Mar 25, 2019
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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09:30 AM  10:30 AM


Characters of categorical representations: theory and applications
Constantin Teleman (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
I will review the theory of topological group actions on complexlinear categories and their algebraic calculus of characters, with (old and new) applications to GromovWitten theory and dualities in mathematical physics.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


TBA
Bhargav Bhatt (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


Dualizing spheres for padic Lie groups, with applications to chromatic homotopy theory
Vesna Stojanoska (University of Illinois at UrbanaChampaign)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
I will describe a Linearization Conjecture that identifies the spectral dualizing module of a padic Lie group in terms of a representation sphere built from the Lie algebra. We can prove this when the action is restricted to certain small finite subgroups. These results are enough to determine SpanierWhitehead duals of some chromatically interesting spectra. This is joint work in progress with Beaudry, Goerss, and Hopkins.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Spectral Lie algebras and unstable homotopy theory
Gijs Heuts (Universiteit Utrecht)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
In this talk I will review spectral Lie algebras, an adaptation of the theory of Lie algebras to the category of spectra. These generalize differential graded Lie algebras over the rational numbers and admit similar applications in derived algebraic geometry and homotopy theory. As an example of this, I will discuss their role as algebraic models for spaces, generalizing Quillen's rational homotopy theory.
 Supplements




Mar 26, 2019
Tuesday

09:30 AM  10:30 AM


Affine BeilinsonBernstein at the critical level for GL_2
Sam Raskin (University of Texas, Austin)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
There has long been interest in BeilinsonBernstein localization for the affine Grassmannian (or affine flag variety). First, KashiwaraTanisaki treated the socalled negative level case in the 90's. Some ten years later, FrenkelGaitsgory (following work of FeiginFrenkel and BeilinsonDrinfeld) formulated a conjecture at the critical level and made some progress on it. Their conjecture is more subtle than its negative level counterpart, but also more satisfying. We will review the necessary background from representation theory of KacMoody algebras at critical level, formulate the FrenkelGaitsgory conjecture, and outline the proof for GL_2.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Moduli of flat connections on smooth varieties
Tony Pantev (University of Pennsylvania)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
This is a report on a joint work with Bertrand To\""{e}n. We study the derived geometry of the moduli of local systems and flat bundles on a smooth but not necessarily proper complex algebraic variety $X$. In the Betti case we show that these moduli carry natural Poisson structures, generalizing the well known case of curves. We also construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity, and show that a new feature, called strictness, appears as soon as the divisor at infinity has nontrivial crossings. In the de Rham case we introduce the notion of a formal boundary of $X$, and explain how to define a restriction to the boundary map $R$ between derived moduli of flat bundles. I will discuss representability results for the geometric fibers of $R$ and will explain why the morphism $R$ comes equipped with a canonical shifted Lagrangian structure.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Singular support for categories over a scheme
Dmytro Arinkin (University of WisconsinMadison)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
Some geometric objects can be studied `microlocally': instead of just looking at their support (the set of points where an object is nontrivial), one can consider their `singular support', which remembers the `direction' of nontrivial behavior. Examples include the wave front of a distribution, the singular support of a constructible sheaf, and the characteristic variety of a Dmodule. Another important example of such `microlocal' theory is singular support of (ind)coherent sheaves, which plays an important role in the global geometric Langlands program. In my talk, I will present a higher categorical analogue of this: the theory of singular support for categories over a scheme, which is important for the local Langlands program.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


An arithmetic enrichment of the degree of a finite map, and applications to enumerative geometry
Kirsten Wickelgren (Georgia Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
Using the EisenbudKhimshiashviliLevine local degree, which is the A1local degree of Morel in A1 homotopy theory, we define a degree of a finite map between smooth schemes over k. When the target is appropriately connected, this degree is a bilinear form over k. We discuss some applications to enumerative geometry over nonalgebraically closed fields. This is joint work with Jesse Kass and Jake Solomon, and will also contain joint work with Padmavathi Srinivasan.
 Supplements



04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
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Mar 27, 2019
Wednesday

09:30 AM  10:30 AM


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

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
I will discuss some recent results on the padic algebraic Ktheory of padic rings, obtained using the cyclotomic trace from Ktheory to topological cyclic homology. Parts of this are joint with Bhargav Bhatt, Dustin Clausen, and Matthew Morrow.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


On the Ktheory of pullbacks
Georg Tamme (Universität Regensburg)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
I will report on joint work with Markus Land. To any pullback diagram of ring spectra we associate a new square of ring spectra which becomes cartesian upon applying Ktheory, or in fact any localizing invariant. The new square canonically maps to the original one, and this map is an equivalence in three corners. In the fourth corner, this map is generally not an equivalence. However, we understand this map well enough to deduce simple proofs of various excision results, most of which were previously proven by different methods in work of SuslinWodzicki, CuntzQuillen, Cortiñas, and GeisserHesselholt.
 Supplements




Mar 28, 2019
Thursday

09:30 AM  10:30 AM


Character Theory and Tempered Cohomology
Jacob Lurie (Harvard University)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
Complex Ktheory is a generalized cohomology theory introduced by Atiyah and Hirzebruch, which associates to each finite cell complex X the Grothendieck group KU(X) of complex vector bundles on X. However, it also admits a purely algebraic description which makes no mention of vector bundles: it is the complexoriented cohomology theory associated to the multiplicative formal group over Spec(Z). In this talk, I'll discuss a variant of this algebraic picture which can be used to recover equivariant complex Ktheory (as well as equivariant elliptic cohomology), and explain its relationship with the classical character theory of finite groups and various generalizations thereof.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Arithmetic subloci of rank one local systems
Hélène Esnault (Freie Universität Berlin)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
We show that closed subsets of the character variety of a normal complex variety, which are padically integral and Galois invariant, are motivic. Joint with Mortiz Kerz. (Not really derived, clearly geometric, and padic)
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Geometrization of the local Langlands correspondence
Laurent Fargues (Institut de Mathématiques de Jussieu)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
I will review my geometrization conjecture of the local Langlands correspondence and my ongoing work with Peter Scholze about the construction of local Langlands parameters. I will mainly focus on the central object I introduced 4 years ago: the stack Bun_G of Gbundles on the curve.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


TBA
Peter Scholze (Universität Bonn)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements




Mar 29, 2019
Friday

09:30 AM  10:30 AM


The Chern character and categorification
Sarah Scherotzke (Westfälische WilhelmsUniversität Münster)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
The Chern character is a central construction which appears in topology, representation theory and algebraic geometry. In algebraic topology it is for instance used to probe algebraic Ktheory which is notoriously hard to compute, in representation theory it takes the form of classical character theory. Recently, Toen and Vezzosi suggested a construction, using derived algebraic geometry, which allows to unify the various Chern characters. We will categorify this Chern character. In the categorified picture algebraic Ktheory is replaced by the category of noncommutative motives. It turns out that the categorified Chern character has many interesting applications. For instance we show that the DeRham realisation functor is of noncommutative origin.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


$K$theory of division algebras over local fields
Lars Hesselholt (Nagoya University)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
This is joint work with Michael Larsen and Ayelet Lindenstrauss. Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers $\ell \neq p$ and integers $j \geq 1$, there exists a canonical "reduced norm" isomorphism of $\ell$adic $K$groups $\operatorname{Nrd}_{D/K} \colon K_j(D,\mathbb{Z}_{\ell}) \to K_j(K,\mathbb{Z}_{\ell})$ such that $d \cdot \operatorname{Nrd}_{D/K}$ is equal to the norm homomorphism $N_{D/K}$. We prove the analogous statement for the $p$adic $K$groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence $\operatorname{Trd}_{A/S} \colon \operatorname{THH}(A\,\,D,\mathbb{Z}_p) \to \operatorname{THH}(S\,\,K,\mathbb{Z}_p)$ between two $p$complete cyclotomic spectra associated with $D$ and $K$, respectively, from which the statement for the $p$adic $K$groups ensues. Interestingly, we show that if $p$ divides $d$, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, $d \cdot \operatorname{Trd}_{A/S}$ agrees with the trace $\operatorname{Tr}_{A/S}$, although this is possible as maps of spectra with $\mathbb{T}$action.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


TBA
Gabriele Vezzosi (Università di Firenze)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Bokstedt periodicity and Bott periodicity
Dmitry Kaledin (Steklov Mathematics Institute)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
One of the miracles of Topological Hochschild Homology is that THH(F_p) is very simple: it is just the algebra of polynomials in one variable of degree 2. However, the existing proofs of this fact are not simple at all. I will present yet another proof that uses quite a few additional general structures THH is known to have (multiplication, a trace functor structure), but almost nothing specific to F_p.
 Supplements



