Diagonal F-splitting and Symbolic Powers of Ideals

Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$ essentially of finite type over an $F$-finite field. We show that $J^{s+t} \subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})$ for all $s, t, \epsilon > 0$ for which the formula makes sense. We use this to show a number of novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that $P^{(2hn)} \subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.Comment: Many small changes. Notably, I added missing Noetherianity and reducedness assumptions to section 2 and corrected an error in lemma 2.2. Upon reflection, the assumptions on $A$ and $B$ in prop 5.3 were just slightly more general than assuming $A$ and $B$ were fields, so I went ahead and said $A$ and $B$ should be field

Loop diagrams in the kinetic theory of waves

Recent work has given a systematic way for studying the kinetics of classical weakly interacting waves beyond leading order, having analogies with renormalization in quantum field theory. An important context is weak wave turbulence, occurring for waves which are small in magnitude and weakly interacting, such as those on the surface of the ocean. Here we continue the work of perturbatively computing correlation functions and the kinetic equation in this far-from-equilibrium state. In particular, we obtain the two-loop kinetic equation for waves with a cubic interaction. Our main result is a simple graphical prescription for the terms in the kinetic equation, at any order in the nonlinearity.Comment: 34 page

RationalMaps, a package for Macaulay2

This paper describes the RationalMaps package for Macaulay2. This package provides functionality for computing several aspects of rational maps such as whether a map is birational, or a closed embedding.Comment: 8 pages. The current version of the package (and other necessary files) can be accessed at https://github.com/Macaulay2/Workshop-2016-Utah/tree/master/RationalMap

Molecular Biomarkers for the Evaluation of Colorectal Cancer

Objectives: To develop evidence-based guideline recommendations through a systematic review of the literature to establish standard molecular biomarker testing of colorectal cancer (CRC) tissues to guide epidermal growth factor receptor (EGFR) therapies and conventional chemotherapy regimens

Molecular Biomarkers for the Evaluation of Colorectal Cancer: Guideline From the American Society for Clinical Pathology, College of American Pathologists, Association for Molecular Pathology, and American Society of Clinical Oncology

OBJECTIVES: - To develop evidence-based guideline recommendations through a systematic review of the literature to establish standard molecular biomarker testing of colorectal cancer (CRC) tissues to guide epidermal growth factor receptor (EGFR) therapies and conventional chemotherapy regimens. METHODS: - The American Society for Clinical Pathology, College of American Pathologists, Association for Molecular Pathology, and American Society of Clinical Oncology convened an expert panel to develop an evidence-based guideline to establish standard molecular biomarker testing and guide therapies for patients with CRC. A comprehensive literature search that included more than 4,000 articles was conducted. RESULTS: - Twenty-one guideline statements were established. CONCLUSIONS: - Evidence supports mutational testing for EGFR signaling pathway genes, since they provide clinically actionable information as negative predictors of benefit to anti-EGFR monoclonal antibody therapies for targeted therapy of CRC. Mutations in several of the biomarkers have clear prognostic value. Laboratory approaches to operationalize CRC molecular testing are presented

31st Annual Meeting and Associated Programs of the Society for Immunotherapy of Cancer (SITC 2016) : part two

Background The immunological escape of tumors represents one of the main ob- stacles to the treatment of malignancies. The blockade of PD-1 or CTLA-4 receptors represented a milestone in the history of immunotherapy. However, immune checkpoint inhibitors seem to be effective in specific cohorts of patients. It has been proposed that their efficacy relies on the presence of an immunological response. Thus, we hypothesized that disruption of the PD-L1/PD-1 axis would synergize with our oncolytic vaccine platform PeptiCRAd. Methods We used murine B16OVA in vivo tumor models and flow cytometry analysis to investigate the immunological background. Results First, we found that high-burden B16OVA tumors were refractory to combination immunotherapy. However, with a more aggressive schedule, tumors with a lower burden were more susceptible to the combination of PeptiCRAd and PD-L1 blockade. The therapy signifi- cantly increased the median survival of mice (Fig. 7). Interestingly, the reduced growth of contralaterally injected B16F10 cells sug- gested the presence of a long lasting immunological memory also against non-targeted antigens. Concerning the functional state of tumor infiltrating lymphocytes (TILs), we found that all the immune therapies would enhance the percentage of activated (PD-1pos TIM- 3neg) T lymphocytes and reduce the amount of exhausted (PD-1pos TIM-3pos) cells compared to placebo. As expected, we found that PeptiCRAd monotherapy could increase the number of antigen spe- cific CD8+ T cells compared to other treatments. However, only the combination with PD-L1 blockade could significantly increase the ra- tio between activated and exhausted pentamer positive cells (p= 0.0058), suggesting that by disrupting the PD-1/PD-L1 axis we could decrease the amount of dysfunctional antigen specific T cells. We ob- served that the anatomical location deeply influenced the state of CD4+ and CD8+ T lymphocytes. In fact, TIM-3 expression was in- creased by 2 fold on TILs compared to splenic and lymphoid T cells. In the CD8+ compartment, the expression of PD-1 on the surface seemed to be restricted to the tumor micro-environment, while CD4 + T cells had a high expression of PD-1 also in lymphoid organs. Interestingly, we found that the levels of PD-1 were significantly higher on CD8+ T cells than on CD4+ T cells into the tumor micro- environment (p < 0.0001). Conclusions In conclusion, we demonstrated that the efficacy of immune check- point inhibitors might be strongly enhanced by their combination with cancer vaccines. PeptiCRAd was able to increase the number of antigen-specific T cells and PD-L1 blockade prevented their exhaus- tion, resulting in long-lasting immunological memory and increased median survival

Diagonal F-splitting and Symbolic Powers of Ideals

Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$essentially of finite type over an $F$-finite field. We show that $J^{s+t}\subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})$ for all $s, t, \epsilon> 0$ for which the formula makes sense. We use this to show a number of novelcontainments between symbolic and ordinary powers of prime ideals in thissetting, which includes all determinantal rings and a large class of toricrings in positive characteristic. In particular, we show that $P^{(2hn)}\subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.Comment: Copy edited and formatted in the EpiGA journal's styleshee

Doctor of Philosophy

dissertationTest ideals are an important object of study in positive characteristic commutative algebra. The test ideals of a regular ring in positive characteristic enjoy a property known as subadditivity. This subadditivity property has numerous important applications, such as Ein-Lazarsfeld-Smith's argument showing a uniform boundedness in the growth of symbolic powers of ideals in regular rings. In [Tak06], Takagi finds a subadditivity formula for test ideals in the non-regular setting that uses the Jacobian ideal Jac(R) as a correction term. Both Takagi's proof, as well as the original proof of subadditivity in [HY03], uses the classical perspective on test ideals as annihilators of tight closure. In this dissertation, we use the more modern perspective of test ideals described in [HT04] and [Sch10] to find a new subadditivity formula for so-called "big" or "non-finitistic" test ideals, which are conjecturally the same as ordinary test ideals; this conjecture is known to hold for graded rings and for Q-Gorenstein rings. In particular, we use the theory of Cartier algebras introduced in [Sch11]. We introduce a new set of Cartier algebras, called the diagonal Cartier algebras, that measure the failure of R to be smooth. These Cartier algebras appear as correction terms in various versions of our subadditivity formula. This dissertation is organized as follows. In Chapter 1, we review the basic terminology of test ideals and Cartier algebras, as well as the process of "reduction modulo p." In Chapter 2, we introduce our subadditivity formula (Theorem 2.2.11) and show this formula yields sharper containments than Takagi's subadditivity formula (Theorem 2.3.1). Chapter 3 is devoted to showing that our subadditivity formula yields sharper containments than (the mod p reduction of) a subadditivity formula for multiplier ideals found by Eisenstein in [Eis10] (Theorem 3.2.2). To get there, we generalize earlier constructions of test ideals and multiplier ideals with good restriction properties (Definition 3.1.3, Definition 3.1.22) found in [Sch09, Tak10, Eis10]. In Chapter 4, we use our new subadditivity formula to generalize the symbolic power containment formulas of [ELS01, HH02] (Theorem 4.2.4, Proposition 4.3.5, Theorem 4.4.1). Finally, in Chapter 5, we provide a combinatorial description of the diagonal Cartier algebras of toric rings (Theorem 5.0.4). We briefly examine the singularities of rings with large diagonal Cartier algebras in Section 5.1