged.

]]>

**What is the format of the workshop?**

Students work in groups through the worksheets (see files below). Informal presentations and discussions on the most important problems occur throughout the day. The style is informal, with students working in a collaborative environment.

**How are the worksheets structured?**

Each day’s worksheet contains a list of definitions, theorems, and exercises (about 40 each day). Some of the problems were modified from those in the references, while others are problems written for this working group.

**Day 1 Worksheet**(tex file, tex label index, pdf file)

Projections and the Gram-Schmidt Process

QR Factorization

Least-squares

Linear Models: Regression**Day 2 Worksheet**(tex file, tex label index, pdf file)

Diagonalization

Symmetric matrices

Spectral Theorem

Quadratic Forms

Singular Value Decomposition**Day 3 Worksheet**(tex file, tex label index, pdf file)

Principal Component Analysis and Dimensional Reduction

Brief look at Markov Chains

LU Factorizations

Duals and annihilators

Some multilinear algebra

The “tex label index” files are useful when editing the tex files, as they list the labels used in the tex code.

**Acknowledgment**

These Linear Algebra Workshop materials may be freely used by others. We ask that when materials are re-used, the following statement be included:

These materials were created by Stefan Klajbor Goderich at the University of Illinois, with support from National Science Foundation grant DMS 1345032 “MCTP: PI4: Program for Interdisciplinary and Industrial Internships at Illinois.”

]]>David LeBauer, *Carl R Woese Institute for Genomic Biology*, University of Illinois

Neal Davis, *Department of Computer Science*, University of Illinois

**Teaching Assistant:**

Stefan Klajbor, *Department of Mathematics*, University of Illinois

**Room:**

239 Altgeld Hall

A two week course designed to introduce Math graduate students with little or no programming experience to methods in data analysis and computation. The goal is to prepare students to apply their understanding of math to solve problems in industry.

Courses from previous years – 2016, 2015, 2014 – focused on numerical analysis. This year the focus is shifting to the use and analysis of large and complex data.

Although the course is aimed at students with limited experience using software, you are expected to complete two introductory courses in order to become familiar with the basic syntax and operations in R and Python. Two free courses from DataCamp are **Required***; completion certificates must be mailed to the instructors by midnight May 25. *Each of these courses should take just a few hours to complete:*

*Students who have significant experience with R and / or Python may elect to substitute a more advanced course.

- May 26: Computing Basics
- May 30-June 2: Data and Statistics in R
- June 5-June 8: Data and Machine Learning with Python
- June 9: Conclusion and Project Presentations

**Linear Algebra Working Group**

May 22-24, in 239 Altgeld Hall

**Computational Mathematics Bootcamp:**

May 26-June 9, in 239 Altgeld Hall; note Memorial Day holiday on Monday May 29

**Prepare and Train Group – Machine Learning: Algorithms and Representations**

Dates: Monday June 12 through Friday July 21; note Independence Day holiday on Tuesday July 4

Location: TBD

Instructors: Maxim Raginsky (Department of Electrical and Computer Engineering) and Matus Jan Telgarsky (Department of Computer Science)

The program is a “Research Experience for Graduate Students” style endeavor: after a series of introductory lectures, the students form small groups (2-5 people) to work on open-ended interconnected problems.

The goal is to guide students through the transition from working on “canned” problems to tackling open-ended problems and formulating the problems themselves. We expect the group work to involve a mixture of computational experiments (to generate conjectures) and theory (to prove them).

The topics will focus on probabilistic and approximation-theoretic aspects of machine learning, with emphasis on neural networks. We will introduce the probabilistic formulation of machine learning and relate the performance of commonly used learning algorithms (such as stochastic gradient descent) to the concentration of measure phenomenon. Problems of varying levels of difficulty will revolve around several open questions pertaining to stability and convergence of stochastic gradient descent. We will also cover several results characterizing neural network function classes, for instance results saying that neural networks can fit continuous functions, that neural networks gain in power with extra

layers, and that neural networks can model polynomials. Open questions will cover more nuanced aspects of adding layers, as well as other neural net architectures, for instance convolutional and recurrent neural networks.

**Internships**

Various dates. Hosts to be arranged.

]]>

Tuesday June 7 – Saturday June 18 (no meetings on Sat-Sun June 11-12)

Location: 239 Altgeld Hall

Instructor: Anil Hirani.

More information is here.

**Prepare and Train**

Monday June 20 – Friday July 29

Location: 159 Altgeld Hall

Instructor: Maxim Arnold (U of Texas at Dallas), assisted by Yuliy Baryshnikov (U of Illinois at Urbana-Champaign)

The program is a “Research Experience for Graduate Students” style endeavor: after a series of introductory lectures, the students form small groups (2-5 people) to work on open-ended interconnected problems.

This year the overall theme is *Topologically constrained problems of statistical physics.*

The goal is to guide students through the transition from working on “canned” problems to tackling open-ended problems and formulating the problems themselves. We expect the group work to involve a mixture of computational experiments (to generate conjectures) and theory (to prove them).

More information is here.

Monday June 6 – Thursday July 28 (no meetings July 4-8)

Times: 5:30-7:00pm on Mondays and Thursdays

Location: 145 and 159 Altgeld Hall

Instructor: Derek Jung

Each session will begin with a 40-45 minute lecture. Then after a break, participants will work together on a problem set for 40-45 minutes. No work is expected outside of class.

Prerequisites: some linear algebra and proof techniques from prior math courses, along with enthusiasm and willingness to work with others. All undergraduate and graduate students are welcome to participate, whether or not you are involved in the other PI4 activities.

More information is here.

**Internships**

Various dates. Hosts to be arranged. Companies and faculty members interested in mentoring a student in this program, please register here.

**Application procedure**

Students: click here to apply to PI4 for Summer 2016.

]]>**Computational Mathematics Bootcamp**

Tuesday May 26 – Friday June 5

Location: 239 Altgeld Hall

Instructor: Sean Shahkarami

Coordinator: Anil Hirani

**Prepare and Train**

Monday June 15 – Monday July 27

Location: 159 and 173 Altgeld Hall

Instructor: Boris Shapiro, Univ. of Stockholm

Topics: Random matrix pencils, their related sorting networks, monodromy, and applications to physics. Students will explore open problems using theoretical and computational methods – the problems will cover several levels of difficulty, and so both beginning students (“Prepare”) and continuing students (“Train”) are welcome. The mathematical methods will be drawn from linear algebra, spectral theory, probability, geometry, and complex analysis.

**Internships**

Various dates. Hosts to be arranged.

Those interested in mentoring a student in this program, please register at https://illinois.edu/fb/sec/5413644

]]>* Jed Chou and Anna Weigandt* The image segmentation problem is the following: given an image, a vector of pixels each with three associated red, green, and blue (RGB) values, determine which pixels should belong to the background and which should belong to the foreground. For example, in the following picture, the pixels which make up the dog should belong to the foreground, and the pixels making up the wood planks should belong to the background.

First we studied the grabCut algorithm and some general theory of image segmentation; we also played with some implementations of the grabCut algorithm to get a feel for how it worked. Anna wrote her own implementation of grabCut in matlab based on Gaussian Mixture Models which would take as input an image along with a manually-drawn matte and spit out a segmentation of the image into foreground and background. Our first goal was to get a clean segmentation of the dog picture shown above. Over the weeks we experimented with the dog picture by adjusting the algorithm parameters and we also discussed ways to extract more information from the color data to enhance the performance of Anna’s implementation. We added several matlab functions to do this. In the dog picture above, the problem mostly seemed to be that cutting along the long, upside-down U-shape between the dog’s hind legs and front legs was more expensive than cutting along the shorter distance spanned by a straight line from the bottom of the hind legs to the bottom of the front legs. We suspected that we needed to make it easier to cut along the this upside-down U-shape by decreasing the neighborhood weights along that arc. However, it wasn’t enough to just manually adjust the neighborhood weights–we wanted a way to automatically adjust the neighborhood weights so that such problems wouldn’t occur for other images. So our problem was to figure out which neighborhood weights to adjust and how to adjust them. First we tried using the directionality of the color data. The intuition was that if the colors in a subset of the image were running along a particular direction, then there should be an edge along that direction in the subset of pixels. We tried to capture the directionality in the color data by calculating the covariance matrix over small square windows of the image (the variance-covariance matrix of the variables X and Y where X was a vector consisting of the differences in color intensity in the horizontal direction and Y was a vector consisting of the differences in color intensity in the vertical direction). We then computed the spectrum of the covariance matrix centered at each pixel in the image. If the absolute value of the largest eigenvalue of the matrix was much greater than the absolute value of the smaller eigenvalue (which indicates a strong directionality along the direction of the corresponding eigenvector in that particular window), and also the largest eigenvector was almost horizontal or almost vertical, then we would decrease the corresponding horizontal or vertical neighborhood weight proportionally to the ratio of the eigenvalues. In theory this would make it less expensive to cut along the edges in the image. To check if the code was picking up directionality properly, we placed a vector field over the image where the vector at a pixel p was the eigenvector of the largest eigenvalue of the covariance matrix centered at p. Though the vector field seemed to capture the directionality properly, tweaking the edge weights in this way seemed to have little effect on the output of the algorithm: the large piece of wood flooring between the dog’s legs was still being categorized as foreground, even after we greatly magnified the adjustments to the edge weights based on the ratio of eigenvalues.

We next experimented with modifiying select node and neighborhood weights on the finger image. We manually marked certain pixels of the finger of the test image and adjusted the weights so that the selected pixels were forced into the foreground. This strategy only ensured the selected pixels were placed in the foreground. It failed to pull the rest of the finger into the foreground.