Higher Structures Research Group

Meetings are held on Wednesdays or Thursdays at 18:00 in the Ikeda room.

• Mar. 04 (Wed) Intro to higher category theory Speaker: Muhammed Şen

  Reference: Dyckerhoff Notes

  Our ultimate goal is to establish some background for CH-1 and 2 of Gaitsgory and Rozenblyum's book.

• Mar. 12 (Thu) Intro to higher category theory-II Speaker: Muhammed Şen

  (Shifted from Wednesday due to Pi Day events). Muhammed will keep discussing categorical constructions and key tools.

• Mar. 27 (Fri) Condensed TQFTs Guest Speaker: Kazım İlhan İkeda, BOUN-FGC

  He will be visiting our department and giving a talk in the department's general seminar. He is also one of the founders of the Higher Structures group at FCG.

• Apr. 01 (Wed) Intro to higher category theory-III Speaker: Muhammed Şen

  Muhammed will keep discussing categorical constructions and key tools.

• Apr. 09 (Thu) Intro to higher category theory-IV Speaker: Efe İzbudak

  Efe will take the wheel and continue with categorical constructions, following our main reference.

• Apr. 17 (Fri) Extended Functorial Field Theories and neighboring subjects Guest Speaker: Kürşat Sözer, Université de Lille

  He will be visiting our department and has kindly agreed to organize an activity with our group at 14:00.

• Apr. 22 (Wed) Intro to higher category theory-V Speaker: Efe İzbudak

  Efe will continue with categorical constructions, following our main reference.

• Apr. 30 (Thu) Intro to higher category theory-VI Speaker: Efe İzbudak

  Efe will continue his lecture on higher categorical constructions, following our main reference.

• May 15 (Fri) Intro to higher category theory-VII Speaker: Efe İzbudak

  Efe will continue his lecture on higher categorical constructions, following our main reference.

  HS Production proudly presents: Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem, arXiv:2605.08394.

• May 28 (Thu) On the Ludewig-Stoffel definition of geometry Speaker: A. Emirhan Gür

  Abstract pending.

• May 30 (Sat) Construction of the stack of geometric bordisms Speaker: A. Emirhan Gür

  In this talk, we will discuss the moduli stack of geometric bordisms $\mathcal{B}ord_n^{(X, \omega)}$, focusing on its shifted symplectic structure of degree $k = 2-n$. We will explore the functor $Z: \mathcal{B}ord_n \to \mathcal{C}$ and its implications in mathematical physics.

• AKSZ Construction for Shifted Contact Structures (2026). Efe İzbudak, Kadri İlker Berktav. arXiv:2606.11179.
  This paper establishes the AKSZ theorem for shifted contact structures and its applications. In brief, to resolve certain obstructions, we first define the quotient mapping stack as the quotient of the symplectified mapping space by the constant multiplicative group action. We then prove that if $X$ is an $n$-shifted contact derived Artin stack and $Y$ is an $\mathcal{O}$-compact, $d$-oriented derived stack, the quotient mapping stack $$[\mathrm{Map}(Y, \widetilde{X})/\mathbb{G}_m]$$ admits an $(n-d)$-shifted contact structure.

  In addition, by formalizing the derived analogue of the graded contact AKSZ formalism, we also introduce the notion of weak shifted contact structures in derived algebraic geometry and prove that under global trivialization of the contact line bundle, the unmodified mapping stack inherits a weak contact structure.

  Extending this setup to spaces with boundary, we demonstrate that derived fillings naturally induce Legendrian morphisms between quotient mapping stacks, and that topological gluing of cobordisms evaluates to derived Legendrian intersections. Furthermore, we trace the transgression of the canonical shifted $1$-form to prove that our quotient mapping stacks satisfy the derived Classical Master Equation (CME).

  As applications of our quotient mapping stack formalism, we define the derived analogues of specific topological field theories, including the Jacobi, Courant-Jacobi, and Loop Space Sigma Models. Finally, by composing this geometric construction with the perverse linearization in a companion paper, we elevate these moduli spaces to generate Cohomological Contact Extended Topological Field Theories.

  @misc@misc@misc{i̇zbudak2026akszconstructionshiftedcontact,
        title={AKSZ Construction for Shifted Contact Structures}, 
        author={Efe İzbudak and Kadri İlker Berktav},
        year={2026},
        eprint={2606.13866},
        archivePrefix={arXiv},
        primaryClass={math.AG},
        url={https://arxiv.org/abs/2606.13866},
  }

• Equivariant Contact Darboux Quotients and Perversely Categorified Legendrian Correspondences (2026). Efe İzbudak. arXiv:2606.11179.
  Prior work has shown that shifted contact derived Artin stacks admit smooth Darboux atlases. However, establishing enumerative invariants and linearizing these categorical structures requires equivariant local models. Working over an algebraically closed field $\mathbb{K}$ of characteristic zero, we establish an equivariant Darboux theorem for $-1$-shifted contact derived Artin stacks. We prove that, in the smooth topology, these stacks admit smooth atlases by the derived contact Darboux scheme $\Delta\mathrm{loc}(s)$ associated to the derived discriminant locus of a relative section $s$. In the presence of reductive stabilizers $G$, this refines to the equivariant geometric quotient stack $[\Delta\mathrm{loc}(s)/G]$. By applying the BBDJS minimal model to the derived symplectification and descending algebraically along the structural free $\mathbb{G}_m$-action, we construct an $\ell$-adic perverse sheaf on any oriented $-1$-shifted contact stack. We utilize Verdier's specialization equivalence for monodromic sheaves to equip this perverse sheaf with a tame geometric monodromy automorphism $T$. This structure allows for the extraction of derived enumerative invariants via the $\ell$-adic Grothendieck-Lefschetz trace, thereby resolving the issue of generic topological acyclicity.

  The content of the other main results in this paper relies on a prior work, in which we have shown that derived intersections of $n$-shifted Legendrians yield $(n-1)$-shifted contact stacks and formulated the non-linear 2-categories of Legendrians $\mathcal{F}_c(X)$ and $Leg_n$. Using this geometric setup, we formulate in this paper a contact analogue of Joyce's conjecture to linearize these structures. We then construct the categorified Legendrian 2-categories $\mathfrak{L}\mathcal{F}c(X)$ and $LLeg_0$ via $\ell$-adic Fourier-Mukai pull-push functors, connecting the study of derived contact moduli spaces to microlocal sheaf theory.

  @misc@misc{i̇zbudak2026equivariantcontactdarbouxquotients,
        title={Equivariant Contact Darboux Quotients and Perversely Categorified Legendrian Correspondences}, 
        author={Efe İzbudak},
        year={2026},
        eprint={2606.11179},
        archivePrefix={arXiv},
        primaryClass={math.AG},
        url={https://arxiv.org/abs/2606.11179},
  }

• A Derived Legendrian Category for Shifted Contact Stacks (2026). Efe İzbudak, Kadri İlker Berktav. arXiv:2605.13792.
  We construct the derived Legendrian category $\mathcal{F}_c(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\infty, 2)$-category $Leg_n$ of Legendrian correspondences in the context of derived algebraic geometry, with several applications to moduli theory. In brief, the objects of the category $\mathcal{F}_c(X)$ are Legendrian morphisms; the morphism spaces and composition operations are defined using equivariant descent. We also establish that $\mathcal{F}_c(X)$ embeds into an $(\infty, 2)$-category of spans defined by the AKSZ construction. We further evaluate topological cobordisms as Lagrangian correspondences to define derived Legendrian surgery.

  @misc{izbudak2026derived,
      title={A Derived Legendrian Category for Shifted Contact Stacks},
      author={Efe İzbudak and Kadri İlker Berktav},
      year={2026},
      eprint={2605.13792},
      archivePrefix={arXiv},
      primaryClass={math.AG}
  }

• Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem (2026). Efe İzbudak, Kadri İlker Berktav. arXiv:2605.08394.
  The classical transversality lemma of contact geometry constructs a contact structure on a hypersurface transverse to a Liouville vector field using point-set topology and local flows. This paper translates the classical transversality lemma into the context of derived algebraic geometry and provides the derived Legendrian intersection theorem, along with various applications to moduli theory.

  In brief, we first prove that taking the quotient of a derived symplectic space descends the symplectic data to a contact structure, avoiding a transverse hypersurface, where the fundamental vector field of a weight 1 $\mathbb{G}_m$-action, in the derived setting, replaces the classical Liouville vector field. Secondly, the derived Legendrian intersection theorem is proven using base change, an $\infty$-categorical descent cube, and $\mathbb{G}_m$-equivariant lifts along the symplectification projection.

  As applications of the main results, we first examine the derived geometry of the discriminant loci of 1-jet bundles and show that these loci carry a $(-1)$-shifted contact structure. In addition, we show that our results apply to certain moduli problems, including projective Higgs bundles, $\ell$-adic local systems, and Lie 2-groups, and we provide further examples of contact derived moduli stacks.

  @misc{izbudak2026equivariant,
      title={Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem},
      author={Efe İzbudak and Kadri İlker Berktav},
      year={2026},
      eprint={2605.08394},
      archivePrefix={arXiv},
      primaryClass={math.AG}
  }

• On higher structures in mathematical physics (2026). Kadri İlker Berktav. IOP Journal of Physics: Conference Series.
  Abstract pending.