Fall 2023
September 5: Shuddhodan Kadattur Vasudevan (IHES)
Title: Perverse filtrations via Brylinski-Radon transformations
Abstract: In 2010, de Cataldo-Migliorini showed how to compute the perverse filtration on the cohomology of an affine variety with values in a constructible sheaf using generic flags.
In this talk, I shall introduce the Brylinski-Radon transformation, discuss its properties and derive consequences for the perverse filtration. Time permitting we shall also discuss some arithmetic applications of our results. This is joint work with Ankit Rai.
September 12: Felix Janda (University of Illinois Urbana-Champaign)
Title: Fixed-point loci in moduli problems
Abstract: Whenever a smooth variety X admits the action of a torus T, the Atiyah-Bott localization formula allows reducing computations in the cohomology of X to computations on the fixed locus X^T. This is particularly powerful when X^T is more explicit than X, as it is often the case in moduli theory.
In my talk, I will explain why in the context of localization, it is essential to work with the moduli stack as opposed to the moduli space, and I will discuss joint work with J. Alper that describes the global structure of torus fixed loci in algebraic stacks.
September 19: Bruce Reznick (University of Illinois Urbana-Champaign)
Title: Hilbert’s Seventeenth Problem
Abstract: If p is a real polynomial in n variables, then p is “psd" if it only takes non-negative values, and p is sos if it can be written as a sum of squares of real polynomials. Every sos polynomial is psd, and for small values of n and the degree, every psd polynomial is sos. In 1888, Hilbert proved that there exist psd polynomials which are not sos, and in 1900 asked whether every psd polynomial is a sum of squares of rational functions (he had proved this for n=2). Artin gave a non-constructive proof in the 1920s using the theory of real algebra, which he had developed.
September 26: Ravi Fernando (University of Illinois Urbana-Champaign)
Title: Saturated de Rham-Witt complexes
Abstract: The de Rham-Witt complex, introduced by Illusie in 1979, is a complex that plays an analogous role in computing crystalline cohomology to that of the de Rham complex for algebraic de Rham cohomology. In 2018, Bhatt-Lurie-Mathew introduced a variant called the saturated de Rham-Witt complex, which refines and provides a new perspective on the classical de Rham-Witt complex. In this talk, we will discuss the approach of Bhatt-Lurie-Mathew, as well as some recent work to generalize it to nontrivial coefficient systems.
October 3: Sheldon Katz (University of Illinois Urbana-Champaign)
Title: Enumerative Geometry of Determinantal Octic Double Solids
Abstract: The double cover of P^3 branched along a smooth degree 8 surface is a Calabi-Yau threefold originally studied by Herb Clemens in the early 1980s. Its genus zero Gromov-Witten invariants were computed in the early 1990s shortly following the famous calculation of the GW invariants of the quintic threefold by Candelas et al, and these predictions were proven in the mid-1990s by Givental and Lian-Liu-Yau, with the Gromov-Witten invariants themselves having been rigorously defined in the intervening years. The higher genus Gromov-Witten invariants have been computed for the octic double solid for g \le 60 using B-model techniques. These predictions are not yet proven. The integer-valued Gopakumar-Vafa invariants can be inferred from the GW invariants. Intrinsically, GV invariants are invariants of moduli spaces of 1-dimensional sheaves.
A determinantal octic double solid is a double cover of P^3 branched along the degree 8 determinant of a symmetric matrix of homogeneous forms on P^3. B-model techniques can also be used to compute enumerative invariants of determinantal octic double solids up to g \le 32. These threefolds have isolated nodes and do not have a projective small resolution, so the standard algebro-geometric definitions of Gromov-Witten or Gopakumar-Vafa invariants do not apply. Nevertheless, when computational methods for GV invariants are applied to this situation, the results agree with the B-model predictions whenever they can be carried out. There are two developing proposals for rigorously defining these invariants. One is based on moduli spaces of 1-dimensional twisted sheaves on (non-algebraic) small resolutions, with the novel feature that all small resolutions must be considered simultaneously to properly define the invariants. The other proposal is based on moduli spaces of sheaves of modules over a certain locally free sheaf of noncommutative algebras.
This talk is based on joint work with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe.
October 10 (joint with AGC seminar): Laura Escobar (Washington University in St. Louis)
Title: Geometric constructions from abstract wall-crossing
Abstract: Theory of Newton-Okounkov bodies has led to the extension of the geometry-combinatorics dictionary from toric varieties to certain varieties which admit a toric degeneration. In a recent paper with Megumi Harada, we gave a wall-crossing formula for the Newton-Okounkov bodies of a single variety. Our wall-crossing involves a collection of lattices $\{M_i\}_{i\in I}$ connected by piecewise-linear bijections $\{\mu_{ij}\}_{i,j\in I}$. In addition, Kiumars Kaveh and Christopher Manon analyzed valuations into semifields of piecewise linear functions and explored their connections to families of toric degenerations. Inspired by these ideas in joint work in progress with Megumi Harada and Christopher Manon we propose a generalized notion of polytopes in $\Lambda=(\{M_i\}_{i\in I},\{\mu_{ij}\}_{i,j\in I})$, where the $M_i$ are lattices and the $\mu_{ij}:M_i\to M_j$ are piecewise linear bijections. Roughly, these are $\{P_i\mid P_i\subseteq M_i\otimes \mathbb{R}\}_{I\in I}$ such that $\mu_{ij}(P_i)=P_j$ for all $i,j$. In analogy with toric varieties these generalized polytopes can encode compactifications of affine varieties as well as some of their geometric properties. In this talk, we illustrate these ideas.
October 17: Ben Bakker (University of Illinois Chicago)
Title: Hwang's theorem on the base of a Lagrangian fibration revisited
Abstract: Irreducible hyperkahler manifolds are higher dimensional analogs of K3 surfaces; their geometry is tightly controlled by the existence of a nowhere degenerate holomorphic 2-form. The only nontrivial fibration structure f:X -> B a hyperkahler manifold X admits is a fibration by Lagrangian tori, and for such Lagrangian fibrations the base B is conjectured to always be isomorphic to projective space. In 2008 Hwang proved that this is the case if B is assumed to be smooth by using the theory of varieties of minimal rational tangents on Fano manifolds. In this talk I will present a simpler proof of this result which leans more heavily on Hodge theory. Specifically, the main input is a basic functoriality result coming from Hodge modules. This is joint work with C. Schnell.
October 24: Martin Bishop (Northwestern)
Title: The integral Chow ring of $\mathcal M_{1,n}$ for $n=3,\dots,10$
Abstract: Chow rings give a way to algebraically track the geometric structure of a space. We will explore the structure of $\mathcal M_{1,n}$, the moduli space of smooth $n$-pointed elliptic curves, and use a stratification of it to compute its integral Chow ring for $n=3,\dots,10$.
October 31: Donu Arapura (Purdue)
Title: Hodge theory of the moduli of vector bundles of degree 0.
Abstract: I will talk about some joint work with Dick Hain about the moduli space of rank n semistable bundles of degree 0 over a complex curve. As a topological space, this can be identified with the character variety of representations of the curve into GL(n) (or SL(n)), and cohomology of this space has a natural action by the mapping class group. When n=2, Cappell-Lee-Miller showed that the action of the Torelli subgroup on this is nontrivial. We found a partial generalization in higher rank. These results can be applied to study the mixed Hodge structure on the cohomology. In many cases, we can see that this is genuinely mixed (nonsplit).
November 7: Yunfeng Jiang (University of Kansas)
Title: The construction of virtual fundamental class on the moduli space of general type surfaces
Abstract: Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the moduli stack of lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class.
This can be taken as a generalization of the tautological invariants on the moduli space of stable curves to the moduli space of stable surfaces. If time permits, we also talk about the possible methods to construct a virtual fundamental class on the Alexeev moduli space of stable maps from semi-log-canonical surfaces to projective varieties.
November 14: Gwyneth Moreland (University of Illinois Chicago)
Title: Positive cycles on Hilbert schemes of points
Abstract: Algebraic geometers are often interested in certain “positive” cones that one can associate to varieties, such as the nef and effective cones. While traditionally studied in the divisor and curves cases, cones of cycles of intermediate dimension have recently been the subject of much study. We discuss some results on nef and effective cones of intermediate dimension cycles in the case of Hilbert schemes of points. In particular, we look at the Hilbert scheme of 3 points in P^3.
November 28: no seminar
December 5: Ming Hao Quek (Stanford)
Title: On the motivic monodromy conjecture for non-degenerate hypersurfaces
Abstract: I will discuss my ongoing effort to comprehend, from a geometric viewpoint, the motivic monodromy conjecture for any complex polynomial f that is non-degenerate with respect to its Newton polyhedron. This conjecture, due to Denef--Loeser, states that for every pole s of the motivic zeta function associated to f, exp(2πis) is a "monodromy eigenvalue" associated to f. On the other hand, the non-degeneracy condition on f ensures that the singularity theory of f is governed, up to a certain extent, by faces of the Newton polyhedron of f. The extent to which the former is governed by the latter is one key aspect of the conjecture, and will be the main focus of my talk.