UCLA-CSU summer bridge program

Program dates

For 2025, the program dates are the five weeks from July 7 to August 8, Monday through Friday, 40hr per week. The program hours are 1-7pm; the unusual hours are chosen to avoid the rush-hour traffic.

Program location

All classes are in-person at UCLA, at the Conference Room 6943 in the Mathematical Scinces Building at 520 Portola Plaza, Los Angeles, CA 90095; see the UCLA map. The math building is complicated since it is on a hill; there are three main entrances — two on the fifth floor near the eastern and central part of the building, and one on the fourth floor near the western end of the building. If you take the western entry, there is an elevator right there; take it to the sixth floor, and room is right around the elevator on your right. (There are also floor maps near the elevator.)

People

The UCLA organizers for the program are Raphaël Rouquier and Sucharit Sarkar. For 2025, the instructors are Bon-Soon Lin (July 7-18), Kawah Wong (July 21-25), Joaquin Moraga (July 28 - August 1), and Peter Xu (August 4-8). The eight students are from four CSU campuses (Long Beach, Los Angeles, Northridge, and San Bernardino), selected by their professors.

Program format

The first two weeks will consist of a review of advanced undergrad math, and the next three weeks will be on three different advanced topics. For 2025, the topics are:

• Week 1: Real analysis (Bon-Soon Lin)
  Topics:
  Consequences of the real axioms
  Metric topology
  Sequences and series
  Limits and continuity
  Differentiability
  Integrability
  Space of functions
  Stone-Weierstrass approximation
  Arzela-Ascoli
  Baire category theorem
• Week 2: Linear algebra (Bon-Soon Lin)
  Topics:
  Vector spaces, subspaces, linear independence, span, basis, dimension, replacement lemma
  Linear maps, matrix representations, rank-nullity theorem, determinants
  Diagonalizability and eigenvalues/eigenvectors
  Inner product spaces, Gram-Schmidt (QR), spectral theorems, SVD, positive-definite matrices
  Jordan canonical form
  Perron-Frobenius theory
• Week 3: Bayesian inference and Uncertainty Quantification (Kawah Wong)
  In mathematical modeling, it is essential to determine the underlying parameters so that the model's behavior can accurately reflect real-world phenomena. Parameter estimation uses the observed data to infer the most plausible values of the parameters. Instead of finding a single parameter set that best fits our data, we may leverage Bayesian statistics to (1) incorporate prior beliefs or domain knowledge in the inference process, (2) produce a posterior probability distribution over the parameter values, which allows us to quantify uncertainties in the estimated parameters. We will introduce fundamental concepts in Bayesian parameter estimation and uncertainty quantification. You will practice these concepts through hands-on computation labs.
  Topics:
  Inverse problems and parameter identification
  Frequentist vs Bayesian estimation
  Bayes' theorem, priors, likelihoods, posteriors
  Uncertainty and model error
  Monte Carlo integration
  Metropolis-Hastings algorithm
  Hamiltonian Monte Carlo (HMC)
  Bayesian calibration and credible intervals
  Posterior predictive checks, and model selection
  Case studies of parameter identification in biology and weather forecast (tentatively)
  Reference:
  Bayesian Data Analysis (3rd edition) By Andrew Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald Rubin
  Statistical Rethinking (2nd edition) By Richard McElreath
• Week 4: Basic commutative algebra and toric geometry (Joaquin Moraga)
  Topics:
  Commutative rings and basics (ideals, quotients, etc).
  Construction of Spectrum.
  Hilbert Nullstellensatz.
  Polyhedral cones and Gordan's Lemma.
  Algebra associated with a polyhedral cone.
  Affine toric variety associated with a cone.
  Fans of polyhedral cones.
  Toric variety associated with a fan.
  Projective toric variety associated with a polytope.
  Fano varieties and reflexive polytopes.
  Polyhedral decompositions and blow-ups.
  Resolving toric singularities.
• Week 5: Representation theory of symmetric groups (Peter Xu)
  Schedule:
  Monday: Rep theory of finite groups over C: the group algebra, Schur's lemma and Maschke's theorem. Begin character theory.
  Tuesday: Finish character theory; classification of irreps using it.
  Wednesday: Tensor products of representations, decomposition of regular representation, induced representations.
  Thursday: Structure/conjugacy classes of symmetric groups; Specht modules and Young diagrams.
  Friday: Frobenius formula/hook length formula. Schur functions and Littlewood-Richardson, Kronecker coefficients, Schur-Weyl duality.

The classes will be in-person, 1-7pm, usually with one or two 15-30 min breaks. No food is provided, so bring your own food. Some campus eateries might be closed in the summer, check their wesbite. Each day, there is 1-2 hours of instruction, 2-3 hours of group work on problems assigned by the instructor, and the final hour is usually when you present your results and solutions to the instructor; the precise format will depend on your instructor. Bring your own laptops, as some instructors will require working on computers. (If you need a loaner laptop, let us know immediately, and we can arrange for them.) UCLA has free guest Wifi, see their website.

Parking

If you are commuting, the main parking options are campus parking, which costs $16/day, see the Parking website. The closest parking structures are Structure 2 and Structure 4. (I personally prefer Structure 4 since it is easier to get to from the 405.)

Stipend

Your total stipend for all 5 weeks is $2500 plus some allowance for commuting/renting. If you are commuting to campus, you will be paid an additional $750. If you live far away and are renting, you will be paid an additional $2000 (instead of $750) towards your rental cost, but we will require proof of rental. (We realize that these commuting and rental allowances may not completely cover your costs, but hopefully will offset them.) I am not authorized to provide any tax guidance regarding these stipend payments. They might be taxable or non-taxable, depending on your situation (see the IRS website). Someone from the UCLA Math Department will contact you to ask for some basic information that they will need to process your stipend payment. Please note, your stipend payment is conditional on at least 80% in-person attendance in the program.

Additional meetings

We will host an orientation meeting over Zoom using this link at 9am (Pacific time) on June 29, 2025, to answer any additional questions that you may have before the start of the program. On the last day of the program (August 8, 2025), we will host a dinner, at or near UCLA, and discuss your experience with the program, and provide guidance for your future plans.

Accessibility

If you require any accommodations in order to participate fully in the program, please let us know.

Sickness

If you become sick during the program, then you may choose to drop out for the duration of their sickness and then return if possible. You will not lose any stipend for the reason of sickness, but if you have to drop out for a long period, we will ask for proof of sickness in the form of an explanatory note from a medical facility.