Yüksek Yapılar Seminerleri/ Higher Structures Seminars

İki haftada bir gerçekleşen Yüksek Yapılar Seminerleri yaz döneminde de düzenlenebilir.

2025 - 2026

Speaker: Doğancan Karabaş (Temple University, Japan Campus)
Title: Gluing methods for DG categories from geometry
Time: February 17, 2026 at 13:00 Istanbul local time (19:00 Tokyo local time) (Note the unusual time!)
Place: Online
Meeting ID: 935 6390 4955
Passcode: 699568
Abstract: Many dg categories arising in geometry admit local descriptions that can be assembled via homotopy-theoretic gluing. In this talk, I will survey a categorical framework, developed jointly with Sangjin Lee, which provides a model-theoretic and combinatorial description of such gluings, namely homotopy colimits. This approach leads to explicit computations of invariants in symplectic geometry and microlocal sheaf theory. In particular, I will describe our result proving a conjecture of Kontsevich that wrapped Fukaya categories of Weinstein manifolds are Morita equivalent to dg algebras of finite type.

Speaker: Semih Özlem (Mudanya University)
Title: Weakened axioms, idempotent splittings, and the structure of learning: From algebra to AI
Time: February 3, 2026 at 19:00 Istanbul local time
Place: Online
Meeting ID: 935 6390 4955
Passcode: 699568
Abstract: We often think of mathematics as a tower of abstractions, but it begins with something deeply human: the act of telling things apart. In this talk, I’ll explore how this simple idea—splitting and focusing—manifests across different fields, from linear algebra to motives to machine learning. We’ll start with a basic observation: if we relax the unit axiom in a vector space, the scalar multiplication by 1 becomes an idempotent, splitting the space into what is preserved and what is annihilated. This splitting phenomenon appears in surprising places: in the theory of motives, where projectors decompose varieties; in knot theory, where Jones–Wenzl projectors filter diagram algebras; and in deep learning, where attention mechanisms focus on relevant features. I’ll introduce the topos-theoretic model of neural networks (Belfiore–Bennequin) and suggest that learning difficulties like catastrophic forgetting and generalization gaps can be viewed as homotopical obstructions to achieving “nice” (fibrant) network states. Architectural tools like residual connections and attention can then be seen as learned, conditional idempotents—adaptable splitters that help networks organize information. This talk is an invitation to think structurally across disciplines. I won’t present finished theorems, but a framework of connections that links motivic philosophy, categorical algebra, and the practice of machine learning. The goal is to start a conversation: can tools from pure mathematics—obstruction theory, homotopy colimits, derivators—help us design more robust, interpretable, and composable learning systems? No expertise in motives, knots, or AI is required—only curiosity about how ideas weave together.

Speaker: Emanuele Pavia (Université du Luxembourg and SISSA-Trieste)
Title: Categorification of Koszul duality in algebraic topology
Time: January 20, 2026 at 18:00 Istanbul local time (16:00 Trieste/Esch-sur-Alzette local time).
Place: Online
Meeting ID: 935 6390 4955
Passcode: 699568
Abstract: It is well known that for a pointed topological space X both the singular chains on its n-fold loop space C_∙(Ω_*^nX) and its singular cochains C^∙(X) are En-algebras. When X is sufficiently nice and the ground ring is a field, then these E_n-algebras are Koszul dual: for n=1, this means that they are Koszul dual associative algebras. In this latter case, Beilinson, Ginzburg and Soergel proved that their bounded derived categories are equivalent. In this talk, we will see how a similar statement holds for all integers n≥2 after considering categorified modules over the En-algebras C_∙(Ω_*^nX) and C^∙(X). These arise geometrically as higher categories of quasi-coherent sheaves over two (inequivalent) derived stacks, both associated to the topological space X .
This is based on joint work with James Pascaleff and Nicolò Sibilla.

Speaker: Redi Haderi (Ankara Yıldırım Beyazıt University)
Title: Two models for higher operads
Time: November 25, 2025 at 21:00
Place: Online
Meeting ID: 935 6390 4955
Passcode: 699568
Abstract: In this talk we will describe two models for the theory of infinity-operads: Lurie's model and the simplicial lists model (developed in joint work with Özgün Ünlü). We begin by briefly introducing operads as a categorical tool to control and study a variety of algebraic structures. As it is known that higher dimensional categorical structures are presentation-sensitive, different ways of thinking about ordinary operads lead to different formulations of a higher variant. Lurie's model is based on the notion of operator category, while the simplicial lists model is based on a nerve theorem related to the monoidal envelope associated to an operad. Time permitting, we will discuss how the two can be related to each other.

Speaker: Thomas Geisse (University of Rikkyo)
Title: Motivic cohomology theories and applications
Time: November 11, 2025 at 11:00
Place: Online
Meeting ID: 935 6390 4955
Passcode: 699568
Abstract: I will define motivic cohomology, and give an overview over its properties. Then I will discuss applications to arithmetic and algebraic geometry, focusing on special values of zeta-functions and class field theory. The talk is aimed at non-experts.

Speaker: Kadri Ilker Berktav (Middle East Technical University)
Title: Legendrians in derived geometry
Time: October 14, 2025 at 18:00
Place: Online
Meeting ID: 935 6390 4955
Passcode: 699568
Abstract: This talk introduces Legendrian structures in derived contact geometry, covering key concepts and constructions that lead to a tubular neighborhood theorem and examples similar to the classical ones. We start by reviewing Lagrangians in derived symplectic geometry and then introduce analogous structures in the derived contact setting using techniques adapted from the symplectic case.

Speaker: Matthew Young (Utah State University)
Title: Z-hat invariants for Lie superalgebras
Time: October 02, 2025 at 15:00
Abstract: Z-hat invariants of 3-manifolds were introduced in the physics literature by Gukov, Pei, Putrov and Vafa in the context of supersymmetric gauge theory with the goal of categorifying the Reshetikhin--Turaev invariants of 3-manifold. Z-hat invariants depend on the input data of a Lie superalgebra and some discrete geometric data on the 3-manifold, such as a Spin-c structure. The goal of this talk is to explain a connection between Z-hat invariants and 3-manifold invariants associated to non-semisimple categories of representations of quantum supergroups. Based on work with Francesco Costantino, Matthew Harper and Adam Robertson.

2024 - 2025

Speaker: Doğancan Karabaş (Kavli Institute for the Physics and Mathematics of the Universe - University of Tokyo)
Title: A computational approach to the homotopy theory of dg-categories
Time: October 01, 2024 at 15:00
Abstract: The homotopy theory of differential graded (dg) categories plays a significant role in various fields, including algebraic geometry, representation theory, higher categories, and symplectic geometry. In particular, understanding dg-categories is crucial for formulating and interpreting homological mirror symmetry. In this talk, I will present our approach to the homotopy theory of dg-categories by establishing a cofibration structure, which can be viewed as a half-model structure. This structure enables a combinatorial description of derived constructions and offers computational advantages. This is joint work with Sangjin Lee (arXiv:2109.03411 and arXiv:2405.03258). Some key applications of our approach, particularly in symplectic and contact geometry, include:

Combinatorial description of homotopy colimits of dg categories, which gives a local-to-global formula computing wrapped Fukaya categories of symplectic manifolds,
Local-to-global construction of functors between wrapped Fukaya categories that are induced by symplectomorphisms,
A simple description of internal Hom and Hochschild cohomology of dg-categories. This ongoing work aims to provide useful tools for addressing the Weinstein conjecture, which concerns the existence of periodic orbits of Reeb vector fields.

I plan to cover as much of this content as time permits, and according to the audience's interest.

Speaker: Kadri İlker Berktav (Bilkent Üniversitesi)
Title: Constructions of contact derived stacks
Time: October 15, 2024 at 18:00
Abstract: This talk presents several examples of derived Artin stacks with shifted contact structures. We start by reviewing derived symplectic/contact geometry. Next, we outline our constructions: the first one extends classical 1-jet bundles, and the second set of constructions arises from shifted geometric quantization.

October 29, 2024: No seminar! (National Holiday)

Speaker: David Ayala (Montana State University)
Title: Factorization homology of higher categories
Time: November 12, 2024 at 16:00
Abstract: The "alpha" version of factorization homology pairs (framed) n-manifolds with E_n-algebras. This construction generalizes classical homology of a manifold, yields novel results concerning configuration spaces of points in a manifold, and supplies a sort of state-sum model for sigma models (ie, mapping spaces) to (n-1)-connected targets. This "alpha" version of factorization homology novelly extends Poincaré duality, shedding light on deformation theory and dualities among field theories. Being defined using homotopical mathematical foundations, "alpha" factorization homology is manifestly functorial and continuous in all arguments, notably in moduli of manifolds and embeddings between them, and it satisfies a local-to-global expression that is inherently homotopical in nature.

Now, E_n-algebras can be characterized as (\infty,n)-categories equipped with an (n-1)-connected functor from a point. The (full) "beta" version of factorization homology pairs (framed) n-manifolds with pointed (\infty,n)-categories (with adjoints). Applying 0th homology, or \pi_0, recovers a version of the String Net construction of surfaces, as well as of Skein modules of 3-manifolds. In some sense, the inherently homotopical nature of (full) "beta" factorization homology affords otherwise unforeseen continuity in all arguments, and local-to-global expressions.

In this talk, I will outline a definition of "beta" factorization homology, focusing on low-dimensions and on suitably reduced (\infty,n)-categories (specifically, braided monoidal categories). I will outline some examples, and demonstrate some operational practice of factorization homology. Some of this material is established in literature, some a work in progress, and some conjectural — the status of each assertion will be made clear. I will be especially interested in targeting this talk to those present, and so will welcome comments and questions.

All of this work is joint with John Francis.

Speaker: Asgar Jamneshan, TU-Dresden
Time: November 28, 2024 at 17:00
Title: Some Applications of Toposes of Measure-Theoretic Sheaves
Abstract: We construct toposes of sheaves on measure spaces and highlight the usefulness of interpreting certain structures from classical measure theory and functional analysis, combined with a Boolean internal logic, in applications to ergodic structure theory and vector duality.

Speaker: Pranav Pandit (ICTS-TIFR)
Title: Towards categorical Kähler geometry
Time: December 10, 2024 at 15:00
Abstract: The Donaldson-Uhlenbeck-Yau theorem describes a deep relationship between holomorphic vector bundles on Kähler manifolds and solutions to certain partial differential equations. I will report on progress towards formulating and proving an analogue of this theorem in categorical noncommutative geometry. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.

Speaker: Keremcan Doğan (Gebze Teknik Üniversitesi)
Title: Proto bialgebroids for exceptional geometries
Time: December 24, 2024 at 18:00:
Abstract: Recent advancements in the mathematics of dualities seem to be in favor of certain generalizations of differential geometric structures on algebroids. The success of the generalized geometry program for T-duality motivates the construction of analogous exceptional geometries suitable for U-duality. In particular the Drinfeld double structure for Lie bialgebroids is a crucial concept for T-duality. In this talk, we will introduce the notion of bialgebroid extending these ideas in the realm of U-duality. We will present a calculus framework on algebroids; both twistless and twistful cases. In order to focus on exceptional geometries, we extend T-duality notions in a more general class of algebroids, where the setting allows us to work with non-dual vector bundles of arbitrary rank. We will conclude the talk with a specific construction crucial for exceptional Drinfeld algebras in the context of higher Courant algebroids and Nambu-Poisson structure.

Speaker: Mehmet Akif Erdal (Yeditepe University, Istanbul)
Title: Fibration category structures on monoidal and enriched categories
Time: January 07, 2025, Tuesday at 18:00 Istanbul local time
Abstract: We first discuss Brown's category of fibrant objects structures on closed monoidal categories by means of some specific arrows called pseudo-cofibrations. These arrows are the ones whose pullback-power with (acyclic) fibrations are also (acyclic) fibrations; which can be defined whenever the underlying category is closed monoidal or enriched over a category of fibrant objects. Later we discuss the category of fibrant objects structures on enriched categories. If V is a closed monoidal category with a category of fibrant object structure on it and C is enriched over V and powered over a colimit dense subcategory of V, then under mild conditions C can also be made into a category of fibrant objects. We discuss constructions and properties of these structures and their extension to the equivariant setting. Lastly, we give some already existing and some new examples of such categories of fibrant objects and mention some applications.

Speaker: Simona Paoli (University of Aberdeen, UK)
Title: The weakly globular approach to higher categories
Time: January 21, 2025, Tuesday at 18:00 Istanbul local time (15:00 Aberdeen-UK local time)
Abstract: Higher categories are motivated by naturally occurring examples in diverse areas of mathematics, including homotopy theory, mathematical physics, logic and computer sciences. Several different approaches exist to formalize the notion of a higher category. In this talk I will give an overview of an approach to model 'truncated' higher categories: namely those having higher morphisms in dimensions 0 up to n only. These arise naturally in homotopy theory, in modelling the building blocks of topological spaces, called n-types.

Classically, in a higher category we have sets of objects and sets of higher morphisms. This is also called 'globularity condition' as it is the condition that gives rise to the globular shape of the higher morphisms in a higher category. Instead, in the so called weakly globular approach I have introduced, the objects and the higher morphism do not form a set but a structure only equivalent (in a higher dimensional sense) to a set. We call this 'weak globularity condition'.

One advantage of this approach is that it is possible to model a weak n-category using a rather rigid structure, namely an n-fold category satisfying additional conditions. These are the weakly globular n-fold categories. I will mention some applications of these structures to homological algebra, as well as a link between weak globularity and the notion of weak units in the case n=2. I will conclude with some conjectures for general dimension n.

Given the highly technical nature of this work, and in the interest of making the talk broadly accessible, I will concentrate on the main ideas and intuitions, but more details can be found in the references below:

About weakly globular n-fold categories:

· S. Paoli, Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories, Algebra and Applications 26, Springer (2019).

· S. Paoli, D. Pronk, A double categorical model of weak 2-categories, _Theory and Application of categories_, 28, (2013), 933-980.

About weak globularity and weak units:

· S. Paoli, Weakly globular double categories and weak units, arXiv:2008.11180 (2024).

An application of weakly globular n-fold categories to homological algebra:

· D. Blanc, S. Paoli, A model for the Andre-Quillen cohomology of an (\infty,1)-category, arXiv:2405.12674 (2024).

February 04, 2025: No seminar! (winter break)

Speaker: Juan Orendain (Case Western Reserve University)
Title: Crossed products of decorated bicategories and length of equipments
Time: March 6, 2025 at 18:00 Istanbul local time (10:00 Cleveland local time)
Abstract: Equipments are categorical structures of dimension 2 having two separate types of 1-arrows -vertical and horizontal- and supporting restriction and extension of horizontal arrows along vertical ones. Equipments are 2-functors satisfying certain conditions, but they can also be understood as double categories satisfying a fibrancy condition. In the zoo of 2-dimensional categorical structures, they fit nicely between 2-categories and double categories, and are generally considered as the 2-dimensional categorical structures where synthetic category theory is done, and in reasonable cases, where monoidal bicategories are more naturally defined.

In this talk I discuss the problem of lifting a 2-category into an equipment along a given category of vertical arrows, and how this problem allows us to define a notion of length on general double categories. The length of a double category is a number that roughly measures the amount of work one needs to do to reconstruct the double category from a bicategory along its set of vertical arrows. I will review the length of double categories, and I will discuss the crossed product of a decorated bicategory construction -a method for constructing different double categories from a given bicategory is presented-. One of the main ingredients of the construction are so-called canonical squares. I will show that in certain classes of equipments -fully faithful and absolutely dense- every square that can be canonical is indeed canonical. I will explain how from this, it can be concluded that fully faithful and absolutely dense equipments are of length 1, and so they can be 'easily' reconstructed from their horizontal bicategories.

Speaker: Vassily O. Manturov, Moscow Institute of Physics & Technology
Title: 10 years of G^k_n: towards invariants of knots and link
Time: March 18, 2025 at 18:00
Abstract: It has been 10 years since the author introduced groups Gkn depending on two natural numbers n>k and constructed invariants of many configuration spaces valued in such groups. https://www.arxiv.org/abs/1501.05208 . The first two natural invariants dealt with braids on n strands, n>3, valued in G3n and G4n. We shall discuss how to construct similar invariants for n-component links and describe various possible ways what to do with knots (single component). The approach uses closed braids and Markov moves. Many unsolved problems will be formulated.

Speaker: George Raptis
Title: From derivators and ∞-categories to ∞-derivators
Time: April 1, 2025 at 18:00
Abstract: The theory of derivators is an approach to homotopical algebra that focuses on the idea of enhancing the classical homotopy category ho(C) of a homotopy theory C by the collection of the homotopy categories of diagrams in C all at once. The resulting objects turn out to be much richer than the homotopy category alone and this viewpoint has been useful for expressing homotopical universal properties. At the same time, this approach is different from (and less strong than) the methods of higher category theory - which has been developed and used in recent years for related purposes with great impact in various areas of research. I will survey the basic theory, applications and examples of derivators, and then I will discuss the general notion of an ∞-derivator, as a natural higher categorical extension of ordinary derivators. This generalization is based on the use of the homotopy n-category, for 1≤n≤∞, it bridges the gap between derivators and ∞-categories, and it provides a common framework of reference for both types of objects/approaches.

Speaker: Si Li
Title: Stochastic process and algebraic index
Time: April 15, 2025 at 13:00 İstanbul local time (18:00 Beijing local time)
Abstract: We explain a stochastic approach to topological field theory and present a case study of quantum mechanical model and its relation to non-commutative geometry and algebraic index.

Speaker: Alexander Zimmermann (Université de Picardie)
Title: On the ring theory of differential graded algebras
Time: April 29, 2025, Tuesday at 18:00 Istanbul local time (17:00 Amiens local time)
Abstract: Let R be a commutative ring. Following Cartan (1954) a differential graded algebra (A,d) over R is a Z-graded R-algebra A with a homogeneous R-linear endomorphism d of degree 1 with d2=0 satisfying
d(a⋅b)=d(a)⋅b+(-1)∣a∣a⋅d(b)
for any homogeneous a, b ∈ A of degree ∣a∣, resp. ∣b∣. Similarly, a differential graded module is defined as a Z-graded A-module with an endomorphism δ of degree 1 and square 0 satisfying
δ(a⋅m)=d(a)⋅m+(-1)∣a∣a⋅δ(m)
for all homogeneous a ∈ A and m ∈ M . Until very recently the ring theory of differential graded algebras and differential graded modules remained largely unexplored. The case of acyclic differential graded algebras was completely classified by Aldrich and Garcia-Rozas in 2002 and the case of R being a field and A being finite dimensional was considered by Orlov in 2020, basically with geometric motivations in mind. In a more systematic study I studied basic ring theoretical questions, such as a notion of dg-Jacobson radicals, a dg- Nakayama lemma, Ore localisation of dg-algebras, and dg-Goldie's theorem. Most interestingly, several standard properties in general ring theory do not generalise, but some do. We give examples, and further classify dg-division rings and dg-separable dg-field extensions, and also a dg-version of the classical Levitzki-Hopkins theorem on artinian respectively semiprimary algebras.

Speaker: Oliver Lorscheid (University of Groningen)
Title: Zeta and K in F_1-geometry
Time: May 13, 2025, at 18:00 Istanbul local time/17:00 Groningen local time
Abstract: In this talk we give an overview of zeta functions and K-theory in F_1-geometry. We begin with a short historical introduction before we explain what the zeta functions of an F_1-scheme is. In the second part on K-theory, we begin with a reminder of Quillen's Q-construction before we give an impression how it is applied to a specific approach to F_1-geometry via monoid schemes. The talk is accessible without any preknowledge on F_1-geometry, and focus lies on conveying the grand ideas.

2023 - 2024

Speaker: Lakshya Bhardwaj, Mathematical Institute, Univ. of Oxford
Title: TQFTs and Gapped Phases with Non-Invertible Symmetries
Time: September 26, 2023 at 15:00 İstanbul time (13:00 Oxford time)
Abstract: I will discuss classification of topological quantum field theories (TQFTs) with non-invertible generalized/categorical symmetries. From a condensed matter point of view, this is related to the classification of gapped phases of systems with non-invertible symmetries. Although the general
formalism will be applicable to any spacetime dimension, I will provide concrete details in spacetime dimension d=2. As main examples, I will describe the only (1+1)d gapped phase with Ising symmetry which carries 3 vacua along with relative Euler terms, and four possible (1+1)d gapped phases with Rep(S_3) symmetry. Along the way, I will also discuss the order parameters for such gapped phases, which carry generalized charges under non-invertible symmetries.

Speaker: Dmitriy Rumynin, University of Warwick
Title: C_2-Graded groups, their Real representations and Dyson's tenfold way
Time: October 10, 2023 at 18:00 İstanbul time (16:00 Warwick time)
Abstract: A C_2-graded group is a pair: a group G and its index two subgroup H. Its Real representation is a complex representation of H with an action of the other coset G\H of odd elements in another way that needs to be chosen. Different choices lead to different theories.
Such representations appeared independently in three different disciplines: Algebra, Physics and Topology.
The goal of the talk is to review the formalism and various choices, including resulting theories.
The talk is based on my recent works with James Taylor (Oxford) and Matthew B. Young (Utah State).

Speaker: Ross Street, Macquarie University
Title: Could representations of your category be those of a groupoid?
Time: October 24, 2023 at 12:00 İstanbul time (20:00 Sydney time)
Abstract: By a representation of a category ℱ here is meant a functor from ℱ to a category V of modules over a commutative ring R. The question is whether there is a groupoid G whose category [G,V] of representations is equivalent to the category [ℱ,V] of representations of the given category ℱ. That is to say, is there a groupoid G such that the free V - category RG on G is Morita V - equivalent to the free V - category Rℱ on ℱ ? The groupoid G could be the core groupoid ℱinv of ℱ; that is, the subcategory of ℱ with the same objects but with only the invertible morphisms. Motivating examples come from Dold-Kan-type theorems and a theorem of Nicholas Kuhn [see “Generic representation theory of finite fields in nondescribing characteristic”, Advances in Math 272 (2015) 598–610]. The plan is to describe structure on ℱ which leads to such a result, and includes these and other examples.

Speaker: Nick Gurski, Case Western Reserve University
Title: Computing with symmetric monoidal functors
Time: November 07, 2023 at 15:00 İstanbul time (07:00 Cleveland time)
Abstract: Coherence theorems, while often technically complicated, serve a simple role: to make computations easier on the user. Abstract forms of coherence theorems often take one of two forms, either a strictification form or a diagrammatic form. The general, abstract kinds of coherence
theorems that would apply to symmetric or braided monoidal functors are of the strictification variety, but in practice the diagrammatic versions are often what one might need. I will present a general form of a diagrammatic coherence theorem applicable to monoidal functors (of any variety) or any other structure governed by a reasonably nice 2-monad. This is joint work with Niles Johnson.

Speaker: William Donovan, Yau Mathematical Sciences Center, Tsinghua Univ.
Title: Homological comparison of resolution and smoothing
Time: November 28, 2023 at 14:00 İstanbul time (19:00 Beijing time)
Abstract: A singular space often comes equipped with (1) a resolution, given by a morphism from a smooth space, and (2) a smoothing, namely a deformation with smooth generic fibre. I will discuss work in progress on how these may be related homologically.

Speaker: Merlin Christ, Institut de Mathématiques de Jussieu – Paris Rive Gauche
Title: Complexes of stable infinity-categories
Time: December 05, 2023 at 18:00 İstanbul time (16:00 Paris time)
Abstract: Abstract: A complex of stable infinity-categories is a categorification of a chain complex, meaning a sequence of stable infinity-categories together with a differential that squares to the zero functor. We refer to such categorified complexes as categorical complexes. We give a categorification of the totalization construction, which associates a categorical complex with a categorical multi-complex. Special cases include the totalizations of commutative squares or higher cubes of stable infinity categories. This can be used to construct interesting examples of categorical complexes, for instance coming from normal crossing divisors.
The study of categorical complexes can be seen as part of the conjectural/emerging subject of categorified homological algebra. We will also indicate a partial formalisation of this, based on the notion of a lax additive (infinity,2)-category, categorifying the notion of an additive 1-category. This talk is based on joint work with T. Dyckerhoff and T. Walde, see https://arxiv.org/abs/2301.02606.

Speaker: Félix Loubaton, Laboratoire J.A. Dieudonné, Université Côte d’Azur; MPI-Bonn
Title: Lax univalence for $(\infty,\omega)$-categories
Time: December 19, 2023 at 18:00 İstanbul time (16:00 Bonn time)
Abstract: The classical Grothendieck construction establishes an isomorphism between the (pseudo)functor $F:C\to Cat$ and the left Cartesian fibration $E\to C$. We can then show that $E$ is the lax colimit of the functor $F$. This presentation is dedicated to the generalization of this result for $(\infty,\omega)$- categories. After defining $(\infty,\omega)$-categories, we will state the lax univalence for $(\infty,\omega)$-categories. We'll then explain how this result allows us to express a strong link between Grothendieck construction for $(\infty,\omega)$-categories and the lax-colimits of $(\infty,\omega)$ - categories, similar to the classical case.

Speaker: Kadri İlker Berktav, Bilkent Üniversitesi
Title: Shifted contact structures on derived stacks
Time: January 16, 2024 at 18:00 İstanbul/Ankara time
Abstract: In this talk, we outline our program for the development of shifted contact structures in the context of derived algebraic geometry. We start by recalling some key notions and results from derived algebraic/symplectic geometry. Next, we discuss shifted contact structures on derived Artin stacks and report our results regarding their local theory, together with some future directions.

January 30, 2024: No seminar! (winter break)

Speaker: Nils Baas, Norges teknisk-naturvitenskapelige universitet
Title: Beyond Categories
Time: February 13, 2024 at 18:00 İstanbul time (16:00 Trondheim time)
Abstract: My talk will be philosophical. I will motivate the need to go beyond higher categories in order to get a good framework for many types of higher structures. This leads me to the notion of hyperstructures which I will motivate and explain. Initially this is a very general concept in order to cover both mathematical and applied aspects which I will explain. I will also relate to extended Field Theories.

Speaker: Meng-Chwan Tan, National University of Singapore
Title: Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification
Time: February 27, 2024 at 12:00 İstanbul time (17:00 Singapore time)
Abstract: We revisit Vafa-Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa-Witten equations. We physically derive (i) a novel Vafa-Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov-Witten invariants, (iii) a novel Vafa-Witten Floer homology assigned to three-manifold boundaries,(iv) a novel Vafa-Witten Atiyah-Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid-Manolescu in [1] about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. In essence, we will relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.

Speaker: Elena Dimitriadis Bermejo, Université Paul Sabatier
Title: A new model for dg-categories
Time: March 12, 2024
Abstract: Dg-categories have been very important in Algebraic Geometry for a really long time; but they are not without their issues. In order to solve these, current researchers have been turning to different models of infinity-categories for inspiration. Enriched infinity categories, dg-Segal categories, enriched quasi-categories... Following this flourishing field, in this talk we will define a new model for dg-categories inspired in Rezk's complete Segal spaces model for infinity-categories. During this talk we will define dg-Segal spaces, give its relationship to classical Segal spaces, use this to define complete dg-Segal spaces and its model structure and give a sketch of the proof of its equivalence to Tabuada's model structure of dg-categories. If time allows, we will say a word about some possible refinements of the model, and mention some work in progress surrounding its relationship to Mertens and Borges Marques' model of dg-Segal spaces.

Speaker: Walker Stern, Bilkent Üniversitesi
Title: Commutative and Frobenius algebras in span categories
Time: March 26, 2024 at 18:00 İstanbul/Ankara time
Abstract: In this talk, I will discuss the relation of span categories to various versions of the symplectic category. I will then expose the connection between simplicial objects and algebras in span categories, focusing on the 1- and 2-categorical cases to explicate the underlying intuitions. Finally, I will discuss recent work (joint with Ivan Contreras and Rajan Mehta) generalizing this correspondence to algebras with further structure, that is, to commutative and Frobenius algebras.

Speaker: John Huerta, Instituto Superior Técnico-Lisboa
Title: Poincaré duality for families of supermanifolds
Time: April 09, 2024 at 18:00 İstanbul time (16:00 Lisbon time)
Abstract: It is well known to experts, but seldom discussed explicitly, that smooth supergeometry is best done in families. This is also called the relative setting, and it implies that we need relative versions of standard supergeometric constructions. Such constructions include the de Rham complex familiar from ordinary differential geometry, but in the supergeometric setting, they also include more exotic objects, such as the Berezinian line bundle (whose sections are the correct objects to integrate over supermanifolds) and the related complex of integral forms, where the super version of Stokes' theorem lives. To work in families, we introduce relative versions of the de Rham complex and the integral form complex, and we prove that they satisfy a relative version of Poincaré duality. No background in supergeometry will be assumed for this talk.

Speaker: Elena Caviglia, University of Leichester
Title: 2-stacks and quotient 2-stacks
Time: April 23, 2024 at 18:00 İstanbul time (16:00 Leichester time)
Abstract: Stacks generalize one dimension higher the fundamental concept of sheaf. They are pseudofunctors that are able to glue together weakly compatible local data into global data. Stacks are a very important concept in geometry, due to their ability to take into account automorphisms of objects. While many classification problems do not have a moduli space as solution because of the presence of automorphisms, it is often nonetheless possible to construct a moduli stack. In recent years, the research community has begun generalizing the notion of stack one dimension higher. Lurie studied a notion of (∞, 1)-stack, that yields a notion of (2, 1)- stack for a trihomomorphism that takes values in (2, 1)-categories, when truncated to dimension 3. And Campbell introduced a notion of 2-stack that involves a trihomomorphism from a one-dimensional category into the tricategory of bicategories. In this talk, we will introduce a notion of 2-stack that is suitable for a trihomomorphism from a 2-category endowed with a bitopology into the tricategory of bicategories. The notion of bitopology that we consider is the one introduced by Street for bicategories. We achieve our definition of 2 stack by generalizing a characterization of stack due to Street. Since our definition of 2-stack is quite abstract, we will also present a useful characterization in terms of explicit gluing conditions that can be checked more easily in practice. These explicit conditions generalize to one dimension higher the usual stacky gluing conditions. A key idea behind our characterization is to use the tricategorical Yoneda Lemma to translate the biequivalences required by the definition of 2-stack into effectiveness conditions of appropriate data of descent. As a biequivalence is equivalently a pseudofunctor which is surjective on equivalence classes of objects, essentially surjective on morphisms and fully faithful on 2-cells, we obtain effectiveness conditions for data of descent on objects, morphisms and 2-cells. It would have been hard to give the definition of 2-stack in these explicit terms from the beginning, as we would not have known the correct coherences to ask in the various gluing conditions. Our natural implicit definition is instead able to guide us in finding the right coherence conditions. Finally, we will present the motivating example for our notion of 2-stack, which is the one of quotient 2-stack. After having generalized principal bundles and quotient stacks to the categorical context of sites, we aimed at a generalization of our theory one dimension higher, to the context of bisites, motivated by promising applications of principal 2- bundles to higher gauge theory. But there was no notion of higher dimensional stack suitable for the produced analogues of quotient prestacks in the two-categorical context. Our notion of 2-stack is able to fill this gap. Indeed, we have proven that, if the bisite satisfies some mild conditions, our analogues of quotient stacks one dimension higher are 2-stacks.

Speaker: Daniel Tubbenhauer, University of Sydney
Title: Counting in tensor products
Time: May 07, 2024 at 11:00 İstanbul time (18:00 Sydney time)
Abstract: This talk is an introduction to analytic methods in tensor categories with the focus on quantifying the number of summands in tensor products of representations and related structures. Along the way, we'll throw in plenty of examples to keep things interesting!

Speaker: Michael Shulman, The University of San Diego
Title: Higher Observational Type Theory
Time: May 21, 2024 at 20:00 İstanbul time (10:00 San Diego time)
Abstract: Homotopy Type Theory is a new approach to the foundations of mathematics, in which the basic objects of mathematics are not sets but homotopy types. It is natively isomorphism-invariant and well-adapted to computer formalization, and can be interpreted in higher toposes to give a synthetic language for internal constructions and proofs. It can also be explained intuitively to students, giving them access to higher structures while avoiding the complicated machinery of combinatorial homotopy theory; and it can be used as a programming language, to compute certain invariants of higher structures by simply running code derived from their definitions. However, until recently it was not known how to achieve both of these latter two properties simultaneously with a single formal system. In this talk I will introduce Homotopy Type Theory and its applications to higher structures from perspective of Higher Observational Type Theory; this is a new formal system for Homotopy Type Theory that, we hope, is both intuitively natural and computationally adequate. This is joint work in progress with Thorsten Altenkirch and Ambrus Kaposi.

Speaker: Özgün Ünlü, Bilkent Üniversitesi
Title: Infinity Operads as Simplicial Lists
Time: June 04, 2024 at 18:00 İstanbul/Ankara time
Abstract: In this talk, we will present a model for infinity operads. We will start by discussing how the category of colored nonsymmetric operads can be embedded in a category which we call the category of simplicial lists. Within this category, our model for infinity operads will generalize colored nonsymmetric operads in the same way that quasicategories generalize ordinary categories when embedded in the category of simplicial sets. Therefore, it is natural to refer to these infinity operads as quasioperads. Next, we will discuss a homotopy coherent nerve functor from the category of simplicial operads to the category of simplicial lists, which sends Kan complex enriched operads to quasioperads, analogous to the homotopy coherent nerve functor from the category of simplicial categories to the category of simplicial sets. Finally, we will discuss the homology of simplicial lists, and hence quasioperads, and perform some homology computations. This is joint work with Redi Haderi.

2022 - 2023

Speaker: Kadri İlker Berktav, Zurich University
Title: Geometric structures as stacks and geometric field theories
Time: Oct 25, 2022 at 17:00 Istanbul time
Abstract: In this talk, we outline a general framework for geometric field theories formulated by Ludewig and Stoffel. In brief, functorial field theories (FFTs) can be formalized as certain functors from an appropriate bordism category Bord to a suitable target category. Atiyah's topological field theories and Segal's conformal field theories are the two important examples of such formulation. Given an FFT, one can also require the source category to endow with a ''geometric structure''. Of course, the meaning of ''geometry'' must be clarified in this new context. To introduce geometric field theories in an appropriate way, therefore, we first explain how to define ''geometries'' using the language of stacks, and then we provide the so-called geometric bordism category GBord. Finally, we give the definition of a geometric field theory as a suitable functor on GBord.

Speaker: Aaron Mazel-Gee, Caltech
Title: Towards knot homology for 3-manifolds
Time: Nov 08, 2022 at 18:00 Istanbul time
Abstract: The Jones polynomial is an invariant of knots in R3 . Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin--Turaev using quantum groups. Khovanov homology is a categorification of the Jones polynomial of a knot in R3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds. In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (∞,2)-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.

Speaker: Can Yaylalı, Darmstadt University
Title: Derived F-zips
Time: Nov 22, 2022 at 17:00 Istanbul time
Abstract: The theory of F-zips is a positive characteristic analog of the theory of integral Hodge-structures. As shown by Moonen and Wedhorn, one can associate to any proper smooth scheme with degenerating Hodge-de Rham spectral sequence and finite locally free Hodge cohomologies an F- zips, via its n-th de Rham cohomology. Using the theory of derived algebraic geometry, we can work with the de Rham hypercohomology and show that it has a derived analog of an F-zip structure. We call these structures derived F-zips. We can attach to any proper smooth morphism a derived F-zip and analyze families of proper smooth morphisms via their underlying derived F-zip.

Speaker: Kürşat Sözer, McMaster University
Title: Crossed module graded categories and state-sum homotopy invariants of maps
Time: Dec 06, 2022 at 17:00 Istanbul time
Abstract: A well-known fact is that groups are algebraic models for 1-types. Generalizing groups, crossed modules model 2-types. In this talk, I will introduce the notion of a crossed module graded fusion category which generalizes that of a fusion category graded by a group. Then,using such categories, I will construct a 3-dimensional state-sum homotopy quantum field theory (HQFT) with a 2-type target. Such an HQFT associates a scalar to a map from a closed oriented 3-manifold to the fixed 2- type. Moreover, this scalar is invariant under homotopies. This HQFT generalizes the state-sum Turaev-Virelizier HQFT with an aspherical target. This is joint work with Alexis Virelizier.

Speaker: Ödül Tetik, Zurich University
Title: Field theory from [and] homology via [are] “duals”
Time: Dec 20, 2022 at 17:00 Istanbul time
Abstract: I will introduce the notion of the 'Poincaré' or 'Koszul' dual of a stratified space with tangential structure (TS), whose construction in general is as yet an open problem. Then I will outline (the finished part of) ongoing work on defining a functorial field theory, given, as input, a disk-algebra with TS. This recovers the framed case, which was proposed by Lurie (later picked up by Calaque and Scheimbauer): duals of stably-framed bordisms are euclidean spaces with flag-like stratifications. In particular, this notion explains the 'shape' of the higher Morita category of En-algebras when expressed in terms of factorization algebras, and gives a natural definition of Morita categories of disk-algebras with any TS. If time permits, I will propose a simple Poisson-structured version of this procedure which should construct, using Poisson additivity, extended classical gauge theories given only the 1-shifted Poisson algebra of bulk observables.
Video link

January 3, 2023: New Year holiday!

Speaker: Neslihan Gügümcü, İzTech University
Title: On a quantum invariant of multi-knotoids
Time: Jan 17, 2023 at 17:00 Istanbul time
Abstract: Knotoids are immersed arcs in surfaces, introduced by Vladimir Turaev. Knotoids in the 2-sphere can be considered as open knot diagrams with two endpoints that can lie anywhere in S2. In this sense, the theory of spherical knotoids extends the theory of knots in the Euclidean 3-space, and the classification problem of knots generalizes to knotoids in an interesting way with the existence of open ends. In this talk we will present multi-knotoids and an Alexander polynomial type invariant for them by utilizing a partition function involving a solution of the Yang-Baxter equation. This talk is a joint work with Louis Kauffman.

Speaker: David Roberts, University of Adelaide
Title: Low-dimensional higher geometry: a case study
Time: Jan 31, 2023 at 10:00 Istanbul time
Abstract: Considerations from several different areas of mathematics have prompted the development of so-called higher geometry: the study of categorified analogues of geometric structures. Despite being studied for nearly two decades, few examples that capture non-abelian phenomena have been constructed. And here by "constructed", we mean to the level that would satisfy traditional differential geometers, as opposed to the kind of construction that category theorists are comfortable with. To this end, I will describe a new framework to work with bundle 2-gerbes, which from a higher- category point of view are certain types of truncated descent data for ∞-stacks on a manifold. The description is sufficient to undertake concrete computations more satisfying to traditional differential geometers and mathematical physicists. I also describe explicit geometric examples that can be constructed using our framework, including infinite families of explicit geometric string structures.

Speaker: Yusuf Barış Kartal, University of Edinburgh
Title: Frobenius operators in symplectic topology
Time: Feb 21, 2023 at 17:00 Istanbul time
Abstract: One can define the Frobenius operator on a commutative ring of characteristic p as the p th power operation, and this has generalizations to a larger class of commutative rings, and even to topological spaces and spectra. Spectra with circle actions and Frobenius operators are called cyclotomic spectra. The simplest example is the free loop space, and important examples arise in algebraic and arithmetic geometry as topological Hochschild homology of rings and categories. By topological reasons and mirror symmetry, it is natural to expect such a structure to arise in symplectic topology-- more precisely in closed string Floer theory''. In this talk, we will explain how to construct such spectra using Hamiltonian Floer theory, i.e. by using holomorphic cylinders in symplectic manifolds. Joint work in progress with Laurent Cote.
Video link

Speaker: Julia Plavnik, Indiana University, Bloomington:
Title: On the classification of modular categories
Time: Feb 28, 2023 at 18:00 Istanbul time
Abstract: Modular categories are intricate organizing algebraic structures appearing in a variety of mathematical subjects including topological quantum field theory, conformal field theory, representation theory of quantum groups, von Neumann algebras, and vertex operator algebras. They are fusion categories with additional braiding and pivotal structures satisfying a non-degeneracy condition. The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories, and we will also give some concrete examples to have a better understanding of their structures. I will give an overview of the current situation of the classification program for modular categories, with a particular focus on the results for odd-dimensional modular categories, and we will mention some open directions in this field.
Video link

Speaker: Olivia Caramello, Gelfand Chair, IHES and Grothendieck Inst.:
Title: Grothendieck toposes as unifying “bridges” in mathematics
Time: Mar 14, 2023 at 17:00 Istanbul time
Abstract: I will explain the sense in which Grothendieck toposes can act as unifying 'bridges' for relating different mathematical theories to each other and studying them from a multiplicity of points of view. I shall first present the general techniques underpinning this theory and then discuss a number of selected applications in different mathematical fields.

Speaker: Redi Haderi, Bilkent University:
Title: A simplicial category for higher correspondences
Time: Mar 28, 2023 at 17:00 Istanbul time
Abstract: Correspondences between simplicial sets (and ∞-categories) are a generalization of the notion of profunctor between categories. It is known that functors between categories are classified by lax diagram of profunctors. We will present this fact from the lens of double category theory. Then, we will show how simplicial sets, simplicial maps and correspondences are organized in a simplicial category (this is a weak simplicial object in categories). A simplicial category may be regarded as a 2-fold version of a simplicially enriched category, and hence some ideas from double category theory apply. In particular we formulate the fact that simplicial maps are classified by diagrams of correspondences. As a corollary, we obtain a formulation of Lurie's prediction that inner fibrations are classified by diagrams of correspondences between ∞-categories. Reference: https://arxiv.org/abs/2005.11597

Speaker: Theo Johnson-Freyd, Perimeter Institute for Theoretical Physics:
Title: Higher algebraic closure
Time: Apr 11, 2023 at 18:00 Istanbul time
Abstract: Deligne's work on Tannakian duality identifies the category sVec of super vector spaces as the "algebraic closure" of the category Vec of vector spaces (over C). I will describe my construction, joint with David Reutter, of the higher-categorical analog of sVec: the algebraic closure of the n-category of "n-vector spaces". The construction mixes ideas from Galois theory, quantum physics, homotopy theory, and fusion category theory. Time permitting, I will describe the higher-categorical Galois group, which turns out to have a surgery-theoretic description through which it is almost, but not quite, the group PL.

Speaker: Erdal Ulualan, Kütayha Dumlupınar University:
Title: Functors From Simplicial Groups to Higher Dimensional Algebraic Structures
Time: Apr 25, 2023 at 17:00 Istanbul time
Abstract: Bu çalışmada bir simplisel grubun Moore kompleksinde tanımlı olan hiper çaprazlanmış kompleks çiftleri kullanılarak parçalanmış simplisel gruplar ile cebirsel modeller arasındaki ilişkiler verilecektir. 1-parçalanmış simplisel grubun bir çaprazlanmış modülü nasıl modellediği ve 1-parçalanmış bisimplisel grubun bir çaprazlanmış kareyi nasıl modellediği gösterilecektir. Sonuç olarak, bu ilişkileri genelleştirerek 1-parçalanmış n-boyutlu multisimplisel grubun bir çaprazlanmış n-küpü nasıl modellediğini göstereceğiz. In this work, we will explain the connection between truncated simplicial groups and algebraic models in term of hyper-crossed complex pairings in the Moore complexes of simplicial groups.We will show that a 1-truncated simplicial group gives a crossed module and a 1-truncated bisimplicial group gives a crossed square. By generalising these relationships to higher dimensions,we will show that a 1-truncated n-dimensional simplicial group gives a crossed n-cube.

Speaker: Claudia Scheimbauer, Technische Universität München
Title: A universal property of the higher category of spans and finite gauge theory as an extended TFT
Time: May 09, 2023 at 18:00 İstanbul time
Abstract: I will explain how to generalize Harpaz’ universal property of the (∞,1)-category of spans to the higher category thereof. The crucial property is “m-semiadditivity”, which generalizes usual semiadditivity of categories. Combining this with the finite path integral construction of Freed-Hopkins-Lurie-Teleman this yields finite gauge theory as a fully extended TFT. This is joint work in progress with Tashi Walde.

Speaker: Atabey Kaygun, Istanbul Technical Univ.
Title: Dold-Kan equivalence and its extensions
Time: May 23, 2023 at 17:00 İstanbul time
Abstract: The Dold-Kan Correspondence is an equivalence between the category of differential graded objects and the category of simplicial objects on an abelian category. This equivalence is best understood within the context of Quillen model categories. However, a more straightforward interpretation using the representation theory of small categories is possible. We will demonstrate that the Dold-Kan equivalence can be expressed through specific induction and restriction functors, paving the way for similar equivalences for crossed-simplicial objects. There are extensions to the
Dold-Kan Correspondence in this context, with the Dwyer-Kan equivalence between the category of duplicial objects and the category of cyclic objects over an abelian category being a notable example. We will also show that the Dwyer-Kan equivalence can be incorporated into the framework we initially developed for the Dold-Kan Correspondence. Lastly, we will discuss further extensions.
This research is a joint work with my PhD student, Haydar Can Kaya.

2021 - 2022

Speaker: Louis H Kauffman, UIC
Title: Introduction to Virtual Knot Theory
Time: November 8 2021 (11 AM Chicago - 8 PM Istanbul)
Abstract: Virtual knot theory studies knots and links embedded in thickened surfaces. This is a fundamental case of knots and links in three dimensional manifolds, and it includes embeddings in the three dimensional sphere, since knots and links in a thickened two dimensional sphere are the same topologically as their embeddings in the three sphere. We explain a diagrammatic and combinatorial approach to these problems. By using diagrams in the plane, or on the two sphere using virtual crossings we can represent all virtual knots up to 1-handle stabilization in their thickened surfaces. The diagrammatic theory leads to the construction of many new invariants and to the reconsideration of known invariants. The talk will introduce these structures with many examples, and it will be self-contained.
Video link

Speaker: Masanori MORISHITA , Kyushu University, Japan
Title: Arithmetic topology and arithmetic TQFT
Time: January 25 2022 (20.30 Kyushu - 14.30 Istanbul)
Abstract: I will talk about some topics in arithmetic topology, related with class field theory, and then an arithmetic analog of Dijkgraaf-Witten topological quantum field theory
Video link
Seminar Notes

Speaker: Berkan Üze (Boğaziçi University, İstanbul)
Title: Higher Geometry
Time: February 07, 2022 (14:30 İstanbul local time)
Abstract: An informal introduction to the “Higher Geometry” of Sati and Schreiber. The main references are: arXiv:2008.01101 and arXiv:2112.13654.

Speaker: Urs Schreiber, NYUAD Abu Dhabi and Czech Academy of Science, Prague
Title: Higher and equivariant bundles
Time: February 8, 2022 Tuesday at 14:30 Istanbul/12:30 Prague/15:30 Abu Dhabi time
Abstract: The natural promotion of the classical concept of (principal) fiber bundles to “higher structures”, namely to equivariant principal infinity-bundles internal to a singular-cohesive infinity-topos, turns out to be a natural foundation for generalized cohomology theory in the full beauty of "twisted equivariant differential non-abelian cohomology of orbifolds", and as such for much of the higher homotopical mathematics needed at the interface of algebraic topology, geometry and mathematical quantum physics. This talk gives some introduction and overview, based on joint work with H. Sati (arXiv:2008.01101, arXiv:2112.13654). Talk slides and further pointers at.
Video link
Access Passcode: CkXFa4^J
Jiri Narozny's question:
May I ask for the title of Dugger's article about cofibrant resolution of general smooth stack please?
Urs Schreiber's response
I was referring to Cor. 9.4 in Dugger's "Universal Homotopy theories" https://arxiv.org/abs/math/0007070 https://arxiv.org/pdf/math/0007070.pdf#page=21
It's relevance is highlighted in several of my articles, such as in Prop. 3.2.24 on p. 101 of "Equivariant principal infinity-bundles" https://arxiv.org/pdf/2112.13654.pdf#page=101

Speaker: Kadri Ilker Berktav, METU
Title: Symplectic Structures on Derived Schemes
Time: February 22 2022 at 14:30 Istanbul
Abstract: This is an overview of the basic aspects of shifted symplectic geometry on derived schemes. In this talk, we always study objects with higher structures in a functorial perspective, and we shall focus on local models for those structures. We begin with background material from algebra and from derived algebraic geometry. To be more specific, the basics of commutative differential graded K-algebras (cdga's) and their cotangent complexes will be revisited. In the second part of the talk, using particular cdgas as local models, we shall introduce the notion of a (closed) p-form of degree k on an affine derived K-scheme with the concept of a non-degeneracy. As a particular case, we shall eventually define a k-shifted symplectic structure on an affine derived K-scheme and outline the construction of Darboux-like local models.
Video link
Seminar Notes

Speaker: Berkan Üze (Boğaziçi University, Istanbul)
Title: A Glimpse of Noncommutative Motives
Time: 08 03 2022 (International Women's Day!), Tuesday at 14:30 Istanbul local time.
Abstract: The theory of motives was conceived as a universal cohomology theory for algebraic varieties. Today it is a vast subject systematically developed in many directions spanning algebraic geometry, arithmetic geometry, homotopy theory and higher category theory. Following ideas of Kontsevich, Tabuada and Robalo independently developed a theory of “noncommutative” motives for DG-categories (such as enhanced derived categories of schemes) which encompasses the classical theory of motives and helps assemble so-called additive invariants such as Algebraic K-Theory, Hochschild Homology and Topological Cyclic Homology into a motivic formalism in a very precise sense of the word. We will review the fundamental concepts at work, which will inevitably involve a foray into the formalism of enhanced and higher categories.We will then discuss Kontsevich’s notion of a noncommutative space and introduce noncommutative motives as “universal additive invariants” of noncommutative spaces. We will conclude by offering a brief sketch of Robalo’s construction of the noncommutative stable homotopy category, which is directly in the spirit of Voevodsky’s original construction. This talk contains no original work and is intended as an expository recapitulation.
Zoom link
Meeting ID: 979 9971 6753
Passcode: 000110

Speaker: Semih Özlem (Feza Gürsey Center for Physics and Mathematics, İstanbul)
Title: Motivic Galois Groups over Number Fields
Time: March 15, 2022 (16:00 Istanbul – 16:30 Tehran - 18:30 Prayagraj local time)

Speaker: Mehmet Akif ERDAL, Yeditepe University
Title: Homotopy theory of monoid actions via group actions and an Elmendorf style theorem
Time: Tuesday, April 12, 2022 at 14:30 Istanbul time
Abstract: For a group G and a collection of subgroups Y of G, the orbit category O_{Y} is the category whose objects are G-orbits G/H for each Hin Y and whose morphisms are G-equivariant maps in between. Due to Elmendorf's Theorem that the category G-spaces and the category of contravariant O_{Y}-diagrams of spaces have equivalent homotopy theories. This provides a great convenience when studying G-equivariant homotopy theory since one can reduce it to non-equivariant homotopy theory of associated fixed point systems. In this talk, we describe a non-trivial extension of this idea to the actions of monoids. Let M be a monoid and G(M) be its group completion, Z be a collection of submonoids of M and for each Nin Z, Y_N be a collection of subgroups of G(N). First we will show that the category of M-spaces and M-equivariant maps admits a model structure in which weak equivalences and fibrations are determined by the standard equivariant homotopy theory of G(N)-spaces for each N in Z. Then, we will show that this model structure is Quillen equivalent to the projective model structure on the category of contravariant O_{Z,Y}-diagrams of spaces, where O_{Z,Y} is the category whose objects are induced orbits Mtimes_N G(N)/H for each Nin Z and H in Y_N and morphisms are M-equivariant maps. Finally, if time permits, we will state some applications.
Zoom link
Meeting ID: 943 0442 7243
Passcode: 023255

Speaker: Haldun Özgür Bayındır, City University of London
Title: Adjoining roots to ring spectra and algebraic K-theory
Time: Tuesday, April 26, 2022 at 14:30 Istanbul time
Abstract: The category of spectra captures an important part of the complexity of topological spaces while providing generalizations of many important notions in homological algebra.
In this work, we develop a new method to adjoin roots to ring spectra and show that this process results in interesting splittings in algebraic K-theory.
In the first part of my talk, I will provide motivation for algebraic K-theory and highly structured ring spectra. After this, I will discuss trace methods, a program that provides computational tools for algebraic K-theory, and introduce our results.
This is a joint work in progress with Tasos Moulinos and Christian Ausoni.
Video link

Speaker: Juan Orendain, Universidad Nacional Autónoma de México-UNAM
Title: Higher lattice gauge fields and cubical ω-groupoids
Time: Tuesday, May 10, 2022 at 19:00 Istanbul time
Abstract: Gauge fields describe parallel transport of point particles along curves, with respect to connections on principal bundles. This data is captured as a smooth functor from the smooth path groupoid of the base manifold into the delooping groupoid of the structure group, plus gluing data. Lattice gauge fields do this for discretized versions of a base manifold. A lattice gauge field is thus a functor from a discrete version of the path groupoid to a delooping groupoid. Lattice gauge fields are meant to serve as discrete approximations of regular gauge fields.
Higher gauge fields describe parallel transport of curves along surfaces, of surfaces along volumes, etc. Several versions of 2-dimensional gauge field have appeared in the literature. I will explain how to extend these ideas to lattice gauge fields on all dimensions, using Brown's cubical homotopy $omega$-groupoid construction associated to filtered spaces, implementing a discrete notion of thin homotopy along the way.
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Speaker: Tatsuki Kuwagaki, Kyoto University
Title: An introduction to perverse schober
Time: Tuesday, May 24, 2022 at 14:30 Istanbul time (20:30 Kyoto time)
Abstract: A perverse sheaf is the topological counterpart of a differential equation with (regular) singularities. A perverse schober is "a category-valued perverse sheaf". It consists of monodromy of categories and their behaviors around singularities. The notion of perverse schober quite naturally appears in many contexts e.g., mirror symmetry. In this talk, I'll give an introduction to a very elementary part of perverse schober and related topics.
Video link