Platinum-based Chemoradiotherapy and Rigosertib in Head and Neck Cancer

Inclusion Criteria: 1. Pathologically confirmed diagnosis of Squamous Cell Carcinoma of the oropharynx (tonsil, base of tongue, soft palate, or oropharyngeal wall), hypopharynx, or larynx. 2. Patient is an appropriate candidate for definitive chemoradiotherapy. 3. Intermediate-risk Head and Neck Squamous Cell Carcinoma (HNSCC), defined as follows: 1. Clinical stage T2-4, N2a-N3 or T3-4, N0-N3 2. P16 (+) by immunohistochemistry (IHC) or HPV (+) by in situ hybridization (ISH) 3. Smoking status of ≥ 10 pack-years, or \< 10 pack-years and T4 or N2c-N3. 4. If not intermediate-risk HNSCC, is high-risk HNSCC, defined as follows: 1. Clinical stage T2-4, N2a-N3 or T3-4, N0-N3. 2. P16 (-) by IHC or HPV (-) by ISH. 5. No evidence of distant metastases. 6. Clinically or radiographically evident measurable disease (as defined by RECIST v 1.1) at the primary site or nodal stations. 7. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. 8. Adequate hematologic function as defined by absolute neutrophil count (ANC) ≥ 1800/μL; platelet (PLT) ≥ 100,000/μL; Hgb ≥ 8.0 g/dL. 9. Adequate renal function, as defined by serum creatinine ≤ 1.5 x upper limit of normal (ULN) or calculated creatinine clearance ≥ 60 mL/min. 10. Adequate liver function as defined by total bilirubin ≤ 1.5 x ULN; aspartate transaminase (AST)/alanine transaminase (ALT) ≤ 2.5 x ULN; and prothrombin time ≤ 1.5 x ULN, unless receiving therapeutic anticoagulation. 11. Ability to understand the nature of the study and any hazards of study participation, to communicate satisfactorily with the Investigator, and to follow the requirements of the entire protocol. 12. Willingness to adhere to the prohibitions and restrictions specified in this protocol. 13. The patient must sign an informed consent form (ICF) indicating that s/he understands the purpose of and procedures required for the study and is willing to participate in the study. Exclusion Criteria: 1. Gross total excision of the primary and nodal disease. 2. Prior treatment with IV or oral rigosertib. 3. Prior chemotherapy for the study HNSCC cancer. 4. Prior radiotherapy to the region of the study HNSCC cancer or adjacent anatomical sites, or to \> 25% of marrow-bearing area. 5. Synchronous malignancies. 6. Prior invasive malignancy unless the patient is disease-free for a minimum of 3 years; however, patients with prior non-melanoma skin cancer, cervical intraepithelial neoplasia (CIN), or prostate cancer with undetectable prostate-specific antigen (PSA) may be enrolled. 7. Severe, active comorbidity. 8. Known infection with human immunodeficiency virus (HIV). 9. Any uncontrolled condition that, in the opinion of the Investigator, could affect the subject's participation in the study. 10. Major surgery within 3 weeks of enrollment or major surgery without full recovery. 11. Ascites requiring active medical management, including paracentesis. 12. Hyponatremia (defined as serum sodium \< 130 milliequivalent mEq/L) or conditions that may predispose patients to hyponatremia. 13. Uncontrolled hypertension, defined as systolic blood pressure ≥ 160 mmHg and/or diastolic blood pressure ≥ 110 mmHg, despite treatment with 2 antihypertensive agents. 14. New onset of seizures within 3 months prior to enrollment, or poorly controlled seizures. 15. Female patients who are pregnant or lactating. 16. Female patients of childbearing potential and male patients with partners of childbearing potential who are unwilling to follow strict contraception requirements. 17. Female patients of childbearing potential who do not have a negative blood or urine pregnancy test at Screening. 18. History of allergic reactions attributed to compounds of similar chemical or biologic composition to rigosertib. 19. Prior therapy with a phosphatidyl-inositol 3 kinase (PI3K), Akt or mammalian target of rapamycin (mTOR) inhibitor. 20. Any other investigational agent or chemotherapy, radiotherapy, or immunotherapy within 4 weeks of enrollment. 21. Psychiatric illness/social situations that would limit the patient's ability to tolerate and/or comply with study requirements, or inability to comply with study and/or follow-up procedures.