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A classical theorem of Eilenberg--Moore, Dwyer, as well as Bousfield, gives conditions under which passage to homology commutes with certain totalizations arising from cosimplicial resolutions. In homotopy-theoretic terms, one can view this as a statement controlling when stabilization commutes with totalization.

In this talk I will discuss a generalized Eilenberg--Moore theorem and its application in motivic homotopy theory. Roughly, the infinite suspension functor from motivic spaces to SH commutes with the totalization of certain cosimplicial objects. The proof is based on an adaptation of the Eilenberg--Moore argument to a sufficiently general homotopical setting.

The main application concerns completion at the motivic Hopf map $\eta$. Using the generalized Eilenberg--Moore theorem, one can compare unstable $\eta$-completion of motivic spaces with $\eta$-completion after stabilization. This allows one to deduce the convergence of monadic resolutions with respect to motivic cohomology over formally real fields, leading to unstable Adams spectral sequences in this context.

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

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This talk introduces the two starting points of the workshop: deforming de Rham cohomology and assembling quantum invariants at roots of unity. Simple examples of the q-derivative and of evaluation in the Habiro ring illustrate the roles of q = 1 and roots of unity. We then outline how the arithmetic and topological perspectives meet in the study of Habiro cohomology.

Suggested references

Scholze, Canonical q-deformations in arithmetic geometry.

Wagner, q-Hodge complexes over the Habiro ring.

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I will review the notion of "BPS state" as understood by physicists. One particular type of BPS state is related to the theory of generalized Donaldson-Thomas invariants of CY3 categories, which (I believe) overlaps with the subject of Ben Davison's course this term; I will sketch this relation.

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Beginning with the q-de Rham complex of a polynomial algebra, we explain Scholze’s proposed canonical q-deformation and why independence of coordinates is a substantive problem. Specialization at q = 1 and at p-th roots of unity provides a first view of its cohomological meaning. We introduce the prismatic perspective and explain how it resolves coordinate independence in the p-complete setting, distinguishing this from the global theory.

Suggested references

Scholze, Canonical q-deformations in arithmetic geometry.

Bhatt–Scholze, Prisms and prismatic cohomology.

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Let A be the quantized Coulomb branch corresponding to a group G and its representation N. In order to compute the partition function of a 3d sphere in the corresponding TQFT, Gaiotto and Okazaki introduced a linear functional T on A, which we call "sphere trace". In the first part of the talk I will describe the image of A under the localization (abelianization) map under the mild technical assumption and use it to give a mathematical definition of the sphere trace for any quantized Coulomb branch.

The second part is based on joint work in progress with Merrick Cai, Vasily Krylov and Alan Yan. Let A be the truncated shifted Yangian that acts on a minuscule chamber module V. I will explain how to compute the q-character of V by using the sphere trace T.

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Starting from knots and their colored Jones polynomials, we explain how surgery on framed links leads to Witten–Reshetikhin–Turaev invariants of 3-manifolds. For integral homology 3-spheres, Habiro assembles the invariants at roots of unity into a single element of the Habiro ring. We introduce this cyclotomic completion and its basic properties, explaining how evaluation at roots of unity and Taylor expansion connect quantum invariants with arithmetic.

Suggested references

Habiro, Cyclotomic completions of polynomial rings.

Habiro, A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres.

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We explore the questions of Garoufalidis–Zagier and Garoufalidis–Scholze–Wheeler–Zagier through concrete q-series and expansions arising from quantum knot invariants and perturbative Chern–Simons theory. An example will illustrate how number fields enter the picture and why a naive extension of coefficients in the classical Habiro ring is insufficient. This leads to an introduction to Habiro rings of number fields and their gluing conditions, providing an arithmetic example to revisit in the final talk.

Suggested references

Garoufalidis–Zagier, Asymptotics of Nahm sums at roots of unity.

Garoufalidis–Scholze–Wheeler–Zagier, The Habiro ring of a number field.

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

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This talk introduces topological Hochschild homology and explains how filtrations on topological invariants give rise to arithmetic cohomology theories. We discuss the motivic filtration of Bhatt–Morrow–Scholze and the even filtration of Hahn–Raksit–Wilson, emphasizing their purpose and their relationship in the relevant settings. The aim is to prepare the objects and ideas needed for the final talk, rather than to develop the full machinery.

Suggested references

Bhatt–Morrow–Scholze, Topological Hochschild homology and integral p-adic Hodge theory.

Hahn–Raksit–Wilson, A motivic filtration on the topological cyclic homology of commutative ring spectra.

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A theorem of Borel says, in a suitable sense, that the rational K-theory of a number field lives in just one cohomological degree. A classical calculation of Bass and Tate suggests that adjoining a transcendental variable creates exactly one more.

This raises a deceptively simple question: for a finitely generated field, how many motivic cohomological degrees can there be?

I will explain how generic motives make the pattern particularly transparent, why the nonzero groups can nevertheless be enormous, and how the resulting open problem is related to the conjectural t-structure on mixed motives. At the end, curves and their Jacobians will suggest a broader version of the question.

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We first introduce Wagner’s construction of algebraic Habiro cohomology: a suitable q-Hodge filtration produces a q-Hodge complex that descends to the Habiro ring. Building on Talk 5, we then explain how THH over connective complex K-theory ku supplies such filtrations under appropriate hypotheses, and how equivariant structures give a homotopy-theoretic description of Habiro descent. The number-field case recovers the Habiro rings from Talk 4 and brings the arithmetic and topological perspectives together.

Suggested references

Wagner, q-Hodge complexes over the Habiro ring.

Wagner, q-de Rham cohomology and topological Hochschild homology over ku.

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

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This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Dwinelle Hall room 79.

In this course we will approach the theory of Yangians — quantum groups associated with loop algebras — via geometric representation theory. Key topics and constructions will include: Quantum loop groups, Nakajima quiver varieties, symplectic resolutions, RTT formalism, R matrices, BBDG decomposition theorem, cohomological Hall algebras and Borcherds algebras.

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