About

Anar Akhmedov is a professor at the School of Mathematics at the University of Minnesota. His research interests span diverse mathematical disciplines, including low-dimensional topology, symplectic/contact topology, gauge theory, algebraic geometry, group theory, and topological data science. His work has predominantly focused on low-dimensional topology, with contributions to the classification of smooth and symplectic 4-manifolds, exotic smooth structures on 4-manifolds, and Stein fillings of contact 3-manifolds. Akhmedov also explores elliptic curves and higher-dimensional varieties, emphasizing the study of rational points. Recently, he has developed a strong interest in topological data science, investigating its potential applications in data analysis.

Research topics

• Mathematics
• Pure mathematics
• Computer Science
• Geometry
• Biology
• Combinatorics
• Botany

Selected publications

• Geography of symplectic 4-manifolds admitting Lefschetz fibrations and their indecomposability

  Journal of the Mathematical Society of Japan · 2024-02-19

  article1st authorCorresponding

  In this paper, we show that for a given finitely presented group $G$, there exist integers $h_G \geq 0$ and $n_G \geq 4$ such that for all $h \geq h_G$ and $n \geq n_G$, and for all $0 \leq i \leq 2n - 2$, there exists a genus-$(2h + n - 1)$ Lefschetz fibration on a minimal symplectic 4-manifold with $(\chi, c_{1}^{2}) = (n, i)$ whose fundamental group is isomorphic to $G$. We also prove that such a fibration cannot be decomposed as a fiber sum for $1 \leq i \leq 2n - 2$ if $h > (5n - 3)/2$. In addition, we give a relation among the genus of the base space of a ruled surface admitting a Lefschetz fibration, the number of blow-ups and the genus of the Lefschetz fibration.

  PublisherDOI

• Exotic smooth structures on connected sums of S2×S2

  The Quarterly Journal of Mathematics · 2023-01-24

  article1st authorCorresponding

  Abstract We construct infinitely many distinct irreducible smooth structures on $n(S^2\,\times\,S^2)$, the connected sum of n copies of $S^2\,\times\,S^2$, for every odd integer $n\geq 27$.

  PublisherDOI

• Complex Ball Quotients and New Symplectic $4$-manifolds with Nonnegative Signatures

  Taiwanese Journal of Mathematics · 2023-09-28

  articleOpen access1st authorCorresponding

  We construct new symplectic $4$-manifolds with non-negative signatures and with the smallest Euler characteristics, using fake projective planes, Cartwright–Steger surfaces and their normal covers and product symplectic $4$-manifolds $\Sigma_{g} \times \Sigma_{h}$, where $g \geq 1$ and $h \geq 0$, along with exotic symplectic $4$-manifolds constructed in [7, 12]. In particular, our constructions yield to (1) infinitely many irreducible symplectic and infinitely many non-symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1) \mathbb{CP}^{2} \# (2n-1) \overline{\mathbb{CP}}^{2}$ for each integer $n \geq 9$, (2) infinite families of simply connected irreducible nonspin symplectic and such infinite families of non-symplectic $4$-manifolds that have the smallest Euler characteristics among the all known simply connected $4$-manifolds with positive signatures and with more than one smooth structure. We also construct a complex surface with positive signature from the Hirzebruch's line-arrangement surfaces, which is a ball quotient.

  PublisherOA PDFDOI

• Complex Ball Quotients and New Symplectic 4-manifolds with Nonnegative Signatures

  arXiv (Cornell University) · 2021

  1st authorCorresponding
  • Pure mathematics
  • Mathematics

  We present the various constructions of new symplectic $4$-manifolds with non-negative signatures using the complex surfaces on the BMY line $c_1^2 = 9χ_h$, the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain line-arrangement surfaces, along with the exotic symplectic $4$-manifolds constructed in \cite{AP2, AS}. In particular, our constructions yield to (i) an irreducible symplectic and infinitely many non-symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1)CP^{2}\#(2n-1)\bar{CP}^{2}$ for each integer $n \geq 9$, (ii) the families of simply connected irreducible nonspin symplectic $4$-manifolds that have the smallest Euler characteristics among the all known simply connected $4$-manifolds with positive signatures and with more than one smooth structure. We also construct a complex surface with positive signature from the Hirzebruch's line-arrangement surfaces, which is a ball quotient.

  PublisherOA PDFDOI

• The existence of an indecomposable minimal genus two lefschetz fibration

  Osaka City University (Osaka City University) · 2021-01-01

  articleOpen access1st authorCorresponding

  It was shown by Usher that any fiber sum of Lefschetz fibrations over S^2 is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists a genus-2 indecomposable minimal Lefschetz fibration (IMLF for short).

  PublisherOA PDFDOI

• Generalized chain surgeries and applications

  Journal of Symplectic Geometry · 2021-01-01

  article1st authorCorresponding

  We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

  PublisherDOI

• Generalized Chain Surgeries and Applications

  arXiv (Cornell University) · 2020-06-05

  preprintOpen access1st authorCorresponding

  We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

  PublisherOA PDFDOI

• Genus 2 Lefschetz fibrations with b 2 + = 1 and c 1 2 = 1 , 2

  Kyoto journal of mathematics · 2020 · 5 citations

  1st authorCorresponding
  • Mathematics
  • Combinatorics
  • Pure mathematics

  In this article, we construct a family of genus 2 Lefschetz fibrations f n : X θ n → S 2 with e ( X θ n ) = 11 , b 2 + ( X θ n ) = 1 , and c 1 2 ( X θ n ) = 1 by applying a single lantern substitution to the twisted fiber sums of Matsumoto’s genus 2 Lefschetz fibration over S 2 . Moreover, we compute the fundamental group of X θ n and show that it is isomorphic to the trivial group if n = − 3 or − 1 , Z if n = − 2 , and Z | n + 2 | for all integers n ≠ − 3 , − 2 , − 1 . Also, we prove that our fibrations admit − 2 section, that their total spaces are symplectically minimal, and that they have symplectic Kodaira dimension κ = 2 . In addition, using techniques developed over the past decade with other authors, we also construct the genus 2 Lefschetz fibrations over S 2 with c 1 2 = 1 , 2 and χ = 1 via the fiber sums of Matsumoto’s and Xiao’s genus 2 Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small 4 -manifolds with b 2 + = 1 and b 2 + = 3 .

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• Geography of simply connected nonspin symplectic 4-manifolds with positive signature. II

  Canadian Mathematical Bulletin · 2020 · 3 citations

  1st authorCorresponding
  • Computer Science
  • Mathematics
  • Pure mathematics

  Abstract Building upon our earlier work with M. C. Hughes, we construct many new smooth structures on closed simply connected nonspin $4$ -manifolds with positive signature. We also provide numerical and asymptotic upper bounds on the function $\lambda (\sigma )$ that was defined in our earlier work.

  PublisherDOI

• Geography of simply connected spin symplectic 4-manifolds, II

  Comptes Rendus Mathématique · 2019-03-01 · 4 citations

  article1st authorCorresponding

  Building upon our early work, we construct infinitely many new smooth structures on closed simply connected spin 4-manifolds with nonnegative signature.

  PublisherDOI

Frequent coauthors

• B. Doug Park

  University of Waterloo

  35 shared

• Scott Baldridge

  27 shared

• Paul M. Kirk

  26 shared

• R. Baykur

  University of Massachusetts Amherst

  25 shared

• Naoyuki Monden

  Okayama University of Science

  11 shared

• Sümeyra Sakallı

  9 shared

• Ludmil Katzarkov

  University of Miami

  7 shared

• Burak Özbağcı

  5 shared

Awards & honors

• American Mathematical Society Fellow for 2020
• Humboldt Research Fellowship for Experienced Researchers, Ge…
• Simons Research Fellowship, 2018-19
• Alfred P. Sloan Research Fellowship, 2012-16
• University of Minnesota Guillermo E. Borja Award (in recogni…