![]() ![]() This is joint work with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Svetlana Makarova, Enrica Mazzon, and Nivedita Viswanathan. We'll present a classification of smooth projective toric varieties with \(m(X) ≥ dim(X)-2\), and show that projective spaces are the only 2-Fano manifolds among smooth projective toric varieties with \(m(X)\) equal to 1, \(dim(X)-2\), \(dim(X)-1\), or \(dim(X)\). Binghamton University Topology Seminar (Invited). ![]() This invariant \(m(X)\) captures the minimal degree of a dominating family of rational curves on \(X\) or, equivalently, the minimal length of a centrally symmetric primitive relation for the fan of \(X\). Graduate Student Conference in Algebra, Geometry and Topology, June 2017, Temple University. Motivated by the problem of classifying toric 2-Fano manifolds, we will introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension, \(m(X)\). In particular we will focus on toric 2-Fano manifolds. Lingjie Ma (UIC, Department of Finance) 4:00 PM in 636 SEO. In this talk we will discuss higher Fano manifolds, which are Fano manifolds with positive higher Chern characters. Geometry/Topology Seminar TBA Jacob Russell (Rice University) view abstract 4:00 PM in 636 SEO. Permutohedral Complexes and Curves With Cyclic Action I also talk about an application of the theory of quasi-F-splitting, which is studied in joint work with Takamatsu-Tanaka-Witaszek-Yobuko-Yoshikawa, to the extension problem. Existing studies depend on geometric tools such as log resolutions, (mixed) Hodge theory, the minimal model program, and vanishing theorems, which are not expected to be true or are not known for higher-dimensional varieties in positive characteristic.įor this reason, I introduce a new algebraic approach to the extension theorem using Cartier operators. In this talk, we discuss the extension theorem in positive characteristic. This is a basic property relating differential forms and singularities, and many results are known over the field of complex numbers. This is joint work with Fulvio Gesmundo.īecause of exams and/or travel, Daniel is unable to attend seminars on Oct 11, Oct 18, Nov 15, and Dec 13.For a normal variety \(X\), we say \(X\) satisfies the extension theorem if, for every proper birational morphism from \(Y\), every differential form on the regular locus of \(X\) extends to \(Y\). As a first step in understanding these varieties, we compute a formula for the degrees of Stiefel manifolds using techniques from classical algebraic geometry, representation theory, and combinatorics. ![]() This viewpoint is known as algebraic frame theory and still many algebro-geometric properties of these varieties remain unknown, in particular, their degrees. The Stiefel manifold, and many other spaces of finite frames, can be viewed as an algebraic variety embedded in the space of k by n matrices. The Stiefel manifold of orthonormal bases for k-planes in an n-dimensional space goes by a different name in the world of frame theory: the space of Parseval n-frames for a k-dimensional space. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. I will introduce the class of completely log-concave polynomials in elementary terms, discuss the beautiful real and combinatorial geometry underlying these polynomials, and describe applications to random walks on simplicial complexes. Log-concave polynomials, matroids, and expandersĬomplete log-concavity is a functional property of real multivariate polynomials that translates to strong and useful conditions on its coefficients. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5. \displaystyle are simply connected, for g at least 1. Geometric vertex decomposition and liaison ![]() Standard Conjecture D for Matrix Factorizations On the cohomology of the moduli space of parabolic connections Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).1 Algebra and Algebraic Geometry Mailing ListĪlgebra and Algebraic Geometry Mailing List. ![]()
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