Connections Between Algebraic Geometry and Algebraic Topology

Invited Speakers

• Aravind Asok
• Ben Bakker
• Johan de Jong
• Sanath Devalapurkar
• Jordan Ellenberg
• Michael Groechenig
• Danny Krashen
• Eduard Looijenga
• Alexander Petrov
• Ravi Vakil
• Kirsten Wickelgren
• Inna Zakharevich

Schedule

Note: All talks on Wednesday and Thursday will be held in ENG 120.

Monday, July 28 ( NATRS 113)

• 1:30–2:20: Ben Bakker — Large local systems in algebraic, analytic, and arithmetic geometry
  [Show abstract]

  A local system V on a complex algebraic variety X is said to be large if its restriction to any positive-dimensional subvariety has infinite monodromy. The prototypical example of a large local system is a local system underlying a variation of Hodge structure whose period map is immersive, and varieties admitting large local systems enjoy many rigidity properties in various categories. I will begin this talk by surveying these properties. I will then describe some recent joint work with Y. Brunebarbe and J. Tsimerman showing that in fact any complex local system on a complex algebraic variety comes from a large local system. Finally, I will describe joint work with A. Shankar and J. Tsimerman showing how related ideas can be used to build integral canonical models of images of geometric period maps and exceptional Shimura varieties.

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• 2:50–3:40: Aravind Asok — Motivic homotopy theory and algebraic vector bundles
  [Show abstract]

  I will survey progress on various problems about algebraic vector bundles, some of which involve tools of motivic homotopy theory.

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• 4:10–5:00: Michael Groechenig — On p-adic integration for linear moduli problems: the symplectic case
  [Show abstract]

  This online talk will be devoted to ongoing work in progress with Wyss & Ziegler, which concerns the relation of p-adic integrals on moduli spaces of linear nature and Donaldson–Thomas theory. The talk will commence with a review of p-adic integration, including our past work, which can be applied to certain (singular) moduli spaces of meromorphic Higgs bundles and representations of a symmetric quiver.
  
  Subsequently, I will describe work in progress, which aims to generalize our previous results to the symplectic (or Hyperkähler) case, including Nakajima quiver varieties and moduli spaces of sheaves on a K3 surface.

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Tuesday, July 29 ( Lory Student Center Ballroom)

• 1:30–2:20: Jordan Ellenberg — Homological stability and arithmetic statistics, 20 years later
  [Show abstract]

  At the third week of the 2005 SRI in Seattle, I gave a talk called Asymptotics and upper bounds for rational points on algebraic varieties, in which I speculated about some interesting similarities between homological stability conjectures in algebraic topology and conjectures in the then-unnamed field of “arithmetic statistics,” like the Cohen–Lenstra conjectures on the distribution of class groups and the Batyrev–Manin conjectures about counting points of bounded height on varieties.
  
  The fundamental idea is that a question in arithmetic statistics over ℚ can be analogized to a question in arithmetic statistics over a function field over a finite field 𝔽_q, which can be rephrased as a question about counting points (or, more generally, summing Frobenius eigenvalues on stalks) on a moduli space over 𝔽_q, which finally via the Grothendieck–Lefschetz theorem can be related to the topology of some natural lift of that moduli space to the complex numbers — and at that point we suddenly find ourselves doing topology.
  
  In the last twenty years, this connection has developed into a rich vein of theorems and new ideas; I will try to survey the main developments, including some very recent ones.

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• 2:50–3:40: Danny Krashen — Complexity of algebraic objects over fields and the period-index problem
  [Show abstract]

  Many algebraic objects of interest (central simple algebras, quadratic forms, torsors for linear algebraic groups) can be studied through the lens of Galois cohomology, and in this way, the structure of the Galois cohomology ring of a field has a strong influence on the potential arithmetic complexity of algebraic objects.
  
  The central problem of the talk will be to understand how easy or hard it is to construct explicit representatives of Galois cohomology classes from various perspectives. These include finding versal elements (universal families that specialize to all elements in a cohomology class), determining symbol length (the minimum number of symbols needed to express an element in Milnor K-theory), and investigating the period-index problem (understanding the relationship between the period and index of a cohomology class).
  
  After exploring the connections between these different approaches to measuring arithmetic complexity, we will describe the emerging landscape of new results on the period-index problem, both in the case of the Brauer group (degree 2 cohomology) as well as for higher degree classes.

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• 4:10-5:00: Ravi Vakil — Naive homotopical techniques in algebraic geometry
  [Show abstract]

  I’ll present a framework for making simple but effective homotopical arguments in algebraic geometry. These methods apply only in certain fortunate situations, but when they do, they yield surprising (to me) facts about basic geometric objects. My goal is to share clear, concrete examples—proofs you can follow and take home. This is based on work with Hannah Larson, Jim Bryan, and Ben Church, and connects to ideas developed by many others.

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Wednesday, July 30 ( ENG 120)

• 1:30–2:20: Inna Zakharevich — Towards a trace on the Grothendieck spectrum of varieties
  [Show abstract]

  In this talk we discuss an alternate model of topological Hochschild homology. This model has the advantage that it is largely combinatorial, and thus exists independently of a choice of enrichment. The hope is that this model can lead to a meaningful trace and model of THH for the Grothendieck spectrum of varieties. This is joint work with Jonathan Campbell and Kate Ponto.

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• 2:50–3:40: Sanath Devalapurkar — Calculus and cohomology
  [Show abstract]

  I will explain a simple reformulation of the Hochschild–Kostant–Rosenberg theorem, describing the algebra of (crystalline) differential operators on a smooth scheme over a commutative ring k in terms of (higher) Hochschild cohomology.
  
  When k is replaced by a commutative ring spectrum, essentially the same construction gives a way of assigning to each 1-dimensional formal group H an algebra of “H-differential operators” on affine space. When H is the additive (resp. multiplicative) formal group, this recovers usual differential operators (resp. q-difference operators).
  
  Many aspects of ordinary (and q-) calculus generalize to H-differential operators (and admit nice homotopy-theoretic explanations), like the Gauss–Manin connection, Frobenius-constant quantizations via p-curvature, Lagrangian p-support, Mellin + Fourier transforms, “H-analogues” of various special functions like hypergeometric functions, etc.
  
  This story has two (mirror dual!) motivations: relative Langlands duality + quantum generalized cohomology; and the theory of prismatic cohomology. If there is time, I will discuss applications to both.
  
  Part of this picture is based on joint work with various subsets of Jeremy Hahn, Max Misterka, Arpon Raksit, and Allen Yuan.

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• 4:10–5:00: Eduard Looijenga — Ball quotients and finite simple groups
  [Show abstract]

  Many ball quotients that are defined in terms of the Eisenstein ring of integers have a modular interpretation. Such ball quotients admit what we call a reflection cover: a cover universal for the property that we allow order 2 ramification over the discriminant divisor. Assuming the discriminant divisor is nonempty, the Galois group of such a cover is generated by involutions and, surprisingly, tends to be finite.
  
  For the 13-dimensional ball quotient found by Allcock, this Galois group is conjectured (by Allcock) to be the Bimonster studied by Conway et al. This conjecture leads to many more potential examples involving finite simple groups (some of them sporadic).
  
  For the moduli space of stable cubic threefolds, this is given by imposing a level 2 structure on a cubic threefold, and this interpretation led to finding new algebro-geometric properties of that moduli space.
  
  The talk will survey some recent developments.

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Thursday, July 31 ( ENG 120)

• 1:30–2:20: Johan de Jong — Integrality of the Betti Moduli Space
  [Show abstract]

  This is a report on joint work with Hélène Esnault. Let X be a smooth projective variety over the complex numbers ℂ. Let M be the moduli space of irreducible representations of the topological fundamental group of X of a fixed rank r. Then M is a finite type scheme over the spectrum of the integers ℤ. We may ask whether M is pure over ℤ in the sense of Raynaud–Gruson; for example, we can ask if the irreducible components of M which dominate Spec(ℤ) actually surject onto Spec(ℤ).
  
  We will explain what this means, present a weak answer to this question, apply this to exclude some abstract groups as the fundamental groups of smooth projective varieties over ℂ, and we discuss what other phenomena can be studied using the method of proof.

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• 2:50–3:40: Kirsten Wickelgren — Quadratic Gromov–Witten invariants of rational del Pezzo surfaces of degree >5
  [Show abstract]

  Quadratic Gromov–Witten invariants allow one to obtain an arithmetically meaningful count of curves satisfying constraints over a field k without assuming that k is the field of complex or real numbers. They were developed in joint work with Kass, Levine, and Solomon in genus 0 for del Pezzo surfaces.
  
  In this talk we will compute these invariants for rational del Pezzo surfaces of degree >5. To do this, we give these invariants the structure of an unramified Witt invariant for any fixed surface and degree. We then construct a multivariable unramified Witt invariant which conjecturally contains all of these invariants for k-rational surfaces. We prove this conjecture in degree >5.
  
  This is joint work with Erwan Brugallé and Johannes Rau.

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• 4:10–5:00: Alexander Petrov — Characteristic classes of étale local systems
  [Show abstract]

  Given a local system of vector spaces over a field F on a topological space X, a cohomology class of the group GL_n(F) naturally gives rise to a cohomology class of X that can be thought of as a characteristic class of the local system.
  
  For F = ℚ_p for some prime p (and starting with a continuous group cohomology class), this is moreover applicable to ℚ_p-étale local systems on algebraic varieties. The resulting theory of characteristic classes turns out to be most interesting when the variety in question is defined over a non-closed field.
  
  I will overview what is currently known about these characteristic classes, mainly focusing on:

  • (1) Computation of characteristic classes for crystalline local systems on varieties over p-adic fields via the Chern classes of vector bundles on the Fargues–Fontaine curve;
  • (2) Canonical extensions of these classes: for a local system on a smooth variety X over a finitely generated field, characteristic classes with rational coefficients lift naturally to cohomology of any good compactification of X.

  All of the results in the talk are based on joint works with Lue Pan and George Pappas.

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Organizers

• Daniel Litt
• Kirsten Wickelgren
• Akhil Mathew

Additional Information

For travel, accommodation, and registration details, please visit the main AMS Summer Research Institute website.