Schedule for: 24w5258 - Enumerative Geometry Beyond Spaces

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

An old cliche in enumerative geometry is that "there are only two techniques in the subject: degeneration and localization". It's a bit hyperbolic, but there is a slice of truth to it. I'll start by sharing the picture that people have in mind when they stay this, and then tell the degenerate story. I will try to highlight the theoretical development over the last 20 years, some of the successes, unexpected difficulties, and a few lessons we have learned. Some of the characters we will meet along the way will include tropical geometry, orbifolds, and logarithmic structures.

"Mirror symmetry comes in two distinct flavors: enumerative (matching counts of curves on a Calabi-Yau manifold with invariants obtained from a variation of Hodge structures) and homological (stated as an equivalence between a Fukaya category and a family of derived categories). In 1994 Kontsevich predicted that the former incarnation of mirror symmetry should tautologically follow from the latter: there should exist enumerative invariants associated to a Calabi-Yau category, which specialize to curve counts for the Fukaya category, and to the invariants obtained from a variations of Hodge structures for the family of derived categories. Kontsevich's prediction has come closer to reality in recent years, following the introduction of Categorical Enumerative Invariants in works of Costello (2005) and Caldararu-Costello-Tu (2020). I will give a general introduction to the subject and discuss open problems and future directions of study. No previous knowledge of mirror symmetry, symplectic geometry, or Calabi-Yau geometry required."

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will explain how several moduli spaces of bundles on curves have abelian motives and how this can be harnessed to provide motivic lifts of known cohomological phenomena, such as chi-independence and mirror symmetry. This is joint work with Simon Pepin Lehalleur.

Using tropical geometry, Block-Göttsche defined polynomials with the remarkable property to interpolate between Gromov-Witten counts of complex curves and Welschinger counts of real curves in toric del Pezzo surfaces. I will describe a generalization of Block-Göttsche polynomials to arbitrary, non-necessarily toric, rational surfaces and propose a conjectural relation with refined Donaldson-Thomas invariants. This is joint work in progress with Hulya Arguz.

The Donaldson-Thomas invariants of a Calabi-Yau threefold X can be categorified to a perverse sheaf on the moduli stack of coherent sheaves on X. I will report on work in progress towards a new microlocal description of this perverse sheaf through a new theory of microsheaves on derived stacks. The microlocal perspective seems to clarify certain conjectural aspects of cohomological Donaldson-Thomas theory. This builds on a previous work, joint with Tasuki Kinjo, where we gave a microlocal description in the case of local surfaces.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk is an introduction to motivic homotopy theory, aimed at non-experts. We will discuss basic definitions and fundamental constructions in unstable and stable motivic homotopy theory, giving some intuition for motivic talks later this week.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

With Alexey Ananyevsky, following a question of Umberto Zannier’s, we consider the followng question: given a smooth quadric hypersurface over a field k (of characteristic not 2), when does the tangent bundle admit a nowhere zero section? We find an explict and (often) computable necessary condition, which is also sufficient in case the hypersurface admits a k-rational point. Even in the case of a k-rational point, we do not have an explicit expression for a nowhere vanishing section if the criterion is satisfied. However, in the first ``interesting’’ case, the algebraic 2-sphere x^2+y^2+z^2=1 over the field of 2-adic rationals, Peter Müller, motivated by our result, has found such an expression by a computer search. Our proofs use applications of motivic homotopy theory to the question of splitting vector bundles and a computation of the A^1-Euler characteristic of a smooth projective hypersurface, much in the spirit of the classical proof for the tangent bundle of an even dimensional sphere, together with some classical resutls on quadratic forms.

We will discuss a generalization of the BKK theorem for toric varieties over arbitrary fields, a correspondance theorem between the quadratically enriched count of rational curves and a combinatorial tropical counterpart, and a wall-crossing formula for such invariants.

We discuss two forms of Grothendieck-Riemann-Roch theorems for derived algebraic stacks. The first one compares the lisse extended G-theory with Chow groups and specializes to a higher equivariant Grothendieck-Riemann-Roch theorem. The second one compares G-theory of the stack and Chow group of the inertia stack in the case of derived Deligne-Mumford stacks. As an application, this gives a virtual Kawasaki-Riemann-Roch formula. This is based on joint works (partially in progress) with Adeel Khan.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Self-dual algebras are to finite algebras what SO is to SL group. We construct a *projective* moduli space for them (and some degenerate objects on the boundary) and discuss some of the enumerative aspects. This is joint work in progress with Andrea Ricolfi and Reinier Schmiermann.

A fundamental problem in algebraic topology is the study of homotopy groups of spheres. The projective line is a sphere in motivic homotopy theory, and its A^1-homotopy classes of endomorphisms is analogue to the fundamental group of the circle in classical topology. Morel computed this group using abstract methods. In this talk we will describe the group of A^1-homotopy classes of endomorphisms of the projective line by using basic algebraic geometry. This is joint work with Viktor Balch Barth, Gereon Quick, and Glen Matthew Wilson.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert's principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any smooth symmetric cubic surface, and discuss connections to monodromy computations for symmetric enumerative problems.

I will discuss the obstructions to the definition of Gromov-Witten type curve counting invariants over a general field, and how they can be dealt with in the case of rational curves on a del Pezzo surface of degree ≥3 over a perfect field of characteristic ≠2,3. This is joint work with Kass, Levine and Wickelgren.

The quadratic Euler characteristic of an algebraic variety is a (virtual) symmetric bilinear form which refines the topological Euler characteristic and contains interesting arithmetic information when the base field is not algebraically closed. For smooth projective varieties, it has a quite concrete expression in terms of the cup product and Serre duality for Hodge cohomology. However, for singular varieties, it is defined abstractly (using either cut and paste relations or motivic homotopy theory) and is still rather mysterious. I will first introduce this invariant and place it in the broader context of quadratic enumerative geometry. I will then explain some progress on concrete computations, first for certain symmetric powers (joint with Lenny Taelman) and second for conductor formulas for hypersurface singularities (older results with Marc Levine and Vasudevan Srinivas on the one hand, and joint work in progress with Ran Azouri, Niels Feld, Yonathan Harpaz and Tasos Moulinos on the other).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The concept of a *residual intersection* arises when an intersection is "not transverse enough" and then decomposes into a "big, bad" piece and a "small, good" piece. We think of the inclusion of the "small, good piece" as a generalization of a regular immersion. I will begin by explaining these concepts more fully, and justifying the intuition by exhibiting certain duality behavior. This follows work of Eisenbud-Ulrich. Afterwards I will explain how the duality theory can be used to reinterpret work of van Straten-Warmt on almost complete intersections as computing an Euler class via algebraic surgery. (Joint with Kirsten Wickelgren.)

I will discuss some constructions in quadratic forms of classical integer invariants such as a Chern-Schwarz-Macpherson class map (the speaker, Jin-Yang, Abe), invariants related to the Euler characteristic of nearby cycles (Levine-Pepin Lehalleur-Srinivas, the speaker), virtual fundamental class (Levine, Khan); I will describe some known results and coputations in comparison with works by Behrend, Behrend-Bryan-Szendroi, and propose questions on possible developments.

I will talk about joint work with Kirsten Wickelgren defining a global degree in stable equivariant homotopy theory and showing it is equal to a sum of local equivariant degrees. We define what it means for an equivariant map of G-manifolds to be oriented relative to an equivariant ring spectrum, and show our definition of the equivariant degree recovers the classical equivariant degree definition of G. Segal for an equivariant map between representation spheres. I will also talk about how we use the local degree to count orbits of rational cubics through a G-invariant set of 8 general points on a sufficiently nice surface.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Motivic Euler characteristic is an invariant whose roots are in motivic homotopy theory, but can be defined in a much more classical way. It plays an important role in the quadratic refinement of curve counting I will report on ongoing research about its birational invariance for Calabi-Yau varieties, and its computation via the theory of Galois invariants and unramified cohomology.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

Fulton's intersection theory provides a home for enumerative geometry for smooth varieties over a field. In 2010, Srinivas conjectured the existence of an extension of the Chow ring to possibly singular varieties, with the expected relationship to the Grothendieck group of vector bundles. In particular, Fulton's "ring of operators" is not such a theory and, as it turns out, neither is the Zariski/Nisnevich cohomology with coefficients in K-theory (ala Bloch-Ogus-Quillen). In joint work with Morrow, we solved Srinivas conjecture by suggesting that the (2n,n)-line of our new motivic cohomology satisfies all of Srinivas' desired properties. I will explain a history of this problem and explain how our motivic cohomology works.