Abstracts

Mini Course 2026

Courses:


  1. Course of Paul Balmer:

    We shall investigate the derived category of permutation modules for a finite group over a field of positive characteristic. Our approach employs the framework of tensor triangular geometry to analyze the internal structure of this category. In particular, we construct a stratification of the category via Brauer quotients, which reflect how permutation modules decompose under restriction to subgroups and passage to quotient systems.

    This stratification allows us to isolate local pieces corresponding to simpler group-theoretic data, providing a bridge between the global behavior of the category and the local information encoded in its subquotients.

    Building on this, we determine the spectrum of compact objects in the derived category. The analysis reduces to the case of elementary abelian subgroups, where we can apply a twisted form of group cohomology to express the local structure of the spectrum in terms of the graded endomorphism ring of the unit object. These local computations assemble into a global description of the spectrum, yielding a complete classification of both thick and localizing tensor ideals. The results provide new insight into how homological and cohomological invariants govern the organization of module categories in modular representation theory.


  2. Course of Yonatan Harpaz:

    Like topological spaces in classical algebraic topology, commutative rings and schemes are often studied via various types of cohomology theories. In doing so, it is often useful to distinguish between two types of cohomological information: the ``on characteristic'' type, such as de Rham cohomology, which for a scheme over a base field k, for example, gives k-linear information, and the ``off characteristic'' type, such as l-adic cohomology, where the interesting information lies away from, and is in some sense insensitive to, the characteristic of the base.

    Many naturally arising invariants of schemes are not purely of one type or the other. Such is the case for algebraic K-theory, a fascinating and complex invariant, bringing together homotopy theory, algebraic geometry and arithmetic. An important tool in studying algebraic K-theory is given by trace methods, which consist in constructing trace maps from algebraic K-theory to topological Hochschild homology and related invariants, such as the cyclotomic trace map to topological cyclic homology. These trace maps capture the part of algebraic K-theory that is on characteristic, or in some sense, supported along it. The celebrated Dundas-Goodwillie-McCarthy theorem says that the fibre of the cyclotomic trace map is invariant under nilpotent extensions, and is hence (at least morally) off characteristic. This separation of information has proven an invaluable tool in accessing and understanding algebraic K-theory.

    A closely related invariant is hermitian K-theory, or K-theory of quadratic forms, whose relation to algebraic K-theory can be compared to the relation between real and complex K-theory as cohomology theories on topological spaces. In recent years there has been growing interest in studying hermitian K-theory through the lens of trace methods. Here, the prime 2 plays a special role, and the separation to on and off characteristic information at the prime 2 is of particular interest. The ability to efficiently study hermitian K-theory at the prime 2 has itself considerably advanced in recent years due to the proliferation of powerful homotopy theoretical technology, and notably higher category theory. In this mini-course we will survey and explain these ideas, putting the emphasis on the geometric and arithmetic applications, while at the same time trying to give an idea of the homotopy theory that underlines it all.


  3. Course of Thomas Unger:

    Real algebra arguably started with Artin's solution of Hilbert's 17th problem in the late 1920s, which kickstarted the theory of ordered fields. In the special case of the field of real numbers a lot of work (in particular concerning counting real zeroes of polynomials) had already been done in the 19th century, and Sylvester's inertia theorem brought quadratic forms into play.

    Early in the 20th century, valuations were thrown into the mix. The triad quadratic forms-orderings-valuations played an important role in the development of real algebra. In this context, results of note include Pfister's local-global principle and the Baer-Krull theorem.

    The theory of orderings was later extended to commutative rings. Here, the discovery of the real spectrum by Coste and Roy in the late 1970s was a game-changer. In the early 1980s, Mahé showed that the connected components of the real spectrum of a commutative ring A are in bijective correspondence with the signatures, i.e., the ring homomorphisms from the Witt ring of A to the ring of integers.

    A bit more than a decade ago, Astier and I moved into the noncommutative direction. We have been developing the real algebraic aspects of central simple algebras with involution over fields, and Azumaya algebras with involution over commutative rings, from the point of view of hermitian forms over these algebras. In my lectures, I will report on our work. My tentative plan is to discuss signatures, positive cones and their related Artin-Schreier theory, links with the Tignol-Wadsworth gauges, Garrel's involution trace pairing, and ending with Sylvester's theorem and Pfister's local-global principle for Azumaya algebras with involution over semilocal rings.



Short talks:

  1. Talk of Oumaima El Aouny:

    (Joint work with Pr. Hicham Yamoul). Spectral sequences play a fundamental role in homotopy theory and derived algebra, providing powerful computational tools for extracting invariants from filtered objects and diagrams. In higher categorical settings, such as (∞,1) categories or homotopical model categories, the construction and interpretation of spectral sequence differentials require homotopy invariant methods.

    In this talk, we explore the role of derived Kan extensions in the construction and interpretation of higher differentials in spectral sequences arising from simplicial and cosimplicial objects. Derived Kan extensions provide a homotopically meaningful framework for extending diagrams and computing derived limits and colimits, which naturally appear in the formation of spectral sequences.

    We discuss how these constructions interact with homotopy limits, derived functors, and higher categorical structures. In particular, we illustrate how higher differentials can be understood through derived categorical mechanisms, highlighting their relationship with homotopy coherent diagrams and derived mapping spaces.

    These ideas contribute to a more conceptual understanding of spectral sequences in higher algebra and homotopical contexts, with potential applications to derived categories, higher homotopy structures, and modern homotopical alge bra.

    Keywords: Derived Kan extensions, higher differentials, (∞,1)-categories, spectral sequences, homotopy limits, derived functors.

    References
    [1] Hans-Joachim Baues and David Blanc : Higher order derived functors and the Adams spectral sequence, (2014).
    [2] Marcin Cha lupnik: Derived Kan Extension for strict polynomial functors, (2014).
    [3] Georges Maltsiniotis : Quillen’s adjunction theorem for derived functors, revisited, (2006).
    [4] David Blanc , Nicholas Meadows : Spectral Sequences in (∞,1)-Categories, (2020).
    [5] Saunders Mac Lane : Categories for the Working Mathematician, (1971).
    [6] Philip S. Hirschhorn : Model Categories and Their Localizations, (2003).
    [7] Ronald brown : Topology and Groupoids , (2006).
    [8] Markus Land : Introduction to ∞-Categories, (2019).


  2. Talk of Yorick Fuhrmann:

    When G is a finite group, a G-spectrum is Borel complete if it comes from a spectrum with a G-action. When G is profinite, one needs more refined notions of Borel completeness. We will explain these, express the resulting Borel complete categories in terms of étale sheaves and representations, and finally relate them to certain categories of Nisnevich and étale motives controlled by the étale fundamental group of the base scheme.

  3. Talk of Samuel Lerbet:

    A classical theorem of Murthy states that a vector bundle of rank d on a smooth affine variety of dimension d over an algebraically closed field splits a free rank 1 summand if, and only if, its top Chern class vanishes. The statement that the same holds for vector bundles of rank d-1 has become known as Murthy's conjecture; it is now a theorem (at least in characteristic 0), whose proof relies on motivic techniques. Moving instead to real closed base fields, the vanishing of the top Chern class is insufficient to guarantee the splitting of a free rank 1 summand as it does not account for phenomena detected by topology over the real locus. We will explain how several natural analogues of Murthy's conjecture formulated to take topology into account fail in this context, showing that the situation over the field of real numbers is rather delicate. This is joint work with A. Asok and J. Fasel.

  4. Talk of Ilan Levin:

    In this talk, we shall define and investigate an invariant for k-linked Pfister forms. Namely, given n1,…,nt-fold Pfister forms that share a common k-fold Pfister form, we associate an n1 + … + nt-k(t-1)-fold Pfister form, and examine in which cases this invariant becomes hyperbolic. Based on joint work with Adam Chapman.

  5. Talk of Fatima Maayane:

    Tierney and Vogel introduced a simplicial method to derive an arbitrary functor E between categories, using projective resolutions built from simplicial kernels. This framework unifies several classical constructions in homological algebra, including Eilenberg-Moore derived functors and Dold-Puppe type functors.

    In this work, we develop a two-dimensional (bisimplicial) refinement of the Tierney-Vogel construction. By applying the resolution procedure inside the category of simplicial objects, we produce bisimplicial projective resolutions that support a general comparison theorem, ensuring the independence of the resulting derived functors from the chosen resolution.

    Applying the functor E levelwise yields a bisimplicial object and a natural double complex. We define bisimplicial derived functors via the homology of the associated total complex, and provide an equivalent formulation using diagonal constructions. The row and column filtrations give rise to convergent spectral sequences, organizing the computation in a functorial two-directional framework.

    When the projective class comes from a cotriple (comonad), we construct a bisimplicial bar-type model and compare the resulting functors with the classical Barr-Beck cotriple derived functors. In the additive setting, we recover the classical relative Eilenberg-Moore derived functors and the hyperhomology of double complexes.

  6. Talk of Daniel Marlowe:

    The existence of a good theory of bifibre sequences, aka Verdier sequences, of stable ∞-categories gives rise to the rich formalism of localising invariants. In this talk, we report on joint work with Yonatan Harpaz, in which we generalise the notion of Verdier sequence to the realm of exact ∞-categories. We will introduce the main technical tool of heart structures on stable categories, discuss the consequences for connective K-theory, and speculate about applications to other K-theoretic invariants.

  7. Talk of Jean Paul Schemeil: The derived category of permutation modules over the absolute Galois group of a field enters the motivic world as Artin motives, sitting inside the derived category of Voevodsky motives. In this way, the Balmer–Gallauer computation of its tt-spectrum provides a fundamental lower bound for the still largely mysterious Balmer spectrum of Voevodsky motives.

    In this talk, I will discuss other lower bounds coming from complementary motivic subcategories. The guiding philosophy is the study of the comparison map for cellular motivic tt-subcategories: in various situations, this map identifies their Balmer spectra with the homogeneous spectra of the bigraded endomorphism rings of the tensor unit. I will illustrate this principle through joint work with Fraser Sparks on Tate motives, where we obtain a description of the Balmer spectrum in terms of the Milnor K-theory of the base field.

    I will then turn to the tt-subcategory generated by motives of smooth projective quadrics. This subcategory is a natural next case to consider: it contains the currently known part of the Picard group of Voevodsky motives, and it complements the Artin and Artin–Tate cases studied by Balmer–Gallauer. Over real closed fields, I will explain how its tt-geometry can be reduced to computations of certain cellular motivic tt-categories parametrised by Pfister quadrics, leading to an explicit description of the Balmer spectrum of the original category.