Inclusion Criteria:

- Patients must have histologically confirmed relapsed or refractory mantle cell
lymphoma or low grade B-cell non-Hodgkin lymphoma (NHL); patients with evidence of
transformation to a high grade histology will not be eligible

- Measurable or evaluable disease, as defined in 2008 Revised Response Criteria for
Malignant Lymphoma; baseline scans used for measurement should be obtained within 30
days of registration, and baseline bone marrow biopsy and/or aspiration should be
obtained with 90 days of registration

- Patients may have received one or more lines of prior chemotherapy with/without
rituximab (including high dose therapy plus stem cell transplant which is counted as
one regimen); prior bortezomib is allowed; patients must not have received bortezomib
in the past 6 months

- Patients with human immunodeficiency virus (HIV) who are not receiving cytochrome
p450 inhibitors, and who have a minimum of 300+ CD4+ cells/mm3, an undetectable viral
load, and no history of acquired immune deficiency syndrome (AIDS) indicator
conditions

- Patients must be able to take oral medication and to maintain a fast as required for
2 hours before and 1 hour after MLN8237 administration

- Eastern Cooperative Oncology Group (ECOG) performance status 0, 1, or 2 (Karnofsky >=
60%)

- Leukocytes >= 3,000/mcL

- Absolute neutrophil count >= 1,500/mcL

- Platelets >= 100,000/mcL

- Total bilirubin within normal institutional limits (may be elevated if direct
bilirubin normal)

- Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
[SGOT])/alanine aminotransferase (ALT) (serum glutamic pyruvate transaminase [SGPT])
=< 3 X institutional upper limit of normal

- Creatinine within normal institutional limits OR creatinine clearance >= 60
mL/min/1.73 m^2 for patients with creatinine levels above institutional normal

- Women of child-bearing potential and men must agree to use adequate contraception
(hormonal or barrier method of birth control; abstinence) prior to study entry and
for the duration of study participation; should a woman become pregnant or suspect
she is pregnant while she or her partner is participating in this study, she should
inform her treating physician immediately; men treated or enrolled on this protocol
must also agree to use adequate contraception prior to the study, for the duration of
study participation, and 4 months after completion of MLN8237 administration

- Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

- Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for
nitrosoureas or mitomycin C, 3 weeks for rituximab) prior to entering the study or
those who have not recovered from adverse events due to agents administered more than
4 weeks earlier

- Patients who are receiving any other investigational agents

- Patients with known brain metastases should be excluded from this clinical trial

- History of allergic reactions attributed to compounds of similar chemical or biologic
composition to MLN8237, bortezomib or rituximab

- Uncontrolled intercurrent illness including, but not limited to, ongoing or active
infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
arrhythmia, or psychiatric illness/social situations that would limit compliance with
study requirements

- Known history of uncontrolled sleep apnea syndrome and other conditions that could
result in excessive daytime sleepiness, such as severe chronic obstructive pulmonary
disease; requirement for supplemental oxygen; or any conditions that could result in
excessive toxicity associated with the benzodiazepine-like effects of MLN8237

- Inability to swallow oral medication or to maintain a fast as required for 2 hours
before and 1 hour after MLN8237 administration or any condition that would modify
small bowel absorption of oral medications, including malabsorption, or resection of
pancreas or upper bowel

- Pregnant women are excluded from this study; breastfeeding should be discontinued if
the mother is treated with MLN8237

- HIV-positive patients on combination antiretroviral therapy which include cytochrome
p450 inhibitors are ineligible; patients with CD4 counts less than 300 CD4+ cells/mm3
and or a high viral load are at increased risk of lethal infections when treated with
marrow-suppressive therapy, and are ineligible

- Grade 2 or greater neuropathy

- The following agents are not permitted while patients are taking MLN8237, and should
be discontinued at prior to registration if patients are taking them:

- Patients must stop using the proton pump inhibitor (PPI) for at least 4 days
prior to the first dose of MLN8237; administration of PPI while on study is not
permitted

- Histamine-2 (H2) receptor antagonists are not permitted from the day prior
through to the end of MLN8237 dosing, except as required for premedication for
rituximab; constant dosing of H2 blockers is not permitted

- Antacid preparations are not permitted for 2 hours before or 2 hours after
administration of MLN8237

- Administration of pancreatic enzymes is not permitted at any time while on study

- Co-administration of enzyme-inducing antiepileptic drugs, rifampin, rifabutin,
rifapentine or St. John's wort is not permitted

- Concurrent bisphosphonate therapy is allowed if it was started before study
entry and is maintained at recommended dosing intervals; if bisphosphonate
therapy is initiated after study entry, bone lesions will not be considered
evaluable for disease response

- Patients must be willing not drive, operate dangerous tools or machinery, or
engage in any other potentially hazardous activity that requires full alertness
and coordination if they experience sedation; if a patient experiences excessive
sedation believed to be related to MLN8237, treatment with MLN8237 should be
interrupted

- Patients must be willing to limit alcohol consumption to no more than 1 standard
unit of alcohol (12 oz beer [350 mL], 1.5 oz [45 mL] of 80- proof alcohol, or
one 6-oz [175 mL] glass of wine per day during the study and for 30 days from
the last dose of MLN8237; minimize the use of agents with central nervous system
(CNS) effects

- Benzodiazepine use is discouraged but not prohibited

- Because of their structural and pharmacological similarity to MLN8237, benzodiazepine
use is discouraged but not prohibited