ODTÜ-BİLKENT Algebraic Geometry Seminar
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**** 2020 Fall Talks ****

This semester we plan to have all our seminars on Zoom

466. Zoom, 9 October 2020, Friday, 15:40

     Alexander Degtyarev-[Bilkent] - Counting 2-planes in cubic 4-folds in $\mathbb{P}^5$
         
     

     Abstract: (work in progress joint with I. Itenberg and J.Ch. Ottem) We use the global Torelli theorem for cubic 4-folds (C. Voisin) to establish the upper bound of 405 2-planes in a smooth cubic 4-fold. The only champion is the Fermat cubic. We show also that the next two values taken by the number of 2-planes are 357 (the champion for the number of *real* 2-planes) and 351, each realized by a single cubic. To establish the bound(s), we embed the appropriately modified lattice of algebraic cycles to a Niemeier lattice and estimate the number of square 4 vectors in the image. The existence is established my means of the surjectivity of the period map. According to Schütt and Hulek, the second best cubic with 357 planes can be realized as a hyperplane section of the Fermat cubic in $\mathbb{P}^6$. If time permits, I will also explain that essentially the same arithmetical reduction answers another geometric question, viz. the maximal number of conics in a sextic surface in $\mathbb{P}^4$. (It would be nice to find a geometric relation between the two.)  The two best numbers of conics are 285 (a single surface) and 261 (three Galois conjugate surfaces; one of them maximizes the number of real conics in a real sextic surface).

      
       
     
467. Zoom, 16 October 2020, Friday, 15:40

     Emre Can Sertöz-[Max Planck-Bonn] - Separating periods of quartic surfaces
         
     

     Abstract: Kontsevich--Zagier periods form a natural number system that extends the algebraic numbers by adding constants coming from geometry and physics. Because there are countably many periods, one would expect it to be possible to compute effectively in this number system. This would require an effective height function and the ability to separate periods of bounded height, neither of which are currently possible. In this talk, we introduce an effective height function for periods of quartic surfaces defined over algebraic numbers. We also determine the minimal distance between periods of bounded height on a single surface. We use these results to prove heuristic computations of Picard groups that rely on approximations of periods. Moreover, we give explicit Liouville type numbers that can not be the ratio of two periods of a quartic surface. This is ongoing work with Pierre Lairez (Inria, France).

         
     
     
468. Zoom, 23 October 2020, Friday, 15:40

     Sinan Ünver-[Koç] - Infinitesimal regulators
         
     

     Abstract: We will describe a construction of infinitesimal invariants of thickened   one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the  Milnor range. The construction also allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders.  Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy,  the infinitesimal version  cannot be obtained from their classical counterparts through a limiting process.

       
       
     
469. Zoom, 6 November 2020, Friday, 15:40

     Kâzım İlhan İkeda-[Boğaziçi] - Yoga of the Langlands reciprocity and functoriality principles
         
     

     Abstract: I shall describe my reflections on the Langlands reciprocity and functoriality principles. Those principles of Langlands are one of the fundamental driving forces of current mathematical research. Here, the term ``yoga'' appearing in the seminar title, which is introduced and used extensively by Grothendieck, means ``meta-theory''. Let $K$ is a number field. The local Langlands group $L_{K_\nu}$ of $K_\nu$ is defined by $L_{K_\nu}=WA_{K_\nu}=W_{K_\nu}\times\mathrm{SL}(2,\mathbb C)$ if $\nu\in\mathbb h_K$, and by $L_{K_\nu}=W_{K_\nu}$ if $\nu\in\mathbb a_K$, where $W_{K_\nu}$ denotes the local Weil group of $K_\nu$. For each $\nu\in\mathbb h_K$, fix a Lubin-Tate splitting $\varphi_{K_\nu}$. The local non-abelian norm-residue homomorphism \begin{equation*} \{\bullet,K_\nu\}_{\varphi_\nu}:{}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}\xrightarrow{\sim}W_{K_\nu} \end{equation*} of $K_\nu$ is defined and studied in the papers by E. Serbest and the author, where ${}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}$ is a certain topological group constructed using Fontaine-Wintenberger theory of fields of norms. Fix $\underline{\varphi}=\{\varphi_{K_\nu}\}_{\nu\in\mathbb h_K}$ and define the non-commutative topological group $\mathscr {WA}_K^{\underline{\varphi}}$, which depends only on $K$, by the ``restricted free topological product'' \[ \mathscr {WA}_K^{\underline{\varphi}}:= {\ast_{\nu\in\mathbb h_K}}' \left({}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}\times\mathrm{SL}(2,\mathbb C): {}_1{\nabla_{K_\nu}^{(\varphi_{K_\nu})}}^{\underline 0}\times\mathrm{SL}(2,\mathbb C) \right)\ast W_\mathbb R^{\ast r_1}\ast W_\mathbb C^{\ast r_2}. \] Here, $r_1$ and $r_2$ denote the numbers of real and the pairs of complex conjugate embeddings of the global field $K$ in $\mathbb C$. Note that, ${\mathscr {WA}_K^{\underline{\varphi}}}^{ab}=\mathbb J_K$. Let $L_K$ denote the hypothetical Langlands group $L_K$ of $K$. The existence problem of $L_K$ is one major conjecture in Langlands Program. For $\nu\in\mathbb h_K\cup\mathbb a_K$, an embedding $e_\nu:K^{sep}\hookrightarrow K_\nu^{sep}$ determines a continuous homomorphism $e_\nu^{\mathrm{Langlands}}:L_{K_\nu}\rightarrow L_K$ unique up to conjugacy, which in return defines a continuous homomorphism \[ \mathsf{NR}_{K_\nu}^{(\varphi_{K_\nu})^{\mathrm{Langlands}}}: {}_\mathbb Z\nabla_{K_\nu}^{\varphi_{K_\nu}}\times\mathrm{SL}(2,\mathbb C) \xrightarrow[\sim]{\{\bullet,K_\nu\}_{\varphi_{K_\nu}}\times \mathrm{id}_{\mathrm{SL}(2,\mathbb C)}} L_{K_\nu}\xrightarrow{e_\nu^{\mathrm{Langlands}}} L_K \] unique up to conjugacy, for each $\nu\in\mathbb h_K$. Fixing one such morphism for each $\nu\in\mathbb h_K$, the collection $\{\mathsf{NR}_{K_\nu}^{(\varphi_{K_\nu})^{\mathrm{Langlands}}}\}_{\nu\in\mathbb h_K}$ defines a unique continuous homomorphism \[\mathsf{NR}_K^{\underline\varphi^{\mathrm{Langlands}}}:\mathscr {WA}_K^{\underline{\varphi}}\rightarrow L_K,\] which is compatible with Arthur's proposed construction of $L_K$. Let $\mathrm{G}$ be a connected, quasisplit reductive group over $K$. There is a bijection between the set of ``$WA$-parameters'' \[\phi:\mathscr {WA}_K^{\underline{\varphi}}\rightarrow {}^L\mathrm{G}(\mathbb C)= \widehat{\mathrm G}(\mathbb C)\rtimes L_K\] of $\mathrm{G}$ over $K$ and the set $\mathscr P_{\mathrm G}$ whose elements are the collections \[\{\phi_\nu:L_{K_\nu}\rightarrow{}^L\mathrm{G}_\nu(\mathbb C)\}_{\nu\in\mathbb h_K\cup\mathbb a_K}\] consisting of local $L$-parameters of $\mathrm{G}_\nu$ over $K_\nu$ for each $\nu$. Note that, assuming the local reciprocity principle for $\mathrm{G}_\nu$ over $K_\nu$ for all $\nu\in\mathbb h_K\cup\mathbb a_K$, the set  $\mathscr P_{\mathrm G}$ is in bijection with the set whose elements are the collections $\{\Pi_{\phi_\nu}\}_{\nu\in\mathbb h_K\cup\mathbb a_K}$  of local $L$-packets of $\mathrm{G}_\nu$ over $K_\nu$ for each $\nu$. As global admissible $L$-packets of $\mathrm G$ over $K$ are the restricted tensor products of local $L$-packets of $\mathrm G_\nu$ over $K_\nu$, by Flath's decomposition theorem, we end up having the following theorems Theorem 1. Let $\mathrm{G}$ be a connected quasisplit reductive group over the number field $K$. Assume that the local Langlands reciprocity principle for $\mathrm G$ over $K$ holds. Then, there exists a bijection \[ \{\text{$WA$-parameters of $\mathrm G$ over $K$}\}\leftrightarrow\{\text{global admissible $L$-packets of $\mathrm G$ over $K$}\} \] satisfying the ``naturality'' properties. and Theorem 2. Let $\mathrm{G}$ and $H$ be connected quasisplit reductive groups over the number field $K$. Let \[\rho:{}^LG\rightarrow {}^LH\] be an $L$-homomorphism. Assume that the local Langlands reciprocity principle for $\mathrm G$ over $K$ holds. Then, the $L$-homomorphism $\rho:{}^LG\rightarrow {}^LH$ induces a map (lifting) from the global admissible $L$-packets of $G$ over $K$ to the global admissible $L$-packets of $H$ over $K$  satisfying the ``naturality'' properties.

          
     
     
470. Zoom, 13 November 2020, Friday, 15:40

     Deniz Ali Kaptan-[Alfred Renyi] - The Methods of Goldston-Pintz-Yıldırım and Maynard-Tao, and results on prime gaps
         
     

     Abstract: The breakthrough method of Goldston, Pintz and Yıldırım and its subsequent refinement by Maynard and Tao effected a giant leap in our understanding of prime gaps. I will give an overview of the evolution of the ideas involved in these methods, describing various applications along the way.

       
       
     
471. Zoom, 20 November 2020, Friday, 15:40

     Ayberk Zeytin-[Galatasaray] - Continued Fractions and the Selberg zeta function of the modular curve
         
     

     Abstract: Selberg zeta function of a Riemann surface X is known to encode the discrete spectrum of the Laplacian on X via the Selberg trace formula. In this talk, following Lewis-Zagier, we will explain how one obtains the Selberg zeta function of the modular curve as the Fredholm determinant of an appropriate operator  on an appropriate Banach space. Along the way, we will discuss the close relationship between the operators in question and continued fractions. Should time permit, we will mention some ongoing work, partly joint with M. Fraczek, B. Mesland and M.H. Şengün.

          
     
     
472. Zoom, 27 November 2020, Friday, 15:40

     Mustafa Kalafat-[Nesin Math Village] - On special submanifolds of the Page space
         
     

     Abstract: Page manifold is the underlying differentiable manifold of the complex surface, obtained out of the process of blowing up the complex projective plane, only once. This space is decorated with a natural Einstein metric, first studied by D.Page in 1978. In this talk, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces.   Finally, we give information on how the Page space is related to some other metrics on the same underlying smooth manifold. This talk is based on joint work with R.Sarı. Related paper may be accessed from https://arxiv.org/abs/1608.03252 Kalafat, Sarı - On special submanifolds of the Page space. Differential Geom. Appl. To appear. 2020 Despite working on basic submanifolds, we introduce a variety of mutually-independent techniques, like graphic illustrations, physicist computations, Teichmüller space, 3- manifold topology, ODE systems etc. So that should not be confused with dry, computational diff.geo. involving only symbolic manipulations, meaningless mess of equalities followed by equalities. We always consider the global topology of the submanifold for example, and deal primarily with compact examples.

      
       
473. Zoom, 4 December 2020, Friday, 15:40

     Alexander Degtyarev-[Bilkent] - Lines in singular triquadrics
         
     

     Abstract:  (joint work in progress with Sławomir Rams) Thanks to the global Torelli theorem, one can relatively easy bound the number of lines (and even classify the large configurations of lines) in any *smooth* polarized K3-surface. The situation changes if *singular* (necessarily ADE-) surfaces are considered: only partial results are known (mainly due to Davide Veniani) and only for quartics. I will discuss our recent work where we adjust the arithmetical reduction for the singular case; in particular, I will explain why it is difficult to keep track of the lines and exceptional divisors simultaneously. We have tested our approach in the case of octic surfaces in P^5, most notably triquadrics. The sharp upper bound on the number of lines in a *singular* triquadric is 32, as opposed to the 36 lines in a smooth one. For special octics these bounds are 30 and 33, respectively.

        
      
474. Zoom, 11 December 2020, Friday, 15:40

     Ali Sinan Sertöz-[Bilkent] - From Calculus to Hodge
         
     

     Abstract: This is an expository talk mainly for the young Complex Geometry students. I will start with the tangent line to a real parabola, pass to the complex case and then to the projective case. After giving informal descriptions of the de Rham and Dolbeault cohomologies, which are related by the Hodge decomposition theorem, I will describe the Hodge Conjecture with integer coefficients which is known to be false in general despite the strong evidence in its favor given by the Lefschetz (1,1)-theorem. It is known that some torsion integral Hodge classes may exist which are not algebraic. The existence of non-torsion integral Hodge classes contradicting the Hodge conjecture were constructed recently (30 years ago!) by Kollar. I want to end the talk discussing this example and its possible variants.

         
     
     
475. Zoom, 18 December 2020, Friday, 15:40

     Ali Ulaş Özgür Kişisel-[ODTÜ] - On complex 4-nets
         
     

     Abstract: Nets are certain special line arrangements in the plane and they occur in various contexts related to algebraic geometry, such as resonance varieties, homology of Milnor fibers and fundamental groups of curve complements. We will investigate nets in the complex projective plane $\mathbb{CP}^2$. Let $m\geq 3$ and $d\geq 2$ be integers. An $(m,d)$-net is a pencil of degree $d$ algebraic curves in $\mathbb{CP}^2$ with a base locus of exactly $d^2$ points, which degenerates into a union of $d$ lines $m$ times. It was conjectured that the only $4$-net is a $(4,3)$-net called the Hessian arrangement. I will outline our proof together with A. Bassa of this conjecture.

          
     
     
476. Zoom, 25 December 2020, Friday, 15:40

     Sefa Feza Arslan-[Mimar Sinan] - Apery table, microinvariants and the regularity index
         
     

     Abstract: In this talk, I will first explain the concepts of Apery table of a numerical semigroup introduced by Cortedellas and Zarzuela (Tangent cones of numerical semigroup rings. Contemp. Math. 502, 45–58 (2009)) and the microinvariants of a local ring introduced by Juan Elias (On the deep structure of the blowing-up of curve singularities. Math. Proc. Camb. Philos. Soc. 131, 227–240 (2001)). We use these concepts to give some partial results about a conjecture on the regularity index of a local ring and to give some open problems.

          
     
     
     

ODTÜ talks are either at Hüseyin Demir Seminar room or at Gündüz İkeda seminar room at the Mathematics building of ODTÜ.
Bilkent talks are at room 141 of Faculty of Science A-building at Bilkent.
Zoom talks are online.