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Niko Naumann: Quillen stratification in equivariant homotopy theory

31st January, 2023

We prove a variant of Quillen's stratification theorem in equivariant homotopy theory for a finite group working with arbitrary commutative equivariant ring spectra as coefficients, and suitably categorifying it. We then apply our methods to the case of Borel-equivariant Lubin-Tate E-theory. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing tensor-ideals of the category of equivariant modules over Lubin-Tate theory, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.

Or Landesberg: Non-Rigidity of Horocycle Orbit Closures in Geometrically Infinite Surfaces

31st January, 2023

Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider ℤ-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.

Toni Mikael Annala: Stable Homotopy without Homotopy

30th January, 2023

Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy invariant. This is a shame, because it means that the stable motivic homotopy theory of Morel-Voevodsky cannot be employed in studying the deeper aspects of such theories, such as cohomology operations that act on the cohomology groups. In this talk, I will discuss ongoing efforts, joint with Ryomei Iwasa and Marc Hoyois, to set up a workable theory of non-homotopy invariant stable motivic homotopy theory, with the goal of providing effective tools of studying cohomology theories in algebraic geometry by geometric means.

Henrik Holm: The Q-shaped derived category of a ring

24th January, 2023

The derived category, D(A), of the category Mod(A) of modules over a ring A is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch(A) of chain complexes of A-modules equipped with its standard model structure. One can view Ch(A) as the category Fun(Q, Mod(A)) of additive functors from a certain small preadditive category Q to Mod(A). The model structure on Ch(A) = Fun(Q, Mod(A)) is not inherited from a model structure on Mod(A) but arises instead from the "self-injectivity" of the special category Q. We will show that the functor category Fun(Q, Mod(A)) has two interesting model structures for many other self-injective small preadditive categories Q. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the Q-shaped derived category of A. We will also show that it is possible to generalize the homology functors on Ch(A) to homology functors on Fun(Q, Mod(A)) for most self-injective small preadditive categories Q.

Christian Lange: Orbifolds and Systolic Inequalities

13th January, 2023

In this talk, I will first discuss some instances in which orbifolds occur in geometry and dynamics, in particular, in the context of billiards and systolic inequalities. Then I will present topological conditions for an orbifold to be a manifold together with applications to foliations and to Besse geodesic and Reeb flows (joint work with Manuel Amann, Marc Kegel and Marco Radeschi). Here a flow is called Besse if all its orbits are periodic. Such flows are related to systolic inequalities. Namely, I will explain a characterization of contact forms on 3-manifolds whose Reeb flow is Besse as local maximizers of certain 'higher' systolic ratios, and mention other related systolic-like inequalities (joint work with Alberto Abbondandolo, Marco Mazzucchelli and Tobias Soethe).

Thomas Koberda: First-order rigidity of homeomorphism groups of manifolds

9th December, 2022

I will discuss some aspects of the first-order theory of homeomorphism groups of connected manifolds. The main result is as follows. Let M be a compact, connected manifold. There is a sentence S(M) in the language of groups such that if N is an arbitrary manifold and the homeomorphism group of N models S(M) then N is homeomorphic to M. This resolves a conjecture of Rubin from the 1980s. I will illustrate some of the ingredients of the proof, including an interpretation of second order arithmetic in the theory of homeomorphism groups of manifolds.

Laura DeMarco: Lattès maps, bifurcations, and arithmetic

8th December, 2022

In the field of holomorphic dynamics, we learn that the Lattès maps - the rational functions on ℙ1 that are quotients of maps on elliptic curves - are rather boring. We can understand their dynamics completely. But viewed arithmetically, there are still unanswered questions. I'll begin the talk with some history of these maps. Then I'll describe one of the recent questions and how it has led to interesting complex-dynamical questions about other families of maps on ℙ1 and, in turn, new perspectives on the arithmetic side. The new material is a joint project with Myrto Mavraki.

Shreyasi Datta: S-arithmetic Diophantine Approximation

2nd December, 2022

Diophantine approximation deals with quantitative and qualitative aspects of approximating numbers by rationals. A major breakthrough by Kleinbock and Margulis in 1998 was to study Diophantine approximations for manifolds using homogeneous dynamics. After giving an overview of recent developments in this subject, I will talk about Diophantine approximation in the S-arithmetic set-up, where S is a finite set of valuations of

Ayelet Lindenstrauss: Algebraic K-theory and the Cyclotomic Trace

14th November, 2022

Projective modules over rings are the algebraic analogues of vector bundles; more precisely, they are direct summands of free modules. Some rings have non-free projective modules. For instance, the ideals of a number ring are projective, and for some number rings they need not be free. Even for rings like ℤ, over which all finitely generated projective modules are free, the category of such modules contains a wealth of interesting information. In this talk I will introduce algebraic K-theory, which encodes this information. I will also explain why one would try to use some kind of trace from K-theory to simpler invariants, outline the cyclotomic trace and briefly show how it is used in calculations.

Krzysztof Fraczek: Deviation Spectrum of Ergodic Integrals for Locally Hamiltonian Flows on Surfaces

8th November, 2022

The talk will consist of a long historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration with Corinna Ulcigrai and Minsung Kim. New developments include a better understanding of the asymptotic of so-called error term (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime).

Laurent Bartholdi: Dimension series and homotopy groups of spheres

28th October, 2022

The lower central series of a group G is defined by γ1=G and γn = [G,γn-1]. The 'dimension series', introduced by Magnus, is defined using the group algebra over the integers:

δn = { g : g-1 belongs to the nth power of the augmentation ideal }.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δn ≥ γn, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ4 / γ4 cyclic of order 2. On the positive side, Sjogren showed that δn / γn is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proved by Gupta) that only 2-torsion may occur.

In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient δn / γn; this proves that Sjogren's result is essentially optimal.

Even more interestingly, we show that this problem is intimately connected to the homotopy groups πn(Sm) of spheres; more precisely, the quotient δn / γn is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group π2p(S2) due to Serre.

Elizabeth Tatum: On a Spectrum-level Splitting of the BP⟨2⟩-Cooperations Algebra

9th August, 2022

In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of bo ⋀ bo and BP⟨1⟩ ⋀ BP⟨1⟩. These splittings helped make it feasible to do computations using the bo- and BP⟨1⟩-based Adams spectral sequences. In this talk, we will discuss an analogous splitting for BP⟨2⟩ ⋀ BP⟨2⟩ at primes larger than 3.

Mark Behrens: tmf-resolutions  

19th July, 2022

I'll talk about the tmf-based Adams spectral sequence, and how it detects most of the v2-periodic elements in the known range of the 2-primary stable stems.  Parts of the material I will discuss are joint with Dominic Culver, Prasit Bhattacharya, JD Quigley, and Mark Mahowald. 

David Barnes: Sheaf models for rational stable equivariant homotopy theory

28th June, 2022

Sheaves sit at an interface of algebra and geometry. Equivariant sheaves offer even more structure, allowing for different group actions at different stalks. We are interested in the case where both the base space and group of equivariance are profinite (that is, compact, Hausdorff and totally disconnected). This combination provides many useful consequences, such as a good notion of equivariant presheaves and an explicit construction of infinite products. 

The 2019 PhD thesis of Sugrue used equivariant sheaves to give an algebraic model for rational G-equivariant stable homotopy theory, where G is profinite. In this talk I will explain the model and related results, such as the equivalence between equivariant sheaves and rational Mackey functors (for profinite G).

Hans-Werner Henn: On the Brown Comenetz dual of the K(2)-local sphere at the prime 2  

21st June, 2022

The Brown Comenetz dual I of the sphere represents the functor which on a spectrum X is given by the Pontryagin dual of the 0-th homotopy group of X. For a prime p and a chromatic level n there is a K(n)-local version In of I. For a type n-complex X, this is given by the Pontryagin dual of the 0-th homotopy group of the K(n)-localization of X. By work of Hopkins and Gross, the homotopy type of the spectra In for a prime p is determined by its Morava module if p is sufficiently large. For small primes, the result of Hopkins and Gross determines In modulo an "error term". For n=1 every odd prime is sufficiently large and the case of the prime 2 has been understood for almost 30 years. For n>2 very little is known if the prime is small. For n=2 every prime bigger than 3 is sufficiently large. The case p=3 has been settled in joint work with Paul Goerss. This talk is a report on work in progress with Paul Goerss on the case p=2. The "error term" is given by an element in the exotic Picard group which in this case is an explicitly known abelian group of order 29. We use chromatic splitting in order to get information on the error term.     

Sergei Gukov: Surprising VOA Structures from Quantum Topology

6th June, 2022

In quantum topology, one usually constructs invariants of knots and 3-manifolds starting with an algebraic structure with suitable properties that can encode braiding and surgery operations in three dimensions. ln this talk, 1 review recent work on q-series invariants of 3-manifolds, associated with quantum groups at generic q, that provide a connection between quantum topology and algebra going in the opposite direction: starting with a 3-manifold and a choice of Spin-C structure, the q-series invariant turns out to be a character of a (logarithmic) vertex algebra that depends on the 3-manifold.

Jesper Grodal: A guided tour to the Picard group of the stable module category

24th May, 2022

The Picard group Tk(G) of the stable module category of a finite group has been an important object of study in modular representation theory, starting with work Dade in the 1970s. Its elements are equivalence classes of so-called endotrivial modules, i.e., modules M such that End(M) is isomorphic to a trivial module direct sum a projective kG-module. 1-dimensional characters, and their shifts, are examples of such modules, but often exotic elements exist as well. My talk will be a guided tour of how to calculate Tk(G), using methods from homotopy theory. The tour will visit joint work with Tobias Barthel and Joshua Hunt, with Jon Carlson, Nadia Mazza and Dan Nakano, and with Achim Krause.