Mar 23, 2020
Monday

10:45 AM  11:00 AM


Welcome

 Location
 MSRI: Simons Auditorium
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11:00 AM  12:00 PM


Excision and algebraic Ktheory
Markus Land (University of Copenhagen)

 Location
 MSRI: Online/Virtual
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 Abstract
I will report on joint work with Tamme. I will recall that algebraic Ktheory does not satisfy excision and describe an effective way of studying the failure for excision in any localizing invariant. As applications we find a short proof of Suslin's excision theorem, that what we call truncating invariants satisfy excision, and new ways of calculating algebraic Kgroups of certain rings. Examples of truncating invariants include periodic cyclic homology over the rationals and the fibre of the cyclotomic trace. If time permits I will also indicate joint work with Tamme and Meier, where we study chromatic localizations of algebraic Ktheory. As applications of our approach we show that K(1)localized algebraic Ktheory satisfies excision (first proven by BhattClausenMathew) and that the K(n)localized Ktheory of any bounded above connective ring spectrum vanishes for n at least 2.
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12:00 PM  02:00 PM


Break

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02:00 PM  03:00 PM


Twisted topological Hochschild homology of equivariant spectra
Teena Gerhardt (Michigan State University)

 Location
 MSRI: Online/Virtual
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Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic Ktheory. Using norms in equivariant stable homotopy theory, one can define a twisted version of THH, which takes as input an equivariant ring spectrum. In this talk I will discuss categorical approaches to studying twisted THH, as well as computational tools. In particular, I will discuss the twisted THH of Thom spectra and the Real bordism spectrum. This talk includes joint work with Angeltveit, Blumberg, Hill, Lawson, and Mandell, as well as joint work with Adamyk, Hess, Klang, and Kong.
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03:00 PM  03:30 PM


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03:30 PM  04:30 PM


The Beilinson fiber square
Benjamin Antieau (University of Illinois, Chicago)

 Location
 MSRI: Online/Virtual
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If R is an associative ring satisfying some mild technical conditions, Beilinson constructs a fiber sequence of spectra identifying the fiber of the map \lim_n K(R/p^n)\rightarrow K(R/p) with a suspension of the ordinary cyclic homology of R, all up to pcompletion followed by inverting p. Joint work with Akhil Mathew, Matthew Morrow, and Thomas Nikolaus provides a new proof of this fact using recent advances in the theory of cyclotomic spectra due to Nikolaus and Scholze. I will explain the motivation for this fiber sequence, which has to do with the infinitesimal part of the padic variational Hodge conjecture, and then I will give a construction of the fiber sequence and of a more general fiber square.
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Mar 24, 2020
Tuesday

11:00 AM  12:00 PM


padic Ktheory and topological cyclic homology
Akhil Mathew (University of Chicago)

 Location
 MSRI: Online/Virtual
 Video

 Abstract
The cyclotomic trace from algebraic Ktheory to topological cyclic homology is an important computational tool because of the DundasGoodwillieMcCarthy theorem, which states that the trace induces an isomorphism of relative theories with respect to nilpotent ideals. After padic completion, this result can be strengthened to henselian pairs, generalizing also the GabberSuslin rigidity theorem in the ladic context. I will explain this generalization and some consequences. Joint with Dustin Clausen and Matthew Morrow.
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12:00 PM  02:00 PM


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02:00 PM  03:00 PM


Equivariant Atheory & stable hcobordism spaces
Mona Merling (University of Pennsylvania)

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 MSRI: Online/Virtual
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03:00 PM  03:30 PM


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03:30 PM  04:30 PM


Relative Geometric Langlands Duality  I
David BenZvi (University of Texas, Austin)

 Location
 MSRI: Online/Virtual
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 Abstract
I will present joint work with Yiannis Sakellaridis and Akshay Venkatesh in which we apply insights from physics (the understanding of electricmagnetic duality for boundary conditions in gauge theory) to a problem in number theory (the connection between period integrals and Lfunctions). The intermediary between the two is a duality between hamiltonian actions of Langlands dual groups, a ``relative" form of the geometric Langlands correspondence.
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Mar 25, 2020
Wednesday

09:30 AM  10:30 AM


How to zest your modular categories
Julia Plavnik (Indiana University)

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 MSRI: Online/Virtual
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10:30 AM  11:00 AM


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11:00 AM  12:00 PM


The motivic filtration on topological cyclic homology
Akhil Mathew (University of Chicago)

 Location
 MSRI: Online/Virtual
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Algebraic Ktheory and topological cyclic homology are invariants defined very generally (e.g., for stable \inftycategories). When applied to a commutative ring, they inherit an additional important (and slightly mysterious) structure: a motivic filtration. For topological cyclic homology, the motivic filtration has been constructed in the recent work of BhattMorrowScholze using a very direct approach; the graded pieces are a type of filtered Frobenius eigenspace on prismatic cohomology. I will describe the BMS construction and some new structural features of the motivic filtration on TC, in particular, an identification of the graded pieces, either in low weights or with denominators, with syntomic cohomology  a construction which relies only on more classical techniques, and which is easier to compute in practice. Joint work with Benjamin Antieau, Matthew Morrow, and Thomas Nikolaus.
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Mar 26, 2020
Thursday

09:30 AM  10:30 AM


Secondary algebraic Ktheory and traces
Aaron MazelGee (University of Southern California)

 Location
 MSRI: Online/Virtual
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Algebraic Ktheory is an invariant of schemes, which measures them through their vector bundles. As proved by BlumbergGepnerTabuada, when defined on stable ∞categories it may be characterized as "the universal additive invariant". This gives rise to trace maps to other invariants such as topological Hochschild and cyclic homologies (THH and TC), which may be seen as analogs of the Chern character. I will explain this perspective, as well as an interpretation of the cyclotomic trace map to TC in terms of derived algebraic geometry which arises naturally from viewing THH as factorization homology over the circle.
Secondary algebraic Ktheory is a categorification of algebraic Ktheory, which measures schemes through their sheaves of categories. It is inspired by elliptic cohomology and the chromatic redshift conjecture; it is closely related to the Brauer group and to the Ktheory of varieties. As proved in forthcoming work, when defined on stable (∞,2)categories it may be characterized as "the universal 2additive invariant". I will explain what this means, and speculate on the resulting 2dimensional trace maps to higherdimensional forms of factorization homology and TC (inspired by CarlssonDouglasDundas). In particular, I will discuss the notion of a stable (∞,2)category, and highlight a number of intriguing patterns that emerge as one moves up and down the categorical ladder.
This represents joint work with David Ayala, Nick Rozenblyum, and Reuben Stern.
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10:30 AM  11:00 AM


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11:00 AM  12:00 PM


Embedding 2categories into (\infty,2)categories
Martina Rovelli (Australian National University)

 Location
 MSRI: Online/Virtual
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 Abstract
In collaboration with Viktoriya Ozornova.
We will present how, by means of a suitable nerve construction, the established homotopy theory of strict 2categories is fully recovered in the model of (\infty,2)categories given by 2complicial sets.
In the first talk, we will review the basics of 2complicial sets, and we will introduce the nerve construction. We will then discuss some of its formal properties, and explain that it induces an embedding of the Lack's homotopy theory of 2categories into the homotopy theory of Verity's 2complicial sets.
In the second talk, we will give an explicit combinatorial description of the nerve which we can use to study many relevant homotopical properties. For instance, we will show that the nerve commutes up to equivalence with many relevant constructions, such as certain pushouts or suspensions.
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12:00 PM  03:30 PM


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03:30 PM  04:30 PM


Relative Geometric Langlands Duality  II
David BenZvi (University of Texas, Austin)

 Location
 MSRI: Online/Virtual
 Video

 Abstract
I will present joint work with Yiannis Sakellaridis and Akshay Venkatesh in which we apply insights from physics (the understanding of electricmagnetic duality for boundary conditions in gauge theory) to a problem in number theory (the connection between period integrals and Lfunctions). The intermediary between the two is a duality between hamiltonian actions of Langlands dual groups, a ``relative" form of the geometric Langlands correspondence.
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Mar 27, 2020
Friday

09:30 AM  10:30 AM


Embedding 2categories into (\infty,2)categories
Viktoriya Ozornova (RuhrUniversität Bochum)

 Location
 MSRI: Online/Virtual
 Video

 Abstract
In collaboration with Martina Rovelli.
We will present how, by means of a suitable nerve construction, the established homotopy theory of strict 2categories is fully recovered in the model of (\infty,2)categories given by 2complicial sets.
In the first talk, we will review the basics of 2complicial sets, and we will introduce the nerve construction. We will then discuss some of its formal properties, and explain that it induces an embedding of the Lack's homotopy theory of 2categories into the homotopy theory of Verity's 2complicial sets.
In the second talk, we will give an explicit combinatorial description of the nerve which we can use to study many relevant homotopical properties. For instance, we will show that the nerve commutes up to equivalence with many relevant constructions, such as certain pushouts or suspensions.
 Supplements



10:30 AM  11:00 AM


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11:00 AM  12:00 PM


The universal property of bispans
Rune Haugseng (Norwegian University of Science and Technology (NTNU))

 Location
 MSRI: Online/Virtual
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Commutative semirings can be described in terms of bispans of finite sets, meaning spans with an extra forward leg; if we instead take bispans in finite Gsets we get Tambara functors, which are the structure on $\pi_0$ of $G$equivariant commutative ring spectra. Motivated by applications of the $\infty$categorical upgrade of such descriptions to motivic and equivariant ring spectra, I will discuss the universal property of $(\infty,2)$categories of bispans. I will focus on the simplest case of bispans in finite sets, where this gives a new construction of the semiring structure on a symmetric monoidal $\infty$category whose tensor product commutes with coproducts. This is joint work with Elden Elmanto.
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12:00 PM  02:00 PM


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02:00 PM  03:00 PM


Microlocal sheaf categories and the Jhomomorphism
Xin Jin (Boston College)

 Location
 MSRI: Online/Virtual
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For an exact Lagrangian $L$ in a cotangent bundle, one can define a sheaf of stableinfinity categories on it, called the KashiwaraSchapira stack. Assuming the Lagrangian is smooth, the sheaf of categories is a local system with fiber equivalent to Mod(k), where k is the coefficient ring spectrum (at least E_2). I will show that the classifying map for the local system of categories factors through the stable Gauss map L>U/O and the delooping of the Jhomomorphism U/O>BPic(S), where S is the sphere spectrum. Part of the proof employs the (\infty,2)category of correspondences developed by GaitsgoryRozenblyum. If time permits, I will also talk about some applications of this result.
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03:00 PM  03:30 PM


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03:30 PM  04:30 PM


The paracyclic geometry of Fukaya categories
Hiro Tanaka (Texas State University)

 Location
 MSRI: Online/Virtual
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I'll talk about a new way to encode paracyclic structures (a combinatorial structure in Waldahausen's sdot construction). This new way is through sheaves on broken cycles. This stack also parametrizes families of certain symplectic manifolds, so this new way allows us to see why the data for constructing Fukaya categories for exact 2dimensional manifolds is equivalent to the data of a 2Segal object (in the sense of DyckerhoffKaparanov). A bonus is that we can prove theorems like: The K theory of the integers is equivalent to a Floer theory of Lagrangian cobordisms.
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