Abstract: In characteristic zero, the motivic Hopf algebra associated to the Betti realisation is known to be connective by a direct computation relying on the Grothendieck comparison theorem between singular cohomology and algebraic de Rham cohomology. The analogous result in positive characteristic, say for the \ell-adic realisation, is not accessible by a similar method. Instead, we explain an indirect way of proving the connectivity in positive characteristic by reducing to the characteristic zero case. This relies in particular on a new Weil cohomology theory whose construction combines complex analytic geometry (through the Betti realisation) and rigid analytic geometry (through rigid analytic motives).

Abstract: In this talk I will explain how to construct (filtered) ℙ1-spectra in the stable log motivic homotopy category logSH(S) representing (log) prismatic and syntomic cohomology for S a quasi-syntomic scheme, and how to deduce from this some structural properties of such cohomology theories (like Gysin sequences and blow-up formulas). If time permits, I will explain how to use these results to establish a natural compatibility between the analogue of Voevodsky's slice filtration for logarithmic TC and variants and the BMS motivic filtrations. In the case of perfect fields admitting resolution of singularities, the motivic trace map is compatible with the slice and BMS filtrations, yielding a natural morphism from the motivic Atiyah-Hirzebruch spectral sequence to the BMS spectral sequence. This is a report on recent joint works with Merici-Lundemo-Park and Park-Østvaer.

We will revisit classical constructions and proofs of Deligne and Beilinson in order to construct generating families of motivic sheaves that are realized as perverse sheaves in any realization (this constructs a candidate for the perverse motivic t-structure). Over a field, this also constructs a uniform motivic cell homotology of any algebraic variety: a chain complex with values in Voevodsky's mixed motives that realizes in actual chain complexes through any realization, thus providing a uniform bound of Betti numbers of any algebraic variety (or any constructible mixed motive) over any field. We will interpret this construction through a uniform version of Nori motives and discuss the meaning of conjectural period isomorphisms in terms of tannakian properties.

Abstract: The concentration theorem (also called localization theorem) asserts that, upon inverting appropriate elements, the equivariant cohomology of a space endowed with a group action is concentrated on its fixed locus. I will present a purely geometric (as opposed to cohomological) form of this theorem for actions of linearly reductive groups on affine schemes, and discuss consequences for equivariant stable motivic homotopy theory.

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Abstract: In joint work with Toni Annala and Ryomei Iwasa, we generalize to non-𝔸1-invariant motivic spectra the theorem that the suspension spectrum of a smooth projective scheme is dualizable with dual given by the Thom spectrum of the negative tangent bundle. This allows us to adapt several classical results to non-𝔸1-invariant motivic spectra over arbitrary derived schemes, such as the computation of ℙ1-stable operations in algebraic K-theory after Naumann-Spitzweck-Østvær, or the idempotence of rational motivic cohomology after Cisinski-Déglise.

Abstract: This is a sequel to Hoyois’s talk and discusses further applications of the Atiyah duality established in the setting of non-𝔸1-invariant motivic spectra. 𝔸1-colocalization in the title refers to the right adjoint to the inclusion of 𝔸1-invariant motivic spectra. Using Atiyah duality, we see that the 𝔸1-colocalization of any module over the 𝔸1-invariant sphere does not change the value on smooth projective schemes. This gives a new elementary approach to logarithmic cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Joint work with Toni Annala and Marc Hoyois.

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Abstract: I will start with a brief recollection of Grothendieck-Witt theory of Poincaré categories and indicate how this setup allows to understand the Grothendieck-Witt theory of rings in which 2 need not be invertible, including a solution of Thomason’s homotopy limit problem for global Dedekind domains. I will then move on and explain how to define Karoubi-Grothendieck-Witt theory and present a formula for the value of a localizing invariant (of Poincaré categories) on Laurent-polynomials over a fixed Poincaré category, unifying the fundamental theorem in K-theory and what is known as the Shaneson-splitting in L-theory. From this we deduce a presentation of Karoubi-Grothendieck-Witt theory similar to the Bass delooping of K-theory, and of Karoubi L-theory showing that it coincides with the universally decorated L-theory of Ranicki, confirming a conjecture of Williams.

Abstract: In response to a question of Umberto Zannier, Alexei Ananyeskiy and I looked at the following question: given a smooth affine quadric hypersurface Q over a field k, how can one telll if Q admits a nowhere zero vector field? A particular case is the affine n-sphere over k, defined by the equation x_1^2+…+x_{n+1}^2=1, for which a classical result in differential topology tells us that, over the real numbers and for n>0 even, every vector field must vanish somewhere. This classical result is proven using the Gauss-Bonnet theorem, saying that the degree of the Euler class of the tangent bundle is the topoological Euler characteristic, which for an even dimensional sphere is 2, and the fact that the existence of a nowhere zero vector field implies that the Euler class of the tangent bundle is zero. Using results in motivic homotopy theory, we were able to make this line of argument work in the algebraic setting for a general smooth affine quadric hypersurface over a perfect field of characteristic not 2. We then translate this result into a simple criterion using facts about quadratic forms due to Knebusch, Pfister, Lam and Scharlau.

Abstract: The Segre type of a real line on a hypersurface of degree 2n-1 in ℙn+1 is a concept introduced by Finashin-Kharlamov. They extended Segre's earlier work on lines on real cubic surfaces, where he identified two types of real lines - elliptic and hyperbolic - on a cubic surface, independent of the choice of orientation. For any smooth cubic surface, the difference between the number of hyperbolic and elliptic lines is always 3. Kass-Wickgelgren generalized this result for fields of characteristic not equal to 2. In my talk I'll present a further generalization of this concept for arbitrary fields of characteristic not equal to 2, by defining the Segre type in GW(k) of a line on a degree 2n-1 hypersurface in ℙn+1 intrinsically, so that the number of lines, weighted by their types, remains independent of the chosen hypersurface.

Abstract: The equivariant concentration theorem states that given a compact Lie group G and a set S of elements in the equivariant cohomology of the point, the S-localized G-equivariant cohomology of X is isomorphic that of its `S-fixed subspace’. We prove a motivic analog of this statement for varieties with group actions. This is joint work with Adeel Khan.

Abstract: Let k be a perfect field of positive characteristic p and X a smooth proper k-scheme. The theory of cube invariant modulus sheaves with transfers developed by Kahn-Miyazaki-Saito-Yamazaki allows to define de Rham-Witt sheaves on X with modulus along an effective Cartier divisor D. If the support of D has simple normal crossings, then we show that these sheaves correspond under Grothendieck duality for coherent sheaves to de Rham-Witt sheaves with zeros along D. From this we deduce refined versions of Ekedahl - and Poincaré duality for crystalline cohomology generalizing results of Mokrane and Nakkajima for reduced D, and a modulus version of Milne-Kato duality for étale motivic cohomology with p-primary torsion coefficients, which refines a result of Jannsen-Saito-Zhao. We furthermore get new integral models for rigid cohomology with compact supports on the complement of D and a modulus version of Milne's perfect Brauer group pairing for smooth projective surfaces over finite fields. This is joint work with Fei Ren.

Abstract: We introduce a pro-cdh topology on formal schemes and prove that the ∞-topos of procdh sheaves of spaces has an optimal bound of homotopy dimension. This remedies a defect for a procdh topology on schemes. As an application, we give a topos-theoretic interpretation of Weibel's vanishing of negative K-theory and motivic cohomology of Elmanto and Morrow. This is a joint work with Shane Kelly.

Abstract: The ∞-category of noncommutative motives M_loc is defined as the universal finitary localizing invariant. 

In this talk we discuss a new alternative construction of M_loc as a localization of the ∞-category of stable ∞-categories, which in particular implies that the claim in the title is true. 

As a corollary we show that the theorem of the heart for nonconnective K-theory, conjectured by Antieau, Gepner, and Heller, is false. Using recent Efimov's results, we also prove that every ω_1-finitary invariant factors through M_loc. 

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Abstract: Gromov–Witten invariants and Welschinger invariants count curves over the complex and real numbers. In joint work with J. Kass, M. Levine, and J. Solomon, we used Morel's 𝔸1-degree to give arithmetically meaningful counts of rational curves on smooth del Pezzo surfaces over general fields. This talk concerns how these invariants change under an algebraic analogue of surgery along a Lagrangian sphere. We allow certain del Pezzo surfaces to acquire a -2 curve and give an arithmetic enrichment of a formula due to D. Abramovich and A. Bertram over ℂ and due to E. Brugallé and N. Puignau over ℝ. We use this formula to compute the enriched Gromov–Witten invariants for quadrics through points defined over the base or quadratic extensions. This is joint work with Erwan Brugallé.