13 research outputs found
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
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 of 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 correspond to residues of torus
localization weights of a certain quasi-coherent sheaf on
the Hilbert scheme of points on , thereby finding a geometric
interpretation for them. As corollaries, we obtain the polynomiality in 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
Algebra and geometry of link homology
These notes cover the lectures of the first named author at 2021 IHES Summer
School on "Enumerative Geometry, Physics and Representation Theory" with
additional details and references. They cover the definition of
Khovanov-Rozansky triply graded homology, its basic properties and recent
advances, as well as three algebro-geometric models for link homology: braid
varieties, Hilbert schemes of singular curves and affine Springer fibers, and
Hilbert schemes of points on the plane.Comment: 49 page
The Hilb-vs-Quot Conjecture
Let be the complete local ring of a complex plane curve germ and its
normalization. We propose a conjecture relating the virtual weight polynomials
of the Hilbert schemes of to those of the Quot schemes that parametrize
-submodules of . 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 , where is either
coprime to or divides , 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 -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