Exams

This page gives basic information about what is to be covered on the exams and their format. For grading details see the Grading Policies information page.

The exams are closed book exams. No calculators are permitted. You are allowed one page (8 1/2 inch by 11 inch) of notes (front and back). The exams are 1 hour and 20 minutes in length and will be during the regularly scheduled lecture time. Scratch paper will be provided. Bring (multiple) pens or pencils to write with.

Exam 1: Wednesday, March 6th 2:35 - 3:55pm, 56-154.

Resources:

• All the readings listed on the course calendar for Lectures 1-7.
• This website that has tabulated character tables and other useful information for finite groups

Things that you are expected to know / be able to do for Exam 1:

• Understand definitions:
  • group, subgroup, self-congjugate (normal) subgroup, order of a group, conjugacy classes, left coset, right coset, isomorphism between two groups, homomorphism between two groups, abelian vs. non-abelian groups (Lectures 2-3)
  • group representation (matrices + vectorspace), reducible representation, irreducible representation, (left and right) regular representation of a finite group, character of a representation
• Understand results of Lemmas and Theorems proved in class:
  • Rearrangement Theorem
  • Lagrange's Theorem
  • Schur's Lemma
  • The Wonderful Orthogonality Theorem for Representations
  • The Wonderful Orthogonality Theorem for Character
• Know when to use and how to interpret outputs of the functions you've coded in exercsies from the symm4ml class repository
  • First exercise: groups module
  • Second exercises: rep.regular_representation and group_conv module functions
  • Third exercsies: functions from linalg and rep modules

Things that you may be expected to do for Exam 1:

• Generate a group from a subset of elements
• Difference between abelian vs. non-abelian groups, and classifying group based on multiplication table
• Making, completing, and identifying errors in multiplication and character tables
• Use results of Lemmas and Theorems to classify nature of representations
• Interpret the output of the functions you have code in the first 3 exercises in this course (groups, groups2, linalg)
• Construct the Left and Right Regular Representations from the multiplication table of a group
• Use the Wonderful Orthogonality Theorem for Character and basic facts of representations to complete an incomplete character table.
• Given a step in a proof, identify what property of groups / representations or lemma / theorem is being used to e.g. equate the left and right hand sides of an equation.
• Use of np.einsum, broadcasting, and reshape to translate equations into code.
  • There are exercises for np.einsum and Broadcasting here under the 2. Einsum and 3. Broadcasting headings, respectively.

Exam 2: Wednesday, April 17th 2:35 - 3:55pm, 56-154.

Resources:

• Resources for Exam 1 -- Exam 2 is cumulative but will focus on newer material.
• All the readings listed on the course calendar for Lectures 11-17.

Things that you are expected to know / be able to do for Exam 2:

• Understand definitions
  • definitions for Exam 1
  • tensor product, tensor product decomposition, product tables of irreps
  • Lie group, (infinitesimal) generators, commutator, Lie algebra
  • group convoltion, steerable filter
  • Fourier basis, circular harmonics, spherical harmonics
• Understand results proved in class:
  • How similarity transforms, direct sums, and tensor products of representations of Lie group representations can be rewritten as operations on the generators of the Lie group. See end of Lecture 13 Notes.
  • rep.decompose_reps_into_irreps (This is an important one since we use it so much!)
• Know when to use and how to interpret outputs of the functions you've coded in exercises from the symm4ml class repository.
  • group, group_conv, linalg from before Exam 1
  • rep functions from before and after Exam 1.
  • recent exercises lie, so3, and nn.

Things that you may be expected to do for Exam 2:

• Re-write operations on the representations of a Lie group as operations on the generators.
• Determine the Lie algebra for a set of generators.
• Given a step in a proof, identify what property of groups / representations / irrep basis functions is being used to e.g. equate left and right hand sides of an equation.
• Use Schur's Lemma to restrict the parameters / weights of simple neural network operations like linear layers and convolutional filters.