Gedatolisib Plus Talazoparib in Advanced Triple Negative or BRCA1/2 Positive, HER2 Negative Breast Cancers

Inclusion Criteria: Individuals from populations who are underrepresented in clinical research (e.g., racial and ethnic minorities, women, individuals from rural/frontier communities, older individuals) will be enrolled with a goal of ensuring that all eligible patients are given the opportunity to participate in novel clinical trials and that research findings can be generalizable to the entire population. * Male or female ≥ 18 years of age at time of consent. * Subjects with histologically confirmed breast cancer that is advanced (defined as metastatic or unresectable). * Phase II Cohort A: Patients with advanced triple negative breast cancer (TNBC) with negative or unknown germline BRCA status. Variants of undetermined significance in BRCA 1/2 should be considered negative. * Note: most recent tumor biopsy must be ER/PR negative or have ER and PR \<10% and all prior biopsies of metastatic sites cannot have ever had ER or PR ≥20%. * Note: HER2 is considered negative if ISH negative by ASCO/CAP guidelines or HER2+ 0 or 1+ on IHC or 2+ with negative ISH * Phase II Cohort B \[CLOSED\]: Patients with advanced HER2 negative breast cancer and a germline BRCA1 or 2 (1/2) mutation ---Note: HER2 is considered negative if ISH negative by ASCO/CAP guidelines or HER2+ 0 or 1+ on IHC or 2+ with negative ISH * Phase I run-in: meets criteria for either cohort A or B * Phase I run-in: Measurable or evaluable (non-measurable) disease (see section 9.2 for more detail). * Phase II: Measurable disease by RECIST 1.1 is required. * Prior therapy: * Cohort A: At least one line of prior systemic therapy for advanced breast cancer (chemotherapy or other targeted therapy allowed). No more than 3 lines of prior chemotherapy for advanced disease are allowed. No limit on prior endocrine or targeted therapies. * Cohort B \[CLOSED\]: No more than 2 lines of prior chemotherapy for advanced disease are allowed. No limit on prior endocrine or targeted therapies. * Both cohorts: no prior PARP inhibitor for advanced breast cancer * Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 within 28 days prior to study registration. * Life expectancy of 12 weeks or greater as determined by the treating physician. * Demonstrate adequate organ function as defined in the table in the protocol; all screening labs to be obtained within 28 days prior to registration. * Archived tumor tissue available (Metastatic disease from non-bone and non-brain sites preferred, but primary breast or lymph node tissue is permitted. Confirmation of available tissue only- tumor samples do not need to be shipped for eligibility purposes. Tumor samples do not need to be shipped until subject is confirmed eligible and is registered for treatment. * Ability to take oral medications. * No history of type I diabetes. For patients with known type II diabetes, must have controlled diabetes (Hgb A1c \< 7.0 mmol/L within 30 days of study entry) with no more than one oral anti-diabetic agent and no insulin. * No active central nervous system (CNS) metastatic disease. NOTE: Subjects with CNS involvement must meet ALL of the following to be eligible: * At least 28 days from prior definitive treatment of their CNS disease by surgical resection, SBRT or WBRT at the time of registration * Asymptomatic and off systemic corticosteroids and/or enzyme-inducing antiepileptic medications for brain metastases for \>14 days prior to registration. * Provided written informed consent and HIPAA authorization for release of personal health information, approved by an Institutional Review Board (IRB). * NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Women of childbearing potential (WOCP) must not be pregnant or breast-feeding. A negative serum or urine pregnancy test is required within 14 days of study registration. If the urine test cannot be confirmed as negative, a serum pregnancy test will be required. * Women of childbearing potential (WOCP) must be willing to use two effective methods of birth control such as an oral, implantable, injectable, or transdermal hormonal contraceptive, an intrauterine device (IUD), use of a double barrier method (condoms, sponge, diaphragm, or vaginal ring with spermicidal jellies or cream), vasectomized partner, or total abstinence from the time of informed consent, throughout the course of the study, and for at least 7 months after the last dose of study drug. * NOTE; Women are considered to be of childbearing potential unless they are postmenopausal (≥45 years of age and has not had menses for greater than 12 consecutive months) or bilateral oophorectomy or surgically sterile (bilateral tubal ligation or hysterectomy) or not heterosexually active for the duration of the study and willing to continue for at least 7 months after the last dose of study drug. * Men who are not surgically sterile (vasectomy) must agree to use an acceptable method of contraception. Male subjects with female sexual partners who are pregnant, possibly pregnant, or who could become pregnant during the study must agree to use condoms from the first dose of study drug through at least 4 months after the last dose of study drug. Total abstinence for the same time period is an acceptable alternative. * As determined by the enrolling physician or protocol designee, ability of the subject to understand and comply with study procedures for the entire length of the study. Exclusion Criteria: * Active infection requiring systemic therapy. Patients with a known history of HIV must have a CD4 count ≥ the institutional lower limit of normal within 28 days prior to registration. Patients with HIV must also be on a stable anti-retroviral regimen for ≥ 28 days before registration. * Pregnant or breastfeeding (NOTE: breast milk cannot be stored for future use while the mother is being treated on study). * Patients who have had chemotherapy, targeted therapy, or radiotherapy within 2 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from acute effects of any prior therapy to baseline or Grade ≤1. Grade 2 or higher exceptions include alopecia, up to grade 2 neuropathy or other Grade 2 AEs or lab values not constituting a safety risk in the opinion of the treating physician. * Treatment with any investigational drug within 14 days prior to registration. NOTE: Investigational imaging agents are not included in the definition and are allowed. * Subject has had major surgery within 14 days prior to registration or has not recovered from major side effects of the surgery (tumor biopsy is not considered as major surgery). * Any prior or concurrent malignancy whose natural history or treatment has the potential to interfere with the safety or efficacy assessment of this investigational regimen. * Known hypersensitivity to any of the excipients of gedatolisib or talazoparib. * Any impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of talazoparib (e.g., ulcerative diseases, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, or small bowel resection). * Known active hepatitis B or C (testing not mandatory). Patients who have completed curative therapy for HCV are eligible. * Known history of myelodysplastic syndrome or acute myeloid leukemia. * Subjects with any of the following conditions: * History of drug-induced pneumonitis within last 12 months or any history of pneumonitis related to an mTOR inhibitor or current clinically significant pulmonary disease not due to the breast cancer. * History of abdominal fistula, gastrointestinal perforation, or intra- abdominal abscess within 28 days prior to registration. * Any history of cerebrovascular accident (CVA) or transient ischemic attack within 12 months prior to registration. * History of acute coronary syndromes (including myocardial infarction, unstable angina, coronary artery bypass grafting, coronary angioplasty, or stenting) or symptomatic pericarditis within 6 months prior to registration. * Symptomatic congestive heart failure (New York Heart Association III-IV) or documented current cardiomyopathy with left ventricular ejection fraction (LVEF) \<50% * Clinically significant cardiac ventricular arrhythmias (e.g. sustained ventricular tachycardia/ventricular fibrillation) or high-grade AV block (e.g. bifascicular block, Mobitz type II and third-degree AV block) unless a pacemaker is in place. * Long QT syndrome or family history of idiopathic sudden death or congenital long QT syndrome. * Any concurrent severe and/or uncontrolled medical condition that would, in the investigator's judgment, cause unacceptable safety risks, contraindicate subject participation in the clinical study or compromise compliance with the protocol. * Subject is currently receiving warfarin or other coumarin-derived anticoagulant for treatment, prophylaxis or otherwise. Therapy with heparin, low molecular weight heparin (LMWH), direct thrombin inhibitors (such as dabigatran) or novel oral anticoagulants (such as rivaroxaban or apixaban) are allowed as long as the patient has been on this therapy for at least 14 days with no clinically significant bleeding.