16 December

9:30-10:00	Opening remarks
10:00-11:15	Ryo Takahashi: Spectrums in representation theory of commutative rings (1) Subcategory classification has been a central theme across many branches of mathematics for over half a century, including ring theory, homotopy theory, representation theory, and algebraic geometry. In the context of representation theory of commutative rings, spectrums are used to classify certain subcategories of module categories or derived categories, which are often realized as sets of prime ideals or related ones. In this series of talks, I will begin with motivation and a brief historical overview of subcategory classification, and then present my recent work on it.
11:15-11:40	Coffee and light refreshments
11:40-12:40	Paul Balmer: TBD TBD.
12:40-14:30	Lunch
14:30-15:30	Hiroki Matsui: Spectra of triangulated categories and their applications to algebraic geometry In 2005, P. Balmer defined a ringed space, called the Balmer spectrum, for a given tensor triangulated category. In algebro-geometric context, this spectrum provides several reconstruction theorems in areas such as algebraic geometry, representation theory, stable homotopy theory, and so on. In this talk, I will define a ringed space, which we call the triangular spectrum, for a given triangulated category, as a tensor-free analog of the Balmer spectrum. I will also present applications of the triangular spectrum to algebraic geometry, including a new proof of the Bondal--Orlov and Ballard reconstruction theorems. The key result is that, for the perfect derived category of a quasi-projective variety, the Balmer spectrum is an open sub-ringed space of the triangular spectrum.
15:30-15:50	Coffee and light refreshments
15:50-16:50	Anish Chedalavada: Higher Zariski geometry Many notions of "geometry" concern themselves with topological spaces equipped with additional structure. For a given space X, this structure is usually an assignment of an open set of X to an appropriate "ring of functions" on X; these will satisfy a gluing property along open sets, and thus form a sheaf. Different geometries will impose different conditions on the requisite sheaves and spaces; for example, the category of locally ringed spaces is the basic geometric structure considered in algebraic geometry, and geometrizes the study of commutative rings via the Zariski spectrum. In [DAG V], Lurie explains that a category of "locally (-)-spaces" is a generic feature of other categories which carry a small amount of structure, known as a geometry. Based on this, I will present joint work with Aoki--Barthel--Schlank--Stevenson which constructs a category of locally 2-ringed spaces, which will geometrize the study of tensor triangulated-categories via the Balmer spectrum. This will give rise to an affine spectrum -| global sections adjunction exactly analogous to ordinary algebraic geometry. I will then introduce the notion of a 2-scheme, and state an affineness criterion for 2-schemes purely in terms of a topological property. Using this, I will explain how to provide a universal property for the category of perfect complexes on a qcqs (spectral) scheme via a construction known as the relative spectrum, also due to Lurie. Time permitting, I will use this latter result to indicate an application to torsion-free endotrivial modules over finite groups in modular characteristic.
16:50 - 17:10	Coffee and light refreshments
17:10-18:10	Shunya Saito: Classifying KE-closed subcategories over a commutative noetherian ring This talk is based on joint work with Toshinori Kobayashi (Meiji University). Classifying subcategories is an active topic in the representation theory of algebras. In particular, several subcategories of the module category of a commutative noetherian ring have been classified so far and are described in terms of the prime spectrum. In this talk, we give a classification of KE-closed subcategories (additive subcategories closed under extensions and kernels) for a commutative noetherian ring. For this, we introduce a class of functions on the spectrum, called n-Bass functions, and establish a bijection between KE-closed subcategories and 2-Bass functions under mild assumptions.

17 December

10:00-11:15	Ryo Takahashi: Spectrums in representation theory of commutative rings (2) Subcategory classification has been a central theme across many branches of mathematics for over half a century, including ring theory, homotopy theory, representation theory, and algebraic geometry. In the context of representation theory of commutative rings, spectrums are used to classify certain subcategories of module categories or derived categories, which are often realized as sets of prime ideals or related ones. In this series of talks, I will begin with motivation and a brief historical overview of subcategory classification, and then present my recent work on it.
11:15-11:40	Coffee and light refreshments
11:40-12:40	Osamu Iyama: TBD TBD.
12:40-14:30	Lunch
14:30-15:30	Victor Pretti: An application of stability conditions to exceptional collections Although seemingly unrelated concepts on triangulated categories, exceptional collections and Bridgeland stability conditions have been used together numerous times. In this talk, we will discuss one interesting application of stability conditions to the Bondal--Polishchuk conjecture. For that, we will use the algebraic stability conditions, as in the work of Qiu--Woolf. In a particular family of varieties, those with a full nondegenerate exceptional collection, for example the projective space and the odd-dimensional smooth quadric, we will see that the topology of the space of algebraic stability conditions gives some control over the class of all possible full exceptional collections. With this, for such varieties, we prove that any two full exceptional collections can be related by a sequence of mutations and shifts. We also describe the silting objects, tilting objects and simple-minded collections for such triangulated categories.
15:30-15:50	Coffee and light refreshments
15:50-16:50	Hisato Matsukawa: A spectrum of a triangulated category with an action Matsui’s triangular spectrum provides a new framework for studying triangulated categories, extending the philosophy of tensor triangular geometry to contexts without a monoidal structure and enabling a finer classification of thick subcategories. I will discuss its universal property and its connections with classical geometric notions. Recently, the speaker introduced a relative version of Matsui’s triangular spectrum. I will explain its construction, universal property, and the relationships among Matsui’s triangular spectrum, Balmer’s tensor triangular spectrum, and the Zariski spectrum. This perspective provides a new way to view and organize geometric phenomena appearing in categories of interest, such as twisted derived categories, derived categories of matrix factorizations, and categories of singularities associated to schemes.
16:50 - 17:10	Coffee and light refreshments
17:10-18:10	Lidia Angeleri Hügel: TBD TBD.

18 December

10:00-11:15	Ryo Takahashi: Spectrums in representation theory of commutative rings (3) Subcategory classification has been a central theme across many branches of mathematics for over half a century, including ring theory, homotopy theory, representation theory, and algebraic geometry. In the context of representation theory of commutative rings, spectrums are used to classify certain subcategories of module categories or derived categories, which are often realized as sets of prime ideals or related ones. In this series of talks, I will begin with motivation and a brief historical overview of subcategory classification, and then present my recent work on it.
11:15-11:40	Coffee and light refreshments
11:40-12:40	Manuel Reyes: Is there a noncommutative spectrum functor? In many classical situations, the spectrum forms a functor from commutative algebras to spaces. Thus it is natural to ask whether it extends to a noncommutative spectrum functor. Several "no-go" theorems forbid us from naively extending such functors, illustrating that functoriality requires us to revise our understanding of the spectrum at a fundamental level. I will discuss the problem of constructing a noncommutative spectrum functor, including both negative results and partial positive steps toward its resolution.
12:40-12:50	Picture
12:50-14:30	Lunch
14:30-15:30	Ariel Rosenfield: Enriched sieves and coverages under change of base Towards circumventing known obstruction results for functorial spectra on categories of non-commutative algebraic structures, we investigate how change of enriching base category via a faithful, conservative right adjoint functor interacts with enriched coverages and sheaves on a given enriched category. We prove that change of base via such a functor gives rise both to an injective mapping on subobjects in enriched presheaf categories, and to an injective mapping on enriched coverages. In case the base change functor is also full, the enriched associated sheaf construction on a presheaf category commutes with base change.
15:30-15:50	Coffee and light refreshments
15:50-16:50	So Nakamura: A ringed-space-like structure on coalgebras for noncommutative algebraic geometry The prime spectrum of a commutative ring is the underlying set of prime ideals of the ring together with the Zariski topology. A theorem proven by Reyes states that any extension of the set-valued prime spectrum functor on the category of commutative rings to the category of (not necessarily commutative) rings must assign the empty set to the n by n matrix algebra with complex entries when n is greater than 1. This suggests that sets do not serve as the underlying structure of a spectrum of a noncommutative ring. It is argued in his recent paper that coalgebras can be viewed as the underlying object of a noncommutative spectrum. In this talk, we introduce coalgebras equipped with a ringed-space-like structure, which we call ringed coalgebras. These objects arise from fully residually finite-dimensional (RFD) algebras and schemes locally of finite type over a field k. The construction uses the Heyneman--Sweedler finite dual coalgebra and the Takeuchi underlying coalgebra. We will discuss that, if k is algebraically closed, the formation of ringed coalgebras gives a fully faithful functor out of the category of fully RFD algebras, as well as a fully faithful functor out of the category of schemes locally of finite type. The restrictions of these two functors to the category of (commutative) finitely generated algebras are isomorphic. In this way, ringed coalgebras can be thought of as a generalization of RFD algebras and schemes locally of finite type.
16:50-17:10	Coffee and light refreshments
17:10-17:20	Abhishek Banerjee: On some spectral spaces associated to tensor triangular categories We give general approaches to obtain large classes of examples for spectral spaces (i.e., spaces that are homeomorphic to the Zariski spectrum of a commutative ring) using a newly discovered criterion of Finocchiaro that uses ultrafilters to identify spectral spaces along with subbases of quasi-compact open sets. In particular, our classes of examples fall into three distinct categories. (1) At a first level, we consider spectral spaces associated to certain types of abelian categories, which include in particular all Grothendieck categories. For this, we consider the spaces of invariants of closure operators acting on subobjects of a given object. (2) Thereafter, we come to the spectrum Spec(\mathbb{T}) of a tensor triangular category as defined by Balmer. We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in (\mathbb{T},\otimes,1) and the collection of open subsets of Spec(\mathbb{T}) in inverse topology. In fact, we prove a more general ``nullstellensatz like'' result in terms of supports on (\mathbb{T},\otimes,1). (3) Finally, we consider modules \mathbb{M} over a tensor triangular category (\mathbb{T},\otimes,1). By considering a closure operator c of finite type on the space SMod(\mathbb{M}) of thick \mathbb{T}-submodules of \mathbb{M}, we show that the space SMod^c(\mathbb{M}) of fixed points of the operator c is a spectral space that also carries the structure of a topological monoid.
17:25-17:35	Fatemeh Esmaeelzadeh: Representation theory of generalized Weyl--Heisenberg groups: Structure, classification, and applications This presentation delves into the representation theory of the generalized Weyl--Heisenberg group, a rich algebraic structure that extends the classical Heisenberg framework and finds applications across quantum mechanics, signal analysis, and noncommutative geometry. We begin by analyzing the quasi-regular representation of this group, deriving explicit conditions for admissible vectors and exploring their structural properties. Special attention is given to square-integrable representations, for which we establish rigorous criteria for admissibility and integrability using tools from harmonic analysis and operator theory.
17:40-17:50	Zahra Nazemian: Invariant subspaces and the characterization problem of affine n-spaces In a Bourbaki seminar [H. Kraft, Challenging problems on affine n-space], Kraft identified the Characterization Problem for polynomial rings as one of the eight central challenges in affine algebraic geometry, alongside the Jacobian Conjecture, the Automorphism Problem, and the Zariski Cancellation Problem (ZCP). We prove the following result (see [H. Huang, Z. Nazemian and Y. Wang, and J. J. Zhang, Relative Cancellation]): Let K be an algebraically closed field of characteristic zero, and let A be a (not necessarily commutative) algebra. Then A is isomorphism to polynomial ring with m variables, for some m>3, if and only if the following two conditions hold: (1) A is finitely generated and connected graded; (2) A has no nontrivial invariant subspaces. Condition (1) is essential: for example, the Weyl algebras satisfy condition (2), but not condition (1), and are not isomorphic to polynomial rings. It remains an open question whether condition (1) can be replaced with a weaker assumption.
18:30-20:30	Banquet

19 December

10:00-11:00	Luca Pol: On the spectrum of global representations A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family U. These arise in classical representation theory, in the study of representation stability, as well as in global homotopy theory. In this talk, I will discuss how to use tensor-triangular geometry to study the derived category of global representations over fields k of characteristic zero. In particular, I will present some calculations of Balmer spectra for various infinite families of finite groups including elementary abelian p-groups, cyclic groups, and finite abelian p-groups of bounded rank. This is joint work with Miguel Barrero, Tobias Barthel, Neil Strickland and Jordan Williamson.
11:00-11:20	Coffee and light refreshments
11:20-12:20	Sam Miller: Permutation twisted cohomology, remixed Let G be a finite group and k a field of prime characteristic p. The Balmer spectrum of the derived category of permutation kG-modules was recently deduced by Balmer and Gallauer via a myriad of new techniques and constructions. One such construction, the "twisted cohomology ring," was used to deduce the Balmer spectrum for elementary abelian p-groups. In this talk, we introduce a generalization of the twisted cohomology ring that we expect can be used to deduce the Balmer spectrum for all finite p-groups.
12:20-14:00	Lunch
14:00-15:00	Giovanna Le Gros: SemiBousfield classes and perversities For a rigidly-compactly generated triangulated category with a fixed t-structure, we introduce semi-Bousfield classes. Examples of semi-Bousfield classes include both Bousfield classes and coaisles of compactly generated tensor t-structures. We will mainly consider the case of the unbounded derived category of a commutative noetherian ring with the standard t-structure. In this case, we can describe the semi-Bousfield classes which come from perversities, which are integer valued functions on the prime spectrum of the ring. Moreover, this assignment is compatible with the classification of compactly generated t-structures by sp-filtrations due to Alonso--Jeremias--Saorin, and localising subcategories by subsets of the spectrum due to Neeman. This is based on joint work with Dolors Herbera and Michal Hrbek.
15:00-15:20	Coffee and light refreshments
15:20-16:20	Tsutomu Nakamura: TBD TBD.
16:30-16:40	Closing remarks