Immunotherapy + Radiation in Resectable Soft Tissue Sarcoma

Inclusion Criteria:

* Diagnosed with biopsy-proven soft-tissue sarcoma with confirmatory pathology review at University of Rochester Medical Center (URMC)
* 18 years or older.
* Able to provide informed consent.
* Resectable disease per surgical evaluation.
* Neoadjuvant/preoperative radiotherapy has been recommended
* Intermediate to high grade sarcoma on biopsy, tumor \> 5 cm in size by imaging
* Willing to have blood draws for flow cytometry and Serametrix analysis.
* Willing to receive neoadjuvant radiation therapy and subsequent surgical resection.
* Patients with known human immunodeficiency virus (HIV) are eligible if the lymphocytes \> 350 CD4+ cells and no detectable viral load.
* Women of childbearing potential (defined as any woman, who 1) has not undergone tubal ligation, a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months or has had menses at any time in the preceding 24 consecutive months):

  * Must not be pregnant or nursing
  * Must have a negative pregnancy test done within 7 days prior to registration as well as within 72 hrs. prior to receiving first dose of study medication
  * Women of childbearing potential must use at least two other accepted and highly-effective methods of contraception and/or agree to abstain from sexual intercourse for at least 5 months after the last dose of nivolumab and/or ipilimumab.
  * Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately.
* Sexually active males must use at least two other accepted and highly-effective methods of contraception and/or agree to abstain from sexual intercourse for at least 7 months after the last dose of nivolumab and/or ipilimumab
* Eastern Cooperative Oncology Group (ECOG) performance status 0 - 2

Exclusion Criteria:

* Gastrointestinal stromal tumor, dermatofibrosarcoma protuberans, rhabdomyosarcoma, Ewing's sarcoma, low-grade sarcomas, osteosarcoma subtypes
* History of radiation to the affected area
* Evidence of metastatic disease prior to treatment
* Any history of prior therapy with ipilimumab or nivolumab, or any agent targeting PD-1, PD-L1 or CTLA-4.
* History of any the following:

  * Active known or suspected autoimmune disease
  * Active autoimmune colitis
  * Autoimmune pan hypopituitarism
  * Autoimmune adrenal insufficiency
  * Known active hepatitis B or C
  * Known active pulmonary disease with hypoxia defined as:
* Oxygen saturation \< 85% on room air or
* Oxygen saturation \< 88% despite supplemental oxygen
* No systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of registration.
* Second active malignancy, not including localized basal cell skin cancer, squamous cell skin cancer, in situ cervical cancer, ductal or lobular carcinoma in situ of the breast; patients with a history of other malignancies are eligible if they have been continuously disease-free for \> 3 years prior to the time of registration.
* Absolute neutrophil count (ANC) \<= 1,500/mm\^3
* Platelet count \<= 100,000/mm\^3
* AST (Aspartate Aminotransferase) or ALT (Alanine Aminotransferase) =\> 2 x upper limit of normal (ULN)
* Thyroid stimulating hormone (TSH) outside of normal limits; supplementation is acceptable to achieve a TSH within normal limits; in patients with abnormal TSH if free T4 is normal and patient is clinically euthyroid, patient is eligible.
* Life expectancy under 5 years.