This phase II trial evaluates the safety and effectiveness of giving immunotherapy (nivolumab and ipilimumab) before surgery for controlling disease in patients with stage I-IIIa sarcomatoid mesothelioma. Immunotherapy with monoclonal antibodies, such as nivolumab and ipilimumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving immunotherapy before surgery may be more effective at controlling disease in patients with sarcomatoid mesothelioma than giving immunotherapy alone.

PRIMARY OBJECTIVES:

I. To determine the percentage of patients with potentially resectable non-epithelioid mesothelioma who are able to proceed with surgery after neoadjuvant ipilimumab and nivolumab.

II. To determine the progression-free survival rate at 12 months after the initiation of neoadjuvant ipilimumab and nivolumab.

SECONDARY OBJECTIVES:

I. To determine the rate of intra-operative or post-operative complications following neoadjuvant immunotherapy.

II. Best response per modified pleural Response Evaluation Criteria in Solid Tumors (RECIST).

III. Major pathologic response rate. IV. Time to recurrence after surgery.

EXPLORATORY OBJECTIVES:

I. To evaluate the association between the change in peripheral T cell clonality relative to baseline and treatment response.

II. To evaluate the association between PD-L1 expression at baseline and treatment response.

III. To evaluate whether a novel mesothelioma immune signature identified by Dr. Mansfield's laboratory is predictive of response.

OUTLINE:

Patients receive nivolumab intravenously (IV), ipilimumab IV, and may undergo surgery on study. Patients also undergo computed tomography (CT) or magnetic resonance imaging (MRI) and positron emission tomography (PET) throughout the trial.

Inclusion Criteria:

• Sarcomatoid or sarcomatoid-dominant (> 50%) biphasic, pleural mesothelioma

• Stage: I-IIIA disease per Union for International Cancer Control (UICC) TNM Classification of Malignant Tumours 8th edition

• Measurable disease or non-measurable disease as defined

• No prior treatment which would be considered treatment for the primary neoplasm or impact the primary endpoint

• No treatment with hormones or other chemotherapeutic agents except for hormones administered for non-disease-related conditions (e.g., insulin for diabetes and or hormonal therapy for breast, prostate cancer etc.)

• Not pregnant and not nursing, because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown

  * Therefore, for women of childbearing potential only, a negative pregnancy test done =< 14 days prior to registration is required

• Age >= 18 years

• Eastern Cooperative Oncology Group (ECOG) performance status =< 2 or Karnofsky >= 60%

• Absolute neutrophil count (ANC) >= 1,000/mm^3

• Leukocytes >= 2,000/mm^3

• Platelet count >= 100,000/mm^3

• Creatinine =< 1.5 x upper limit of normal (ULN) OR creatinine clearance >= 40 mL/min

• Total bilirubin =<1.5 x ULN, except patients with Gilbert Syndrome who can have total bilirubin < 3.0 mg/dl

• Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 3.0 x ULN

• Alkaline (alk) phosphatase (phos) =< 3.0 x ULN

• No active, known or suspected autoimmune disease except for vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger

• No active systemic infection requiring therapy, as well as positive tests for hepatitis B surface antigen or hepatitis C antibody

• No history of any other condition that may require the initiation of anti-tumor necrosis factor alpha (TNFalpha) therapies or other immunosuppressant medications during the study

• Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial

• Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better

• Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial

• STEP 2 ELIGIBILITY CRITERIA: Completion of at least 1 cycle of treatment and not have an unresolved adverse event that would preclude surgery

• STEP 2 ELIGIBILITY CRITERIA: No evidence of progression that would preclude resection

• STEP 2 ELIGIBILITY CRITERIA: ECOG performance status =< 2 or Karnofsky >= 60%

• STEP 2 ELIGIBILITY CRITERIA: Predicted forced expiratory volume in 1 second (FEV1) > 35% and postoperative predicted diffusion capacity of the lung for carbon monoxide (DLCO) > 35%

• STEP 2 ELIGIBILITY CRITERIA: Registration to step 2 no less than 21 days and no more than 90 days after the last dose of neoadjuvant therapy

Exclusion Criteria:

• No patients deemed to be unresectable or poor surgical candidates
• No patients with chest wall invasion, peritoneal spread, contralateral pleural involvement, mediastinal organ involvement, vertebral involvement, or metastases to contralateral intrathoracic lymph nodes, or any supraclavicular nodes
• No patients with a history of symptomatic interstitial lung disease