# Institute for Mathematics and Its Applications

2014 PI Summer Graduate Program

# Modern Applications of Representation Theory

## The University of Chicago, Chicago, IL

July 20-August 6, 2014

### IMA Program Application Form

### Description

This program is for graduate students from both IMA Participating Institutions as well as other U.S. universities. We expect to be able to fund up to 35 students from IMA Participating Institutions and 15 students from other U.S. universities. In order to participate, students must complete the online application form, provide a personal statement, and submit (1) a letter of nomination from the PI chair (for students from an IMA PI) or (2) a recommendation letter (for students from institutions that are not an IMA PI).

A main portion of this program consists of a three-week summer school for graduate students to be held in Summer 2014 on the campus of the University of Chicago. It will focus on modern applications of representation theory discovered largely within approximately the last 10 years.

Specifically, we intend to cover applications of representation theory in algebraic and geometric computational complexity, cryo-electron imaging, digital signal processing, holographic algorithms and quantum computing, machine learning and pattern recognition, and a few other specialized topics.

The instruction in the summer school comprises several week-long lecture series supplemented by a foundational tutorial and two days of short hour-long lectures on specialized topics. The objectives are to (i) quickly review the basic materials (tutorial); (ii) focus on the developments of the last 10 years (several long lecture series); and (iii) provide a glimpse of the state of current research and open problems (a number of short talks).

Most of the materials intended to be covered in (ii) and (iii) are at this point not easily accessible—not covered in any textbooks, courses, or even survey articles—they are only available in the form of original research papers or preprints. It is our hope that the summer school will bring these materials to graduate students, postdocs, and in general, any nonspecialist.

### **Organizers**

**Risi Kondor**, Departments of Computer Science and Statistics, The University of Chicago**Lek-Heng Lim**, Department of Statistics, The University of Chicago**Jason Morton**, Department of Mathematics, The Pennsylvania State University

**Guest Lecturers**

- Gregory Chirikjian, Department of Mechanical Engineering, Johns Hopkins University
**Shamgar Gurevich**, Department of Mathematics, University of Wisconsin, Madison**Risi Kondor**, Departments of Computer Science and Statistics, The University of Chicago**J. M. Landsberg**, Department of Mathematics, Texas A&M University**Peter McCullagh**, Department of Statistics, The University of Chicago**Cristopher Moore**, Santa Fe Institute**Jason Morton**, Department of Mathematics, The Pennsylvania State University**Ketan Mulmuley**, Department of Computer Science, The University of Chicago**Michael Orrison**, Department of Mathematics, Harvey Mudd College**Daniel Rockmore**, Department of Mathematics, Dartmouth College**Amit Singer**, Department of Mathematics, Princeton University**Chris Umans**, Department of Computing and Mathematical Sciences, California Institute of Technology

Check the **poster** and the IMA program homepage. For questions, write to **conference organizers**.

**Schedule**

July 21 | July 22 | July 23 | July 24 | July 25 | July 28 | July 29 | July 30 | July 31

August 1 | August 4 | August 5

#### MONDAY, July 21, 2014, in Room 112, at 5727 S. University Avenue

MANDATORY SAFETY PRESENTATION, 8:30–9:00 AM

LECTURE 1, 9:00–10:00 AM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Introductory Representation TheoryLECTURE 2, 10:30–11:30 AM,

Michael Orrison(Harvey Mudd College)

"Algebraic Voting Theory"

Slides, Video,Vimeo

Topic Grouping: Learning, Statistics, and OptimizationWELCOME LUNCH, 11:45 AM–1:30 PM, ECKHART 110

LECTURE 3, 1:30–2:30 PM,

Jason Morton(The Pennsylvania State University)

Notes, Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLECTURE 4, 3:00–4:00 PM,

Risi Kondor(The University of Chicago)

"Introduction to Harmonic Analysis On Groups"

Slides, Video, Vimeo

Topic Grouping: Learning, Statistics, and OptimizationLECTURE 5, 4:30–5:30 PM, "Review and Problems, Introductory Representation Theory"

Topic Grouping: Introductory Representation Theory

#### TUESDAY, July 22, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Introductory Representation TheoryLECTURE 2, 10:30–11:30 AM,

Michael Orrison(Harvey Mudd College)

"Adapted Bases and Fast Transforms"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

Jason Morton(The Pennsylvania State University)

Notes, Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLECTURE 4, 3:00–4:00 PM,

Risi Kondor(The University of Chicago)

"Fourier Analysis on the Symmetric Group"

Slides, Video, Vimeo

Topic Grouping: Learning, Statistics, and OptimizationLECTURE 5, 4:30–5:30 PM, "Review and Problems, Introductory Representation Theory"

Topic Grouping: Introductory Representation Theory

#### WEDNESDAY, July 23, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Introductory Representation TheoryLECTURE 2, 10:30–11:30 AM,

Amit Singer(Princeton University)

"Introduction to Three Dimensional Structure Determination of Macromolecules by Cryo-Electron Microscopy"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

Jason Morton(The Pennsylvania State University)

Notes, Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLECTURE 4, 3:00–4:00 PM, Samuel Denton, Cameron Farnsworth, Mee Seong Im,

Deepti Pachauri, Zvi Rosen

Topic Grouping: Short TalksLECTURE 5, 4:30–5:30 PM, "Review and Problems, FFT, Signal Processing, Imaging"

Topic Grouping: FFT, Signal Processing, Imaging

#### THURSDAY, July 24, 2014, in Room 112, at 5727 S. University Avenue

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Introductory Representation TheoryLECTURE 2, 10:30–11:30 AM,

Amit Singer(Princeton University)

"Orientation Assignment in Cryo-EM"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLUNCH BREAK, 11:30–1:30 PM

Jason Morton(The Pennsylvania State University)

Notes, Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLECTURE 4, 3:00–4:00 PM,

Risi Kondor(The University of Chicago)

"Invariants and Optimization"

Slides, Video, Vimeo

Topic Grouping: Learning, Statistics, and OptimizationLECTURE 5, 4:30–5:30 PM, "Review and Problems, Tensor Networks, Categories, Quantum"

Topic Grouping: Tensor Networks, Categories, Quantum

#### FRIDAY, July 25, 2014, in Room 112, at 5727 S. University Avenue

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Introductory Representation TheoryLECTURE 2, 10:30–11:30 AM,

Amit Singer(Princeton University)

"Class Averaging and Symmetry Detection in Cryo-EM"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

Jason Morton(The Pennsylvania State University

Notes, Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLECTURE 4, 3:00–4:00 PM,

Amit Singer(Princeton University)

"Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLECTURE 5, 4:30–5:30 PM, "Computational Complexity Introduction"

Topic Grouping: Computational Complexity

#### MONDAY, July 28, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Computational ComplexityLECTURE 2, 10:30–11:30 AM,

Cristopher Moore(Santa Fe Institute)

"Shor’s Quantum Factoring Algorithm"

Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM, Gregory Chirikjian (Johns Hopkins University)

"Representation Theory and Computational Harmonic Analysis on Euclidean Motion Groups"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, Imaging

LECTURE 4, 3:00–4:00 PM, Gregory Chirikjian (Johns Hopkins University)

"Applications in Robotics and Biology"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLECTURE 5, 4:30–5:30 PM, "Review and Problems, FFT, Signal Processing, Imaging"

Topic Grouping: FFT, Signal Processing, Imaging

#### TUESDAY, July 29, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM, Gregory Chirikjian (Johns Hopkins University)

"Discrete Subgroups, Almost-Uniform Sampling on Lie Groups and Implications for FFTs"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, Imaging

LECTURE 2, 10:30–11:30 AM,

Cristopher Moore(Santa Fe Institute)

"Representations of the Symmetric Grouping and the Hunt for a Quantum Algorithm for Graph Isomorphism"

Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Computational Complexity

LECTURE 4, 3:00–4:00 PM,

Daniel Rockmore(Dartmouth College)

Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLECTURE 5, 4:30–5:30 PM, "Review and Problems, Computational Complexity"

Topic Grouping: Computational Complexity

#### WEDNESDAY, July 30, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Computational ComplexityLECTURE 2, 10:30–11:30 AM,

Cristopher Moore(Santa Fe Institute)

"Estimating the Permanent with Nonabelian Determinants"

Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

Daniel Rockmore(Dartmouth College)

Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLECTURE 4, 3:00–4:00 PM,

Daniel Rockmore(Dartmouth College)

Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLECTURE 5, 4:30–5:30 PM, "Review and Problems, Tensor Networks, Categories, Quantum"

Topic Grouping: Tensor Networks, Categories, Quantum

#### THURSDAY, July 31, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Computational ComplexityLECTURE 2, 10:30–11:30 AM,

Chris Umans(California Institute of Technology)

"I. Approaches to Bounding the Exponent of Matrix Multiplication"

Slides, Video

Topic Grouping: Computational ComplexityLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

Jason Morton(The Pennsylvania State University)

Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLECTURE 4, 3:00–4:00 PM,

Chris Umans(California Institute of Technology)

"II. Group-Theoretic Approach"

Slides, Video

Topic Grouping: Computational ComplexityLECTURE 5, 4:30–5:30 PM, "Review and Problems, Computational Complexity"

Topic Grouping: Computational Complexity

#### FRIDAY, August 1, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,Chris Umans(California Institute of Technology)

"III. Matrix Multiplication Using Coherent Configuations"

Slides, Video

Topic Grouping: Computational ComplexityLECTURE 2, 10:30–11:30 AM,

J. M. Landsberg(Texas A&M University)

Video, Vimeo

Topic Grouping: Computational ComplexityLUNCH BREAK, 11:30–1:30 PM

LECTURE 3, 1:30–2:30 PM, J. M. Landsberg (Texas A&M University)

Special Seminar: "Geometric Complexity Theory and Minimal Free Resolutions"

Video, Vimeo

Topic Grouping: Computational ComplexityLECTURE 4, 3:00–4:00 PM,

Jacob Turner(The Pennsylvania State University)

"Relating Orbit Spaces and Quivers"

Video, Vimeo

Topic Grouping: Tensor Networks, Categories, QuantumLECTURE 5, 4:30–5:30 PM, "Review and Problems, FFT, Signal Processing, Imaging"

Topic Grouping: FFT, Signal Processing, Imaging

#### MONDAY, August 4, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,

Peter McCullagh(The University of Chicago)

"Statistical Models and Representation Theory"

Video1, Video2, Vimeo

Topic Grouping: Learning, Statistics, and OptimizationLECTURE 2, 10:30–11:30 AM,

Ketan Mulmuley(The University of Chicago)

"Kronecker Coefficients and Geometric Complexity Theory"

Video, Vimeo

Topic Grouping: Computational ComplexityLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

Shamgar Gurevich(University of Wisconsin, Madison)

"The Heisenberg Representation and The Fast Fourier Transform"

Slides, Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLECTURE 4, 3:00–4:00 PM,

Samuel Denton(York University)

"Compressed Combinatorics"

Video, Vimeo

Topic Grouping: FFT, Signal Processing, ImagingLECTURE 5, 4:30–5:30 PM,

Shamgar Gurevich(University of Wisconsin, Madison)

"Diagonalization of the Discrete Fourier Transform and the Weil Representation"

Slides, Video

Topic Grouping: FFT, Signal Processing, Imaging

#### TUESDAY, August 5, 2014, in Room 112, at 5727 S. University Avenue

LECTURE 1, 9:00–10:00 AM,

Peter McCullagh(The University of Chicago)

"Statistical Models and Representation Theory"

Video, Vimeo

Topic Grouping: Learning, Statistics, and OptimizationLECTURE 2, 10:30–11:30 AM,

Ketan Mulmuley(The University of Chicago)

"Kronecker Coefficients and Geometric Complexity Theory"

Video, Vimeo

Topic Grouping: Computational ComplexityLUNCH BREAK, 11:30 AM–1:30 PM

LECTURE 3, 1:30–2:30 PM,

Shamgar Gurevich(University of Wisconsin, Madison)

"The Geometric Weil Representation and Pseudo-Random Vectors"

Slides

Topic Grouping: FFT, Signal Processing, Imaging

**Participants**

**Tejal Bhamre**, Department of Physics and Applied Mathematics, Princeton University**Mario Borha**, Department of Mathematics, Northern Illinois University**Sixia Chen**, Department of Computer Science & Engineering, University of Connecticut**Yutong Chen**, Program in Computational and Applied Mathematics, Princeton University**Gregory Chirikjian**, Department of Mechanical Engineering, Johns Hopkins University**Justin Curry**, Department of Mathematics, Duke University**Walter Dempsey**, Department of Statistics, The University of Chicago**Samuel Denton**, Department of Mathematics, York University**Cameron Farnsworth**, Department of Mathematics, Texas A & M University**Jonathan Fernandes**, Department of Mathematics, University of Maryland**Brett Geiger**, Department of Mathematics, University of Houston**Renato Ghini Bettiol**, Department of Mathematics, University of Notre Dame**Jonathan Godbout**, Department of Mathematics, Worcester Polytechnic Institute**William Grodzicki**, Department of Mathematics, University of Minnesota, Twin Cities**Shamgar Gurevich**, Department of Mathematics, University of Wisconsin, Madison**Jing Hao**, Department of Mathematics, University of Wisconsin, Madison**Pamela Harris**, Department of Mathematical Sciences, United States Military Academy**Peng He**, Department of Mathematics, University of Pittsburgh**Mee Seong Im**, Department of Mathematics, University of Illinois at Urbana-Champaign**Shahrzad Jamshidi**, Department of Mathematics, The Pennsylvania State University**Matthew Joseph**, Department of Mathematics, The University of Chicago**Joseph Kileel**, Department of Mathematics, University of California, Berkeley**Hwamog Kim**, Department of Mathematics and Statistics, Mississippi State University**Risi Kondor**, Departments of Computer Science and Statistics, The University of Chicago**Robert Krone**, Department of Mathematics, Georgia Institute of Technology**J. M. Landsberg**, Department of Mathematics, Texas A & M University**Kang-Ju Lee**, Department of Mathematics, Texas A & M University**Lek-Heng Lim**, Department of Statistics, The University of Chicago**Ye Luo**, Department of Mathematics, Georgia Institute of Technology**Peter McCullagh**, Department of Statistics, The University of Chicago**Cristopher Moore**, Santa Fe Institute**Jason Morton**, Department of Mathematics, The Pennsylvania State University**Ketan Mulmuley**, Department of Computer Science, The University of Chicago**Tatiana Orlova**, Department of Computer Science, The University of Chicago**Michael Orrison**, Department of Mathematics, Harvey Mudd College**Deepti Pachauri**, Department of Computer Sciences, University of Wisconsin, Madison**Daniel Rockmore**, Department of Mathematics, Dartmouth College**Zvi Rosen**, Department of Mathematics, University of California, Berkeley**Hans-Christian Ruiz**Department of Biophysics, Radboud Universiteit Nijmegen**John Santerre**, Department of Computer Science, The University of Chicago**Tony Se**, Department of Mathematics, University of Kansas**Amit Singer**, Department of Mathematics, Princeton University**Roberto Soto**, Department of Mathematics, The University of Iowa**Benjamin Strasser**, Department of Mathematics, University of Minnesota, Twin Cities**Yi Sun**, Department of Mathematics, Massachusetts Institute of Technology**Nedelina****Teneva**, Department of Computer Science, The University of Chicago**Bolor Turmunkh**, Department of Mathematics, University of Illinois at Urbana-Champaign**Jacob Turner**, Department of Mathematics, The Pennsylvania State University**Chris Umans**, Department of Computer Science, California Institute of Technology**Chen Wan**, School of Mathematics, University of Minnesota, Twin Cities**Jiuya Wang**, Department of Mathematics, University of Wisconsin, Madison**Tyson Williams**, Department of Computer Sciences, University of Wisconsin, Madison**Ken Sze-Wai Wong**, Department of Statistics, The University of Chicago**Angela Wu**, Department of Mathematics, The University of Chicago**Yifei Xu**, Department of Basic Science, Mississippi State University

### Practical information

All lectures will take place in Room 112, of the Financial Mathematics/Stevanovich Center, at 5727 S. University Avenue, Chicago, Illinois, unless otherwise noted. On-campus accommodation and meals will be provided as well as support towards travel expenses.

**The IMA will need to receive a personal statement describing your background and reasons for wanting to attend, and a letter of nomination from your department chair. Please send letters to Applications. Deadline for applications is April 30, 2014.**

We also thank the IMA for providing the application website and general sponsorhip.