Motivic homotopy theory 2025 - Revision history

Hom54528 at 07:19, 24 March 2025

2025-03-24T07:19:58Z

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	In this talk, I will explain the quadratic Riemann-Roch theorem, in which the Chern character linking K-theory and rational Chow groups is replaced by the Borel character linking Hermitian K-theory (aka higher Grothendieck-Witt groups) with rational Chow-Witt groups. I will also explain how to compute the relevant Todd classes, in link with the formal ternary laws.		In this talk, I will explain the quadratic Riemann-Roch theorem, in which the Chern character linking K-theory and rational Chow groups is replaced by the Borel character linking Hermitian K-theory (aka higher Grothendieck-Witt groups) with rational Chow-Witt groups. I will also explain how to compute the relevant Todd classes, in link with the formal ternary laws.


			'''Shane Kelly: Procdh topologies'''

			We will discuss procdh topologies in the classical setting, and the formal schemes setting (à la EGA), in particular, focussing on applications to algebraic K-theory and making remarks on cohomological dimension. If there is time we may make some comments about the derived schemes setting. This is mostly joint work with Shuji Saito.

Hek43333 at 12:41, 19 March 2025

2025-03-19T12:41:22Z

← Older revision		Revision as of 13:41, 19 March 2025
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	<br>		<br>
	==Conference Picture==		==Conference Picture==
			<div class="res-img">[[File:Hoyois_conf.JPG\| center]]</div>
	<br>		<br>
	==Sponsors of the conference==		==Sponsors of the conference==
	This conference is funded by '''SFB 1085 "Higher Invariants"'''		This conference is funded by '''SFB 1085 "Higher Invariants"'''

Hek43333 at 12:35, 19 March 2025

2025-03-19T12:35:15Z

← Older revision		Revision as of 13:35, 19 March 2025
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	*Longke Tang (Princeton)		*Longke Tang (Princeton)
	*Alexander Vishik (Nottingham)		*Alexander Vishik (Nottingham)
	*~~Kirsten Wickelgren~~ (~~Duke~~)		*Vladimir Sosnilo (Regensburg)
	*Maria Yakerson (CNRS) <br>		*Maria Yakerson (CNRS) <br>

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	\| 14:00 - 15:00<br><b>Tess Bouis</b>		\| 14:00 - 15:00<br><b>Tess Bouis</b>
	\| Free afternoon		\| Free afternoon
	\| 14:00 - 15:00<br><b>~~Kirsten Wickelgren~~</b>		\| 14:00 - 15:00<br><b>Vladimir Sosnilo</b>
	\|		\|
	\|-		\|-

Hom54528 at 18:50, 17 March 2025

2025-03-17T18:50:27Z

← Older revision		Revision as of 19:50, 17 March 2025
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	\| 9:30 - 10:30<br><b>Alexey Ananyevskiy</b>		\| 9:30 - 10:30<br><b>Alexey Ananyevskiy</b>
	\| 9:30 - 10:30<br><b>Shane Kelly</b>		\| 9:30 - 10:30<br><b>Shane Kelly</b>
	\| 9:30 - 10:30<br><b>Jean Fasel</b>		\| 9:00 - 10:00<br><b>Jean Fasel</b>
	\| 9:00 - 10:00<br><b>Josefien Kuijper</b>		\| 9:30 - 10:30<br><b>Josefien Kuijper</b>
	\| 9:30 - 10:30<br><b>Tom Bachmann</b>		\| 9:30 - 10:30<br><b>Tom Bachmann</b>
	\|-		\|-

Hom54528 at 17:00, 17 March 2025

2025-03-17T17:00:03Z

← Older revision		Revision as of 18:00, 17 March 2025
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	\| 9:30 - 10:30<br><b>Alexey Ananyevskiy</b>		\| 9:30 - 10:30<br><b>Alexey Ananyevskiy</b>
	\| 9:30 - 10:30<br><b>Shane Kelly</b>		\| 9:30 - 10:30<br><b>Shane Kelly</b>
			\| 9:30 - 10:30<br><b>Jean Fasel</b>
	\| 9:00 - 10:00<br><b>Josefien Kuijper</b>		\| 9:00 - 10:00<br><b>Josefien Kuijper</b>
	~~\| 9:30 - 10:30<br><b>Jean Fasel</b>~~
	\| 9:30 - 10:30<br><b>Tom Bachmann</b>		\| 9:30 - 10:30<br><b>Tom Bachmann</b>
	\|-		\|-

Hom54528 at 15:09, 14 March 2025

2025-03-14T15:09:08Z

← Older revision		Revision as of 16:09, 14 March 2025
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	'''Elden Elmanto: Symmetric bilinear forms and mod-2 syntomic cohomology'''		'''Elden Elmanto: Symmetric bilinear forms and mod-2 syntomic cohomology'''

	The first successful deployment of motivic homotopy theory was Voevodsky and Orlov-Visihik-Voevodsky's resolution of Milnor's conjecture relating the graded pieces of the Witt ring, mod-2 Milnor K-theory and mod-2 étale cohomology for fields of odd characteristics. Earlier, Kato had discovered the same story for characteristic two fields, with logarithmic de Rham forms in place of étale cohomology. I will report on joint work in progress with Arnab Kundu on this story with mixed characteristics, crucially in the (2,0) case. I will explain the relationship between the graded pieces of a filtration on symmetric L-theory (in the sense of the 9 authors) of a 2-henselian local ring, with mod-2 syntomic cohomology (in the sense of Bhatt-Morrow-Scholze). Time permitting, I will explain how this relates to an extension of Milnor-Witt motivic cohomology to qcqs schemes following work of myself with Morrow and Bouis.		The first successful deployment of motivic homotopy theory was Voevodsky and Orlov-Visihik-Voevodsky's resolution of Milnor's conjecture relating the graded pieces of the Witt ring, mod-2 Milnor K-theory and mod-2 étale cohomology for fields of odd characteristics. Earlier, Kato had discovered the same story for characteristic two fields, with logarithmic de Rham forms in place of étale cohomology. I will report on joint work in progress with Arnab Kundu on this story with mixed characteristics, crucially in the (2,0) case. I will explain the relationship between the graded pieces of a filtration on symmetric L-theory (in the sense of the 9 authors) of a 2-henselian local ring, with mod-2 syntomic cohomology (in the sense of Bhatt-Morrow-Scholze). Time permitting, I will explain how this relates to an extension of Milnor-Witt motivic cohomology to qcqs schemes following work of myself with Morrow and Bouis.


			'''Jean Fasel: The quadratic Riemann-Roch theorem'''

			In this talk, I will explain the quadratic Riemann-Roch theorem, in which the Chern character linking K-theory and rational Chow groups is replaced by the Borel character linking Hermitian K-theory (aka higher Grothendieck-Witt groups) with rational Chow-Witt groups. I will also explain how to compute the relevant Todd classes, in link with the formal ternary laws.

Hom54528 at 08:12, 11 March 2025

2025-03-11T08:12:33Z

← Older revision		Revision as of 09:12, 11 March 2025
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	To compute such counts, one can translate the problem into tropical geometry, where it reduces to a weighted count of certain graphs with point conditions. This significantly simplifies the computation. In this talk, I will present a tropical correspondence theorem for quadratic plane curve counts.		To compute such counts, one can translate the problem into tropical geometry, where it reduces to a weighted count of certain graphs with point conditions. This significantly simplifies the computation. In this talk, I will present a tropical correspondence theorem for quadratic plane curve counts.


			'''Oliver Röndigs: Low integral Milnor-Witt stems over the integers'''

			Work of Calmes, Harpaz, and Nardin, based on their collaboration
			with Dotto, Hebestreit, Land, Moi, Nikolaus, and Steimle, provides a reasonable
			motivic ring spectrum KQ representing hermitian K-theory in the motivic stable
			homotopy category of S. Here S can be any regular Noetherian base scheme
			of finite Krull dimension; the inconvenient restriction that 2 be invertible on S is not
			required anymore. Recent joint work with K. Arun Kumar shows that this motivic
			spectrum coincides with the one constructed in Kumar's thesis, and in particular
			is cellular. One consequence, derived in joint work with Håkon Kolderup and Paul
			Arne Østvær, is a determination of various filtrations on hermitian K-theory over
			Dedekind rings. As another consequence, effectivity and connectivity properties
			of the unit map from the motivic sphere spectrum to KQ induce information on
			motivic stable homotopy groups of spheres over the integers.

Hom54528 at 12:12, 6 March 2025

2025-03-06T12:12:10Z

← Older revision		Revision as of 13:12, 6 March 2025
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	To compute such counts, one can translate the problem into tropical geometry, where it reduces to a weighted count of certain graphs with point conditions. This significantly simplifies the computation. In this talk, I will present a tropical correspondence theorem for quadratic plane curve counts.		To compute such counts, one can translate the problem into tropical geometry, where it reduces to a weighted count of certain graphs with point conditions. This significantly simplifies the computation. In this talk, I will present a tropical correspondence theorem for quadratic plane curve counts.


			'''Fabio Tanania: Real isotropic cellular spectra'''

			In this talk, I will introduce the category of isotropic motivic spectra over the real numbers. In brief, this is obtained from SH(R) by annihilating anisotropic quadrics. I will then discuss in more detail the structure of the subcategory of isotropic cellular spectra. This can be described as a one-parameter deformation, whose generic fiber is the classical stable homotopy category, while the special fiber is the derived category of comodules over the topological dual Steenrod algebra. This reveals a striking similarity between real isotropic cellular spectra and F2-synthetic spectra.

Hom54528 at 12:09, 6 March 2025

2025-03-06T12:09:44Z

← Older revision		Revision as of 13:09, 6 March 2025
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	<br>		<br>

			'''Alexey Ananyevskiy: Nowhere vanishing sections of vector bundles revisited'''

			Consider a vector bundle over a smooth affine variety and suppose that the rank of the bundle equals the dimension of the variety. It is a classical result of Murthy that if the base field is algebraically closed, then triviality of the top Chern class in Chow group yields the existence of a nowhere vanishing section. Morel generalized this to arbitrary fields, enhancing top Chern classes to Euler classes valued in Chow-Witt groups and showing that these are universal obstruction classes. Asok and Fasel further developed this approach, in particular, they showed that Murthy's result holds also over the fields of 2-cohomological dimension at most one. In a complementary direction, Bhatwadekar and Sridharan, and Bhatwadekar, Das and Mandal using some delicate commutative algebra showed that Murthy's result holds over the field of real numbers provided that either the rank of the bundle is odd, or the manifold of real points X(R) satisfies some additional assumption. In the talk I will discuss how one can generalize these results to fields of virtual 2-cohomological dimension at most one.


	'''Toni Annala: Atiyah Duality and Logarithmic Cohomology Theories'''		'''Toni Annala: Atiyah Duality and Logarithmic Cohomology Theories'''

Hom54528 at 08:04, 5 March 2025

2025-03-05T08:04:42Z

← Older revision		Revision as of 09:04, 5 March 2025
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	The difference in complexity between algebraic geometry and topology is reflected in the structure of the Balmer spectra of the respective “motivic” categories. While the Balmer spectrum of the topological version is just the Zariski spectrum of integers, its algebro-geometric counterpart is much wilder. The vast majority of known points there are given by isotropic realisations. These are parametrised by a choice of a prime p and a p-equivalence class of field extensions of the base field. Points of characteristic 2 are understood better, since, in this case, it is known that the isotropy of algebraic varieties is controlled by pure symbols over flexible closure (so, every variety is a “norm-variety”, in a sense). This permits to impose certain “coordinates” on the Balmer spectrum coming from pure symbols. In particular, it shows that isotropic points are closed.		The difference in complexity between algebraic geometry and topology is reflected in the structure of the Balmer spectra of the respective “motivic” categories. While the Balmer spectrum of the topological version is just the Zariski spectrum of integers, its algebro-geometric counterpart is much wilder. The vast majority of known points there are given by isotropic realisations. These are parametrised by a choice of a prime p and a p-equivalence class of field extensions of the base field. Points of characteristic 2 are understood better, since, in this case, it is known that the isotropy of algebraic varieties is controlled by pure symbols over flexible closure (so, every variety is a “norm-variety”, in a sense). This permits to impose certain “coordinates” on the Balmer spectrum coming from pure symbols. In particular, it shows that isotropic points are closed.


			'''Kirsten Wickelgren: Equivariantly enriched Gromov–Witten invariants'''

			Abstract: We compute the equivariant Euler characteristic as an Euler number and give several related results on degrees in equivariant homotopy theory. As an application, we consider certain enriched Gromov–Witten invariants. For example, we compute an equivariantly enriched count of rational plane cubics through a G-invariant set of 8 general points in CP^2. When Z/2 acts by pointwise complex conjugation, this recovers a signed count of real rational cubics. This is joint work with Candace Bethea.