Author Archives: Richard Laugesen

Linear Algebra Workshop 2018

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The 2018 Linear Algebra Workshop is offered as part of the PI4 program, and will be led by Stefan Klajbor Goderich. The goal of the workshop is to review and learn some advanced linear algebra with an eye towards applications and the Computational Bootcamp, which runs immediately after this workshop.

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.”

2018 summer program

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Click here to apply to PI4 for Summer 2018.

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.

 

Linear Algebra Workshop 2017

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The 2017 Linear Algebra Workshop was offered as part of the PI4 program. The goal of the workshop was to review and learn some advanced linear algebra with an eye towards applications and the Computational Bootcamp, which ran immediately after this workshop.

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.”

2017 summer program

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Click here to apply to PI4 for Summer 2017.

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.

 

2016 summer program

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Computational Mathematics Bootcamp

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.

Workshop in Linear Algebra

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.