January 15: Christian Schnell (Stony Brook University)
Location: SCHM 112
Title: A Hodge-theoretic proof of Hwang's theorem
Abstract: Compact hyperkähler manifolds are the higher-dimensional generalization of K3 surfaces. There are few examples, but no complete classification is known at the moment. Compact hyperkähler manifolds have almost no morphisms to other projective varieties; Matsushita showed that, with a few trivial exceptions, every morphism is a Lagrangian fibration (= the higher-dimensional generalization of an elliptic fibration on a K3-surface). I will explain a new Hodge-theoretic proof for Hwang's theorem, which says that if the base of a Lagrangian fibration on an irreducible compact hyperkähler manifold is smooth, then it must be projective space. This is joint work with Ben Bakker from UIC.
January 22: Yilong Zhang (Purdue University)
Location: SCHM 112
Title: Lines on Cubic Hypersurfaces
Abstract: The study of lines on cubic hypersurfaces dates back to the 19th century, when Cayley and Salmon discovered the 27 lines on a cubic surface. In the 1970s, Clemens and Griffiths studied lines on cubic threefolds, and used the geometry of its parameter space to prove the irrationality of a cubic threefold. In the 80's, Beauville and Donagi extended this investigation to dimension four, showing that the parameter space of lines on a cubic fourfold form a hyperkähler variety, which is closely related to K3 surfaces. In this talk, I will talk about my recent research on special lines on cubic hypersurfaces starting from dimension four, and their relation to the smoothness of a component of Hilbert scheme containing a pair of disjoint lines.
January 29: Kenji Matsuki (Purdue University)
Location: SCHM 112
Title: Inductive scheme for the problem of resolution of singularities: characteristic zero vs positive characteristic, PART III
Abstract: This is PART III of my talk on the problem of resolution of singularities. I will try to make it accessible to the people who might have missed the previous talks, by writing up the notes for them.
Brief Review
Let us briefly review what we have established so far.
1st Talk: (Reformulation of the problem) We reformulated the problem of resolution of singularities to the problem of resolution of singularities of a basic object.
2nd Talk: (Naive Inductive Scheme) We introduced the “naive” inductive scheme on dimension to solve the problem of resolution of singularities of a basic object via the notion of a hypersurface of maximal contact.
On Wednesday, Jan. 29, after quickly recalling
the notion of a basic object,
the naive inductive scheme as above,
we will discuss
3rd Talk: (Sublimation of the naive inductive scheme into the general inductive scheme) The naive inductive scheme, as the name indicates, only naively applies to the special situation. We will discuss how to “sublimate” the naive inductive scheme into the one which applies to the general situation by the introduction of the invariants. As a result, this finishes and provides a complete algorithm of resolution of singularities in characteristic zero. A hypersurface of maximal contact only exists in characteristic zero.
The talk will be elementary, and should be accessible to the beginning graduate students in Algebraic Geometry and in other related fields of interest.
February 5: Ben McReynolds (Purdue University)
Location: SCHM 112
Title: Nilpotent representations of Galois and fundamental groups
Abstract: In this talk, I will discuss applications of a purely group theoretic result to fundamental groups of hyperbolic manifolds, absolute Galois groups of number fields, and algebraic fundamental groups of curves and surfaces. The main point being that these objects are not determined by their nilpotent representation theory. This is joint work with Milana Golich.
February 12: Siyang Liu (University of South California)
Location: SCHM 112
Title: Hyperplane Arrangements, Hypertoric Variety, and Representation Theory
Abstract: Complements of complex hyperplane arrangements are affine varieties with interesting combinatorial structures and deep relations to geometric representation theory. They encode necessary data for hypertoric varieties coming from algebraic symplectic reduction of quaternion vector spaces by algebraic tori. Braden-Licata-Proudfoot-Webster constructed "hypertoric enveloping algebra" associated to a given toric hypertoric variety and studied the BGG-type category O of this algebra. In the recent joint work with Sukjoo Lee, Yin Li and Cheuk Yu Mak, we showed that universal deformations of the category O can be in fact recovered from the geometry of complexified hyperplane arrangements, in particular simple and standard modules can be realized as certain submanifolds, and endomorphisms provided by intersection points. This also verifies a conjecture of Lauda-Licata-Manion and a proposal by Lekili-Segal.
February 19: Kenji Matsuki (Purdue University)
Location: SCHM 112
Title: Inductive scheme for the problem of resolution of singularities: characteristic zero vs positive characteristic, PART IV
Abstract: This is PART IV of my talk on the problem of resolution of singularities. The main theme of this talk is:
OUR ATTEMPT IN POSITIVE CHARACTERISTIC
Finally I will try to discuss our on-going attempt, together with Hiraku Kawanoue of RIMS at Kyoto, to try to solve the problem in positive characteristic. We have been successful so far up to dimension 3, and will try to explain the difficulty going into higher dimensions.
Brief Outline of Classical Strategy in Characteristic 0
As we discussed in the previous 3 talks, the classical strategy in characteristic 0 is:
Framework: Basic Object
Main tool of induction: A smooth hypersurface of maximal contact
Invariants: (w-ord,s)
Final Stage: Monomial Case (Easy !)
Brief Outline of Our Strategy in Positive Characteristic
Framework: Idealistic Filtration
Main tool of induction: A collection of possibly singular hypersurfaces of maximal contact
Invariants: (µ,s)
Final Stage: Monomial Case (Difficult !)
Our strategy can be successfully carried out up to dimension 3. In dimension 4, the final stage of solving the problem in the monomial case becomes subtle and difficult, facing the so-called Moh-Hauser jumping phenomenon.
Let me also mention that our framework of “the idealistic filtration” has a lot to do with Włodarczyk’s brilliant solution to the globalization problem in characteristic 0 through his novel idea of “homogenization”, where Kawanoue solves the same globalization problem through his idea of “Differential saturation”.
February 26: Vadim Vologodsky (Higher School of Economics)
Location: SCHM 112
Title: Modules with q-connection and the ``sheared'' prismatization.
Abstract: Let X be a smooth p-adic formal scheme over Z_p. Given a ``framing'', that is, an etale map X ----> T to a torus, Scholze introduced the algebra D_q of q-differential operators on X. Though the algebra D_q depends on the choice of a framing, Scholze conjectured in 2016 that the category of (p, q-1)-complete modules over D_q depends on X only and thus can be defined globally. Subsequently, in a joint work with Bhatt, Scholze proved a weaker version of this conjecture, for nilpotent D_q-modules. I will sketch a proof of the strong form of the conjecture (for p>2) and explain its relation to the p-adic Simpson correspondence. The talk is based on a joint work with Bhatt, Kanaev, Mathew, Zhang.
March 5: Emanuel Reinecke (Institut des Hautes Études Scientifiques)
Location: SCHM 112
Title: Relative Poincare duality in rigid geometry
Abstract: While the Z/p-etale cohomology of rigid-analytic varieties is in general hard to control, it becomes more tractable when the varieties are proper. For example, in the smooth and proper case, it is finite-dimensional and has recently been shown in work of Zavyalov and of Mann to satisfy Poincare duality. In my talk, I will explain a relative Poincare duality statement for etale cohomology with finite coefficients which applies to any proper morphism of rigid-analytic varieties over nonarchimedean fields of mixed characteristic, confirming an expectation of Bhatt-Hansen. A key ingredient in the proof will be a new construction of trace maps for proper morphisms. Joint work with Shizhang Li and Bogdan Zavyalov.
March 12: Eyal Markman (University of Massachusetts Amherst)
Location: SCHM 112
Title: Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds.
Abstract: In 1977 Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian varieties are said to be of Weil type and these Hodge classes are known as Weil classes. We prove that the Weil classes are algebraic for all abelian sixfold of Weil type of discriminant -1, for all imaginary quadratic number fields. The algebraicity of the Weil classes follows for all abelian fourfolds of Weil type (for all discriminants and all imaginary quadratic number fields), by a degeneration argument of C. Schoen. The Hodge conjecture for abelian fourfolds is known to follow from the above result.
March 26: Bogdan Zavyalov (Princeton University/Institute for Advanced Study)
Location: SCHM 112
Title: Poincare Duality for pro-etale Q_p-local systems
Abstract: Let X be a smooth rigid-analytic space over C_p. In contrast to algebraic geometry, it turns out that there are many pro-etale Q_p local systems on X that do not admit any Z_p-lattice. Furthermore, cohomology of these local systems often fail to be finite dimensional as Q_p-vector spaces and do not satisfy the naive version of Poincare Duality. At first glance, this may suggest that pro-etale Q_p-local systems (without a Z_p-lattice) are somewhat pathological. However, Kedlaya and Liu observed that these cohomology groups are still finite-dimensional in some precise sense; namely, these cohomology groups admit a natural structure of Banach--Colmez spaces. In my talk, I will discuss that cohomology of pro-etale Q_p-local systems also satisfy a version of Poincare Duality inside the category of Banach--Colmez spaces. Joint work in progress with Shizhang Li, Wieslawa Niziol, and Emanuel Reinecke.
March 31 (Special date): Zhixian Zhu (Capital Normal University)
Location/Time: 10–11AM on Zoom (https://purdue-edu.zoom.us/j/97786675764?pwd=ZUxkQzFsck5IaFBBZnhiOVNTZys4UT09)
Title: Positivity of line bundles on toric variety
Abstract: In this talk, we first characterize the jet ampleness of line bundles on a toric variety, in terms of the lattice length of their associated polytopes. As an application, we prove a k-jet generalization of Fujita’s conjectures. Inspired by the Mukai conjecture, which naturally generalizes Fujita’s conjectures, we make a conjecture regarding the higher syzygies in terms of the lattice length and report our progress in this conjecture. Part of the talks is based on joint work with J. Gonzalez, and joint work in progress with L. Song and H. Wen.
April 2: Ben Tighe (University of Oregon)
Location: SCHM 112
Title: Symmetries and vanishing theorems for symplectic varieties
Abstract: Compact hyperkahler manifolds are distinguished in algebraic geometry for their Hodge theory, as many geometric properties are completely determined by their second cohomology. One reason for this is the existence of a holomorphic symplectic form, which determines a Hodge structure on the entire cohomology ring coming from symmetries on the level of differential form. In this talk, I will outline how these symmetries extend to Hodge-theoretic objects in the derived category for singular analogues: symplectic varieties. Applications of these symmetries include a description of the higher Du Bois (resp. Higher rational) property for symplectic varieties and an extension of the total Lie algebra action on intersection cohomology.
April 9: François Greer (Michigan State University)
Location: SCHM 112
Title: Boundedness for abelian fibrations
Abstract: I will explain recent work with Engel, Filipazzi, Mauri, and Svaldi about the moduli problem of K-trivial fibrations. In particular, we show that in a fixed dimension, there are finitely many deformation classes of holomorphic symplectic varieties with b_2>4 that are deformable to a Lagrangian fibration.
April 16: Masayuki Kawakita (Research Institute for Mathematical Sciences, Kyoto University)
Location: SCHM 112
Title: Minimal log discrepancies on a fixed threefold
Abstract: The minimal log discrepancy is an invariant of singularities related to termination of flips. The ACC for minimal log discrepancies is still unknown in dimension three, and it is one of the most important remaining problems in the minimal model theory of threefolds. I will explain a proof of the ACC for minimal log discrepancies on an arbitrary fixed threefold.
April 23: Amadou Bah (Columbia University)
Location: SCHM 112
Title: The Swan conductor function of an l-adic sheaf on rigid annulus is convex.
Abstract: For a given étale l-adic sheaf F on rigid space X, the Swan conductor, a measure of wild ramification of F along the special fiber of a formal model of X, is the l-adic analog of the radius of convergence of solutions of p-adic differential equations on X. I will show (when X is an annulus) that the function defined by the Swan conductor has properties that mirror those of the radius of convergence function.
April 30: Mircea Mustaţă (University of Michigan)
Location: SCHM 112
Title: A birational description of the minimal exponent
Abstract: The minimal exponent of a hypersurface is an invariant of singularities introduced by Morihiko Saito via D-module theory, which refines the log canonical threshold. I will give an introduction to this invariant and then I will describe a joint result with Qianyu Chen, giving a birational description of this invariant via twisted sheaves of log differentials.
If you are interested in giving a talk, please contact one of the organizers: