Quadratic ideals and Rogers-Ramanujan recursions

We give an explicit recursive description of the Hilbert series and Gr\"obner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers-Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.Comment: 17 page

Shalika germs for tamely ramified elements in GLnGL_n

Degenerating the action of the elliptic Hall algebra on the Fock space, we give a combinatorial formula for the Shalika germs of tamely ramified regular semisimple elements $\gamma$ of $GL_n$ over a nonarchimedean local field. As a byproduct, we compute the weight polynomials of affine Springer fibers in type A and orbital integrals of tamely ramified regular semisimple elements. We conjecture that the Shalika germs of $\gamma$ correspond to residues of torus localization weights of a certain quasi-coherent sheaf $\mathcal{F}_\gamma$ on the Hilbert scheme of points on $\mathbb{A}^2$, thereby finding a geometric interpretation for them. As corollaries, we obtain the polynomiality in $q$ of point-counts of compactified Jacobians of planar curves, as well as a virtual version of the Cherednik-Danilenko conjecture on their Betti numbers. Our results also provide further evidence for the ORS conjecture relating compactified Jacobians and HOMFLY-PT invariants of algebraic knots.Comment: 47 pages, added clarifications on the unramified case and an application to components of affine Springer fibers, fixed typos and reference

The Hilb-vs-Quot Conjecture

Let $R$ be the complete local ring of a complex plane curve germ and $S$ its normalization. We propose a conjecture relating the virtual weight polynomials of the Hilbert schemes of $R$ to those of the Quot schemes that parametrize $R$-submodules of $S$. We prove an identity relating the Quot side to strata in a lattice quotient of a compactified Picard scheme, showing that our conjecture generalizes a conjecture of Cherednik's beyond the unibranch case, and that it would relate the perverse filtration on the cohomology of the Picard side to the stratification. We also lift our work to a parabolic refinement where we track partial flags. We propose a Quot version of the Oblomkov-Rasmussen-Shende conjecture, relating the parabolic Quot side to Khovanov-Rozansky link homology. It becomes equivalent to the original Hilbert version under our Hilb-vs-Quot conjecture, but is more tractable. For germs of the form $y^n = x^d$, where $n$ is either coprime to or divides $d$, we prove our Quot version in its full form. No similar result keeping all three gradings is known for the Hilbert version. Finally, we enhance the Quot version to incorporate a polynomial action on the link homology, as well as its $y$-ification; neither has a Hilbert analogue.Comment: 51 pages, 2 figures. Comments welcome

Implementing a Functional Precision Medicine Tumor Board for Acute Myeloid Leukemia

We generated ex vivo drug-response and multiomics profi ling data for a prospective series of 252 samples from 186 patients with acute myeloid leukemia (AML). A functional precision medicine tumor board (FPMTB) integrated clinical, molecular, and functional data for application in clinical treatment decisions. Actionable drugs were found for 97% of patients with AML, and the recommendations were clinically implemented in 37 relapsed or refractory patients. We report a 59% objective response rate for the individually tailored therapies, including 13 complete responses, as well as bridging five patients with AML to allogeneic hematopoietic stem cell transplantation. Data integration across all cases enabled the identifi cation of drug response biomarkers, such as the association of IL15 overexpression with resistance to FLT3 inhibitors. Integration of molecular profi ling and large-scale drug response data across many patients will enable continuous improvement of the FPMTB recommendations, providing a paradigm for individualized implementation of functional precision cancer medicine. SIGNIFICANCE: Oncogenomics data can guide clinical treatment decisions, but often such data are neither actionable nor predictive. Functional ex vivo drug testing contributes signifi cant additional, clinically actionable therapeutic insights for individual patients with AML. Such data can be generated in four days, enabling rapid translation through FPMTB.Peer reviewe

Implementing a Functional Precision Medicine Tumor Board for Acute Myeloid Leukemia

We generated ex vivo drug-response and multiomics profi ling data for a prospective series of 252 samples from 186 patients with acute myeloid leukemia (AML). A functional precision medicine tumor board (FPMTB) integrated clinical, molecular, and functional data for application in clinical treatment decisions. Actionable drugs were found for 97% of patients with AML, and the recommendations were clinically implemented in 37 relapsed or refractory patients. We report a 59% objective response rate for the individually tailored therapies, including 13 complete responses, as well as bridging five patients with AML to allogeneic hematopoietic stem cell transplantation. Data integration across all cases enabled the identifi cation of drug response biomarkers, such as the association of IL15 overexpression with resistance to FLT3 inhibitors. Integration of molecular profi ling and large-scale drug response data across many patients will enable continuous improvement of the FPMTB recommendations, providing a paradigm for individualized implementation of functional precision cancer medicine. SIGNIFICANCE: Oncogenomics data can guide clinical treatment decisions, but often such data are neither actionable nor predictive. Functional ex vivo drug testing contributes signifi cant additional, clinically actionable therapeutic insights for individual patients with AML. Such data can be generated in four days, enabling rapid translation through FPMTB.Peer reviewe

Koszulin algebrat ja resoluutiot

A standard graded k-algebra R is called Koszul, if the residue class field k = R=R+ has a linear R-resolution. This characterization is equivalent to a number of conditions, and implies that R is first of all a quadratic algebra. In addition, it can be shown that the existence of a quadratic Gröbner basis for the defining ideal implies Koszulness. These implications are easily shown to be strict, and little precise information is known what makes an algebra lose or gain the Koszul property. This thesis patches various definitions of Koszulness appearing in the literature together, and as an example of a class of Koszul algebras, considers the resolutions of k over algebras formed from binomial edge ideals of graphs. We give general ranks of the first syzygies, describe explicit resolutions for certain classes of graphs and discuss the combinatorial properties that make algebras formed from edge ideals lose Koszulness on addition of edges.Koszulin algebrat ovat luokiteltuja k-algebroja R, joiden jäännösluokkakunnalla k = R=R+ on lineaarinen resoluutio. Tämä luonnehdinta on ekvivalentti usean muun kanssa, ja sen seurauksena R on esimerkiksi neliöllinen algebra. Tämän lisäksi voidaan osoittaa, että neliöllisen Gröbnerkannan olemassaolosta algebran R määrittelevälle ideaalille seuraa Koszul-ominaisuus. Nämä implikaatiot ovat helposti osoitettavissa yksisuuntaisiksi, ja vain vähän tiedetään siitä mikä saa algebran menettämään ko. ominaisuus. Tämä diplomityö tuo yhteen eri määritelmiä Koszulin algebroille kirjallisuudesta, ja esimerkkinä luokasta Koszulin algebroita käsittelee graafien reunoihin liitettävien binomisten reunaideaalien määrittelemiä algebroja. Ko. ideaaleille lasketaan ensimmäiset syzygimoduulit ja usealle eri graafiluokalle rakennetaan koko vapaa resoluutio, sekä pohditaan mikä saa algebran menettämään Koszul-ominaisuutensa graafiin reunoja lisättäessä