TITLES AND ABSTRACTS

For a formal scheme, we relate the de Jong fundamental group of its rigid generic fiber with the pro-etale fundamental group of its special fiber. This is joint work with Marcin Lara and Alex Youcis.

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Using a geometric definition of logarithmic Hochschild homology of derived pre-log rings, we construct an André-Quillen type spectral sequence and show a logarithmic version of the Hochschild-Kostant-Rosenberg theorem. We use this to show that (log) Hochschild homology is representable in the category of log motives. Among the applications, we deduce a generalized residue sequence involving blow-ups of log schemes. This is a joint work with Tommy Lundemo, Doosung Park and Paul Arne Østvær.

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Hilbert’s Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry. In this talk, I will discuss a version of Hilbert’s Nullstellensatz in chromatic homotopy theory, where Lubin-Tate theories play the role of algebraically closed fields. Time permitting, I will then indicate some of the applications of the chromatic nullstellensatz including to redshift for the algebraic K-theory of commutative algebras. This is joint work with Tomer Schlank and Allen Yuan.

Homotopy at infinity is a classical invariant, famously related to the Poincaré conjecture and Whitehead's counterexample. The usefulness of an analogue in motivic homotopy theory was highlighted by Aravind Asok and Paul Arne Østvaer, with a roadmap proposed by Aravind in a 2008 talk. Recently, we have discovered how this project connects to a priori unrelated topics: links of singularities in geometry and interior cohomology appearing in the cohomological study of Shimura varieties. In this talk, I will explain a motivic homotopical construction obtained in collaboration with Adrien Dubouloz and Paul Arne Østvaer which ties in these three types of invariants. Our main result is an abstract Mumford's plumbing formula, an analogue of the construction appearing in the famous characterisation of isolated singularities of normal surfaces. Along the way, I will highlight nice properties such as analytical invariance, autoduality and the link with (unoriented) fundamental classes of the diagonal.

We will discuss the following topics:

1. The proof of the strict homotopy invariance theorem for framed sheaves over an arbitrary field.

2. Presentation of the result and the framed motives theory in an abstract form called above homotopy logical spaces or diagrams.

3. (joint with Urzabaev) The Gersten complex associated with an abstract cohomology theory that is defined for an arbitrary field and is exact on complements to normal crossing divisors in smooth local schemes, and further, the equivalence of the heart of DM(k) and Rost cycle modules with integral coefficients.

4. (joint with Kolderup and Ostvaer) We touch briefly on the base scheme generalisations.

We indicate above the parts that include joint results.

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In this talk I will present a series of geometric examples to illustrate the properties and the computation of stable punctured tubular neighbourhoods using appropriate specializations of the general abstract motivic plumbing formula presented in Frederic Deglise's talk. I will put a particular focus on punctured tubular neighbourhoods of trees of orientable rational curves on smooth surfaces. In this setting, the plumbing formula takes the form of a quadratic variant of Mumford-Ramanajuam computations of local topological fundamental groups of certain classes of isolated singularities and fundamental groups at infinity of families of smooth affine surfaces. Time permitting, I will give an illustration of the use of these quadratic invariants to distinguish certain discrete families of pairwise A1-weakly equivalent smooth affine surfaces appearing in the context of Zariski Cancellation Problem.

We calculate motivic Hochschild homology of the prime field $\mathbb F_p$ over an algebraically closed field of characteristic different from $p$. The result displays interesting $\tau$-divisibility and torsion with consequences for the theory of commutative motivic ring spectra. This is joint work with Mike Hill, Kyle Ormsby and Paul Arne Østvær.

Slides

We prove Voevodsky's conjecture that the zeroth slice of the sphere is HZ using prismatic cohomology. This is joint with Tom Bachmann and Matthew Morrow.

I will describe recent work with Aravind Asok and Tom Bachmann on the motivic analogue of the classical Freudenthal Suspension Theorem. Time permitting some intended applications will be indicated.

For semistable varieties, we construct log-motivic complexes Z_log(n) that agree with the motivic complexes of Suslin-Voevodsky on the smooth locus. Then we prove the deformational part of a p-adic variational Hodge-conjecture

for projective regular W(k)-schemes with semistable reduction: In analogy to a result of Bloch-Esnault-Kerz who treat the good

case we show that a rational log-motivic cohomology class in bidegree (2n,n) on the closed fibre lifts to a continuous proclass if and only if its Hyodo-Kato class lies in the n-th Hodge filtration under the Hyodo-Kato isomorphism. The case n=1 (logarithmic Picard group) was already treated by Yamashita. Along the way, we relate the n-th cohomology sheaf of Z_log(n) to a logarithmic Milnor K-sheaf and to the logarithmic Hyodo-Kato sheaf.

This is joint work with Oliver Gregory.

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We have modified Atiyah-Bott localization methods to apply to invariants in quadratic enumerative geometry. We will describe these results and say a bit about some of the recent applications.

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There are eight circles that are mutually tangent to a given trio of circles. As with most problems in enumerative geometry, this count does not always hold over fields that are not algebraically closed. Enriched enumerative geometry corrects this issue by giving a bilinear form-valued count. We will demonstrate at least two ways to geometrically interpret the bilinear form associated to a tangent circle. We will also discuss how these distinct geometric interpretations fit into a speculative phylogenetic tree of enumerative geometry.

This talk is mostly based on a joint work with Anand Sawant. In a previous work we observed that for a smooth variety X over a field k, one may define cellular A^1-homology H^{cell}_n(X) which are in general pro objects in the category of strictly A^1-invariant sheaves of abelian groups on (Sm_k)_{Nis}. We conjecture they should be indeed actual sheaves. For X admitting a cellular structure it is true. In general the sheaves H^{cell}_n(X) behave much better than the A^1-homology sheaves, for instance it is known they vanish if n is negative or bigger than dim(X) . They are also very much computable: the sheaves H^{cell}_n(P^r) for any projective space are entirely known, although the A^1-homology (or Suslin homology) shaves are still unkown. I will then address Poincare' duality for X a smooth A^1-connected projective k-scheme, and for instance one of its "consequence", stated as a conjecture: if X is of dimension d, then H^{cell}_d(X) \cong K^{MW}_d <=> X is orientable .

I will give (a lot of) examples where this is true and I will also address related problems: for such an X and U \subset X an open subset, distinct from X, is H^{cell}_d(U) = 0 ? Finally I will also consider Poincare' duality for non orientable smooth A^1-connected projective k-schemes X and and potential use to understand the "signature" of such X's.

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The study of hermitian K-theory has until recently been mostly confined to the case when 2 is invertible on the base ring. Building upon recent work, we will study the property of hermitian K-theory of schemes in characteristic 2 and mixed characteristic, comparing them to the already known cases. We will conclude with a construction of a motivic spectrum KQ representing a version of hermitian K-theory and explain which questions remain at the moment open. This is joint work with Baptiste Calmés and Yonatan Harpaz.

In this talk, I will explain the construction of the A1-local stable motivic homotopy category of log schemes. As an application, cohomology theories of schemes can be extended to log schemes. In particular, motivic cohomology, homotopy K-theory, and algebraic cobordism of log schemes can be defined. For any log smooth log scheme, cohomology of its boundary can be computed in terms of cohomology of schemes. This includes the case of log points.

Classically, tropical geometry can be used to solve problems in enumerative geometry.

In the talk, I will show that one can also use tropical methods in A1-enumerative geometry by presenting a new proof of Stephen McKean’s quadratically enriched Bézout theorem using tropical geometry.

This is joint work with Andrés Jaramillo Puentes.

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Adams showed in 1960 that the homotopy group $\pi_{2n-1}(S^n)$ contains an element of Hopf invariant one if and only if $n=2,4,8$. This result has interesting applications in geometry and algebra, e.g. the nonexistence of H-space structures on spheres. In this talk, I will discuss a motivic analogue of the Hopf invariant one problem, its solution over certain base fields, and consequences for representability of motivic spheres by schemes admitting unital products. This is joint work with William Balderrama and Dominic Leon Culver.

Lecture notes

The classical Atiyah-Bott localization theorem in equivariant singular cohomology for spaces with torus action is one of the main computational tools in enumerative geometry. The need to access general parameter spaces (singular and stacky) and the need for refined counts (in other cohomology theories) motivates the need for a more general localization theorem. In this talk, based on a recent joint work with Dhyan Aranha, Adeel Khan, Alexei Latyntsev and Hyeonjun Park, we will discuss such a unified Atiyah-Bott localization theorem for equivariant cohomology theories of algebraic stacks.

The celebrated and beautiful Weil conjectures connect the Betti numbers of a complex variety whose defining equations can be reduced mod p to the number of solutions mod p using zeta functions. We define a quadratic enrichment of these zeta functions. For cellular varieties, we show a rationality result and a connection to the Betti numbers of the real points. This is joint work with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.

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