|
April 9: Arkadij Bojko (Oxford)
SPECIAL TIME: 11:30am-12pm pretalk, 12pm-1pm main talk. Title: Wall-crossing for Hilbert schemes on fourfolds and Quot-schemes on surfaces Abstract: Virtual counting of coherent sheaves has seen recently a large development in complex dimension four, where it was defined for Calabi--Yau fourfolds by Borisov--Joyce and Oh--Thomas. I will focus on invariants for Hilbert schemes of points as they have not been well understood before. The only known result expressed integrals of top Chern classes of tautological vector bundles associated to smooth divisors in terms of the MacMahon function and Cao--Kool conjectured this holds for any line bundle. To address these questions I discuss the conjectural wall-crossing formulae of Joyce and discuss how to relate them to the conjectures on Hilbert schemes. On the other hand, Arbesfeld--Johnson--Lim--Oprea--Pandharipande studied Quot-schemes on surfaces and their virtual integrals giving explicit expressions for their generating series. Interestingly, these satisfy similar wall-crossing formulae as Hilbert schemes in the fourfold case when the curve class is zero. As a consequence their general invariants share a large similarity. Computing explicitly virtual fundamental classes and integrals on both, we can firstly recover the results in the five author paper from a small piece of data. Moreover, we obtain a universal transformation comparing integrals on Hilbert schemes on fourfolds and elliptic surfaces. |
|
April 16: Evgeny Shinder (University of Sheffield)
SPECIAL TIME: 11:30am-12pm pretalk, 12pm-1pm main talk. Title: Factorization centers, Cremona groups and the Grothendieck ring of varieties Abstract: I will state the question of uniqueness for centers of blow ups and blow downs of birational maps, explain what is currently known and give two applications. The first is to the structure of Cremona groups, namely their nongeneration by involutions in dimension >= 3. The second application is for the Grothendieck ring of varieties, of dimension <= 2, over perfect fields. Based on joint work with H.-Y. Lin, and with H.-Y. Lin and S. Zimmermann. |
|
April 23: Yajnaseni Dutta (Bonn)
SPECIAL TIME: 9-9:30am pretalk, 9:30-10:30am main talk. Title: Holomorphic 1-forms and geometry Abstract:In this talk I will discuss various topological and geometric consequences of the existence of zeros of global holomorphic 1-forms on smooth projective varieties. Such consequences have been indicated by a plethora of results. I will present some old and new results in this direction. One highlight of the topic is an interesting connection between two sets of such 1-forms, one that arises out of the generic vanishing theory and the other that falls out of Hodge theory of algebraic maps. This is joint work with Feng Hao and Yongqiang Liu. |
|
May 7: Jakub Witaszek (University of Michigan)
Title: Global +-regularity and the Minimal Model Program for arithmetic threefolds Abstract: In this talk, I will explain a mixed characteristic analogue of Frobenius regularity and how it can be used to establish the Minimal Model Program for threefolds in mixed characteristic. This is based on a joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron. |
|
May 14: Jiawang Nie (UCSD)
Title: Optimization, Positivstellensatz and Convex Algebraic Geometry Abstract: This talk is about optimizing polynomial functions under constraints. A general method is to apply the Moment-SOS hierarchy of semidefinite programming relaxations. The convergence is based on various Positivstellensatz. Closely related polynomial optimization is convex algebraic geometry. It concerns geometric properties of convex semialgebraic sets through semidefinite programming. We are going to review basic results for these topics. |
|
May 21: Olivier Martin (Stonybrook University)
Title: Effective zero-cycles and the Bloch-Beilinson filtration Abstract:Let X be a smooth projective variety whose algebra of holomorphic forms is generated in degree at most 2, for instance an abelian variety or a hyperKähler manifold. A recent conjecture of Voisin states that given points x and y on X the zero-cycle {x} is rationally equivalent to {y} if and only if {x}-{y} is in F^3CH_0(X), where F is the (conjectural) Bloch-Beilinson filtration. In other words, to ascertain that two points are rationally equivalent to each other one does not need to look very deep in the Bloch-Beilinson filtration. I will present a generalization that allows for higher degree generators of the algebra of holomorphic forms of X and for rational equivalence of effective zero-cycles of higher degree, both at the expense of looking deeper in the Bloch-Beilinson filtration. I will provide some evidence in support of this conjecture and present a related conjecture which predicts when the diagonal of a smooth projective variety X belongs to the subalgebra of CH^*(XxX) generated in degrees ≤ d. |
|
May 28: Rachel Webb (UC Berkeley)
Title: Abelianization and quantum lefschetz for orbifold I-functions Abstract: Let G be a connected reductive group with maximal torus T, and let V and E be two representations of G. Then E defines a vector bundle on the orbifold V//G; let X//G be the zero locus of a regular section. The quasimap I-function of X//G encodes the geometry of maps from P^1 to X//G and is related to Gromov-Witten invariants of X//G. By directly analyzing these maps from P^1, we explain how to relate the I-function of X//G to that of V//T. Our formulas prove a mirror symmetry conjecture of Oneto-Petracci that relates the quantum period of X//G to a certain Laurent polynomial defined by a Fano polytope. Finally, we describe a large class of examples to which our formulas apply, examples that are the orbifold analog of quiver flag varieties. Question for the audience: what else can one investigate with these examples? |
|
June 4:
Title: Abstract: |
|
June 11:
Title: Abstract: |
Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Fall 2017, Winter 2018, Spring 2018, Fall 2018, Winter 2019, Fall 2019, Winter 2020, Spring 2020, Fall 2020, Winter 2021.
The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Samir Canning at srcannin@ucsd.edu about problems with the website or posters.