UCLA-CSU summer bridge program

Program dates

For 2026, the program dates are the five weeks from July 6 to August 7, 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 Sciences 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 2026, the instructors are Bon-Soon Lin (July 6-17), Joaquin Moraga (July 20-24), Advika Rajapakse (July 27-31), and Ali Kashefi (August 3-7). 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 2026, 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: 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 4: Topology and fundamental group (Advika Rajapakse)

  Topics:

  • Point-set topology, paths in space, and homotopy of paths.
  • Group theory and the fundamental group.
  • Computation of the fundamental group of the circle.
  • Van Kampen's theorem.
  • Covering spaces, the universal cover, and classification of covering spaces.
• Week 5: Linear and nonlinear systems of differential equations (Ali Kashefi)

  Topics:

  • One-dimensional flows, flows on the line, fixed points, and stability, with applications.
  • Introduction to bifurcations, including saddle-node, transcritical, and pitchfork bifurcations, with applications.
  • Two-dimensional linear systems, different types of stability, phase portraits, and stable and unstable eigenspaces.
  • Two-dimensional nonlinear systems and phase portraits, nullclines, existence, uniqueness, and strong topological consequences in two dimensions.

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 website. 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 $17/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 1pm (Pacific time) on July 2, 2026, to answer any additional questions that you may have before the start of the program. On the last day of the program (August 7, 2026), 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 your 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.