Seminar talk announcements
BIMSA AG Seminar
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AG BIMSA
Birational and singularity invariants from nc Hodge theory
Speaker: Tony Pantev (University of Pennsylvania)
Time: 10:00-11:00 (GTM+8), 2024-10-10
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
I will explain how a natural amalgam of classical Hodge theory with the nc Hodge structures arising from Gromov-Witten theory gives rise to new additive invariants of smooth projective varieties called Hodge atoms. Combined with Iritani's blow-up formula, Hodge atoms provide obstructions to birational equivalence and novel invariants of singularities. I will discuss applications to classical rationality problems and singularity theory. This is a joint work with L.Katzarkov, M.Kontsevich, and T.Y.Yu.
BIMSA AG Seminar
Two new motivic complexes for non-smooth schemes
Speaker: Shane Kelly
Time: 10:00-11:00 (GMT +8), 2024-10-17
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
The Atiyah-Hirzebruch spectral sequence calculating topological K-theory from singular cohomology has been available since around 1960. The analogous spectral sequence for algebraic K-theory has been more elusive. According to conjectures of Beilinson and Lichtenbaum from the 80s, the E2-page of such a spectral sequence is made of motivic cohomology groups. These are generalisations of Chow groups and are intimately related to a large swath of the open conjectures in algebraic and arithmetic geometry.
For smooth schemes over a field, these were constructed by Bloch in the 80s under the guise of higher Chow groups and later by Voevodsky around 2000 inside a very general and robust theory of A1-homotopy theory. Atiyah-Hirzebruch style spectral sequences have been developed by Friedlander-Suslin and Levine.
Recently, two new generalisations of this motivic cohomology to non-smooth schemes have independently appeared, which turn out to be equivalent. One constructed by Elmanto-Morrow and the other by the speaker in joint work with Shuji Saito. We discuss these two new cohomologies and their comparison.
Algebraic versus homological equivalence of algebraic cycles
Speaker: Arnaud Beauville (Université Côte d'Azur)
Time: 15:00-16:00 (GMT +8), 2024-10-17
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
An algebraic cycle on a smooth projective variety is algebraically trivial if it can be deformed algebraically to zero. This implies that its cohomology class is zero; in 1969 Griffiths showed that the converse is false for many hypersurfaces. A different example is constructed from a curve C embedded in its Jacobian JC : the "Ceresa cycle" [C] - [(-1)*C] in JC is not algebraically trivial if C is general (Ceresa, 1983), while it is if C is hyperelliptic. In the last three years a number of approaches have been developed to find non-hyperelliptic curves for which this cycle is algebraically trivial.
In the talk I will survey the history of the problem, then discuss these recent examples of non-hyperelliptic curves, in particular the approach of Laga and Shnidman (2024).
BIMSA AG Seminar
Neutral representations of finite groups, arithmetic of quotient singularities, and fields of moduli
Speaker: Angelo Vistoli (Scuola Normale Superiore)
Time: 15:00-16:00 (GMT +8), 2024-10-24
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
Fix
a base field k, which for simplicity we will assume to be of
characteristic 0. Fix an algebraic variety X over the algebraic closure
k' of k, possibly with an additional structure, such as a polarization,
or a finite set of marked points; we will always assume that the
automorphism group of X over k' is finite.
Under very general
general conditions there exists a well defined moduli problem of twisted
form of X, giving rise to the so called residual gerbe G_X of X, which
is an étale gerbe over a subfield E of k' containing k, which is known
as the field of moduli of X. A natural question to ask is: when is X
defined over its field of moduli E? This is equivalent to the residual
gerbe G_X being neutral. We are searching for criteria that depend on
the geometry of X over k', and not on arithmetic information.
I am going to discuss two techniques that were introduced by Giulio Bresciani and myself to answer this question.
One, that works particularly well for varieties with a smooth marked point, is based on the concept of R-singularity.
The
other, which applies much more generally, that of neutral
representations. This gives criteria to show that X is defined on its
field of moduli, by studying the action of the automorphism group of X
on the intrinsically defined cohomology groups of X (for example, the
cohomology of the structure sheaf, or the cotangent sheaf).
BIMSA AG Seminar
A fibration formula for characteristic classes of constructible étale sheaves
Speaker: Yigeng Zhao (Westlake University)
Time: 15:00-16:00, 2024-10-31
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
For a constructible étale sheaf on a smooth variety, we first review the constructions of cohomological characteristic classes and non-acyclicity classes. We then show that these invariants satisfy a fibration formula, which can be viewed as a generalization of the classical Grothendieck-Ogg-Shafarevich formula. This is joint work with Enlin Yang.
BIMSA AG Seminar
Classifying spaces in motivic homotopy: from group cohomology to purity questions
Speaker: Matthias Wendt (Bergische Universität Wuppertal)
Time: 15:00-16:00 (GMT +8), 2024-11-07
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
In the talk I will explain what classifying spaces in motivic homotopy theory are, and how they can be used to better understand the topology of algebraic varieties, for example to construct algebraic varieties with certain cohomological properties. As a particular example, I will discuss purity/extension questions for quadratic forms on algebraic varieties. In this case, questions around the computation of (unramified) Witt groups can be reduced to group cohomology questions which are accessible to computer algebra tools. This is joint work in progress with Elden Elmanto.
BIMSA AG Seminar
Toric exoflops and derived categories
Speaker: Aimeric Malter (BIMSA)
Time: 15:00-16:00 (GMT+8), 2024-11-14
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
In this talk, I aim to introduce the notion of toric exoflops and highlight their use in the study of derived categories. We will discuss how to use them to obtain derived equivalences between complete intersection as well as categorical resolutions in sense of Kuznetsov for derived categories of singular complete intersections.
Speaker Intro:
My research lies in the study of derived categories and their implications in Algebraic geometry and Mirror symmetry.
BIMSA AG Seminar
Derived moduli spaces of Hermitian-Einstein connections
Speaker: Dennis Borisov (University of Windsor)
Time: 10:00-11:00 (GMT +8), 2024-11-21
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
I will show that Hermitian-Einstein connections can be described as representations of a Lie algebroid, which is defined using the holonomy Lie algebra of the underlying manifold. This naturally leads to the category of derived representations. The theorem by Donaldson, Uhlenbeck and Yau provides a correspondence between spaces of Hermitian-Einstein connections and moduli spaces of holomorphic vector bundles. The latter carry natural derived structures. I will try to relate the derived structures on both sides of the correspondence, in particular in the special case of a Calabi-Yau four-folds.
BIMSA AG Seminar
On Hodge Polynomials for Non-Algebraic Complex Manifolds
Speaker: Etnesto Lupercio (ICMAT)
Time: 10:00-11:00 (GMT +8), 2024-11-28
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
Hodge theory, with its pivotal role in connecting the geometry of varieties and their cohomology groups, offers profound insights into algebraic varieties' intricate structure. This work explores the extension of Hodge polynomials to a broader range of geometries, particularly non-Kähler complex manifolds.
We investigate the preservation of the intrinsic motivic nature of Hodge polynomials in the context of a large family of non-algebraic manifolds, such as Hopf, Calabi-Eckmann, and LVM manifolds. Our research establishes the preservation of the motivic properties of Hodge polynomials under these new settings, supported by explicit calculations. This work is a collaboration with Ludmil Katzarkov (University of Miami, IMSA and ICMS), Kyoung-Seog Lee (POSTECH), Laurent Meersseman (Université d’Angers), and others.
BIMSA AG Seminar
On the Quantum K-theory of Quiver Varieties at Roots of Unity
Speaker: Peter Koroteev
Time: 15:00-16:00 (GMT +8), 2024-12-05
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
In the framework of equivariant quantum K-theory of Nakajima quiver varieties we construct a q-analog of a Frobenius intertwiner between -equivariant quantum K-theory and the standard conventional quantum K-theory. We prove that this operator has no poles at primitive complex -th roots of unity in the curve counting parameter . As a byproduct, we show that the eigenvalues of the iterated product of quantum difference operators by quantum bundles of quiver variety
evaluated at roots of unity are governed by Bethe equations for with all variables substituted by their -th powers. In the cohomological limit, the above iterated product is conjectured to reduce to the p-curvature of the quantum connection for prime .
BIMSA AG Seminar
Non-linear PDEs and Secondary Calculus via Derived Algebraic Geometry
Speaker: Jacob Kryczka
Time: 15:00-16:00 (GMT +8), 2024-12-12
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
Secondary calculus, due to Alexander Vinogradov, is a formal replacement for the differential calculus on the typically infinite dimensional space of solutions to a non-linear partial differential equation.
On the one hand, one may associate a Variational Bicomplex to any PDE, which roughly speaking, plays the same role as the de Rham complex of an ordinary manifold or scheme. It is a central object of study in the geometric theory of PDEs, the theory of Integrable systems, Variational Calculus as well as Gauge Field Theory, and the objects of Secondary calculus are obtained from its cohomology groups, possibly with coefficients. By studying these groups, we may extract information about the original PDE, such as existence of symmetries, recursion operators, conservation laws etc.
In my talk I will discuss a vast generalization of this object to the derived and stacky setting using the language of (relative) homotopical algebraic geometry.
The refined object we obtain- the so called “Derived Variational Tricomplex” then provides a powerful tool for equipping moduli spaces of non-linear PDEs with shifted symplectic structures, as well as approaching the problem of formulating a global, hence non-perturbative, approach to the BV-BRST formalism.
Time permitting I will discuss applications to these topics.
This is based on a joint work with A. Sheshmani and S-T Yau and a work in progress.
BIMSA AG Seminar
Birational geometry and some applications to physics
Speaker: Caucher Birkar (YMSC & BIMSA)
Time: 15:00-16:00 (GMT +8), 2024-12-19
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
I will discuss some concepts and results in birational geometry with applications to physics.
Speaker Intro:
Professor Birkar, the 2018 Fields Medalist, works in algebraic geometry, in particular, birational geometry. His work involves various topics such as minimal models,Fano and Calabi-Yau and general type varieties, singularity theory, positive characteristic geometry, etc.
BIMSA AG Seminar
h-function of local rings of characteristic p
Speaker: Cheng Meng (YMSC)
Time: 15:00-16:00 (GMT +8), 2024-12-26
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
The local theory of singularities in characteristic p is an important part in commutative algebra. They are deeply connected to singularities in characteristic 0 like the KLT singularities and log canonical singularities. Singularities in characteristic p can be described using the Frobenius map and language of commutative algebra; we use various concepts (like F-regularity and tight closure) and numerical invariants (like Hilbert Kunz multiplicity and F-signature) to characterize them. In this talk, we will mention the relationship between singularities in characteristic 0 and characteristic p. We will then focus on the numerical invariants, especially the Hilbert-Kunz multiplicity. In particular, we will develop a new numerical invariant, called the h-function, which captures many other numerical invariants.
BIMSA AG Seminar
Zeta and L functions of Voevodsky’s motives
Speaker: Bruno Kahn (IMJ-PRG)
Time: 15:00-16:00 (GMT +8), 2025-01-16
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
Schemes of finite type over have a zeta function, and smooth projective varieties over have zeta functions attached to their cohomology groups, generalising the Hasse-Weil zeta function of an elliptic curve; they come with an Euler product factorisation. Both definitions go back to Serre. In the first case, the definition is elementary. In the second case, it uses -adic cohomology, and the local factors at places of bad reduction are in general not known to be independent of , which makes the definition partly conjectural. Il will explain how to attach unconditionally an -function to any (geometric) motive over in the sense of Voevodsky. This definition does not use cohomology, but instead the motivic six functors formalism initiated by Voevodsky and constructed by Ayoub. For the motive of a smooth projective variety, one recovers the correct local factors at places of good reduction, but these factors are in general different at the other places.
BIMSA AG Seminar
Higher genus Gromov-Witten correspondences for smooth log Calabi-Yau pairs
Speaker: Benjamin Zhou (YMSC)
Time: 15:00-16:00 (GMT +8), 2025-02-13
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
We prove higher genus correspondences between open, closed, and logarithmic Gromov-Witten invariants that can be defined from a smooth log Calabi-Yau pair consisting of a toric Fano surface with a smooth elliptic curve . Techniques such as the degeneration formula for logarithmic Gromov-Witten invariants, the Topological Vertex, and constructions from Gross-Siebert mirror symmetry are used. Time permitting, we also describe a link with -refined theta functions defined from and open mirror symmetry of an outer Aganagic-Vafa brane in . This is part of joint work with Tim Gräfnitz, Helge Ruddat, and Eric Zaslow.
BIMSA AG Seminar
Derived symmetries for crepant resolutions of hypersurfaces
Speaker: William Donovan (YMSC & BIMSA)
Time: 15:00-16:00 (GMT +8), 2025-02-20
Venue: A6-101 Online: Zoom 638 227 8222 (BIMSA)
Abstract:
Given a singularity with a crepant resolution, a symmetry of the derived category of coherent sheaves on the resolution may often be constructed using the formalism of spherical functors. I will introduce this, and new work (arXiv:2409.19555) on general constructions of such symmetries for hypersurface singularities. This builds on previous results with Segal, and is inspired by work of Bodzenta-Bondal.
Speaker Intro:
Will Donovan joined Yau MSC, Tsinghua U in 2018. Since 2021 he is an Associate Professor, and Adjunct Associate Professor at BIMSA. His focus is geometry, in particular applying ideas from physics and noncommutative algebra to study varieties, using tools of homological algebra and category theory. He studied at Cambridge U, completed his PhD at Imperial College London, and was postdoctoral researcher at Edinburgh U, UK. From 2014-18 he was research fellow at Kavli IPMU, U Tokyo, where he is now Visiting Associate Scientist. His work is published in journals including Communications in Mathematical Physics and Duke Mathematical Journal. He is supported by China Thousand Talents Plan, and received a Japan Society for Promotion of Science Young Scientist grant award.