Yüksek Yapılar Seminerleri/ Higher Structures Seminars

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

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: John Huerta, Instituto Superior Técnico
Time: January 02, 2024

Speaker: Kadri İlker Berktav, Bilkent Üniversitesi
Time: January 16, 2024

Speaker: John Huerta, Instituto Superior Técnico
Time: January 02, 2024

Speaker: K. İlhan İkeda, Feza Gürsey Center for Physics and Mathematics
Time: January 30, 2024

Speaker: Nils Baas, Norges teknisk-naturvitenskapelige universitet
Time: February 13, 2024

Speaker: Meng-Chwan Tan, National University of Singapore
Time: February 27, 2024

Speaker: Elena Dimitriadis Bermejo, Université Paul Sabatier
Time: March 12, 2024

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.
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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.
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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.
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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.
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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
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Seminar Notes

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.
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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: 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.
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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.
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