### Yüksek Yapılar Seminerleri - Güz 2021

Speaker: Louis H Kauffman, UICTitle: 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.

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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.

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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.

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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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