Pedram Hekmati
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Previous Organizations / Institutes
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Pedram Hekmati has a total of 6 co-authors. Here are the top co-authors by number of shared publications:
- David Baraglia
School of Mathematical Sciences, The University of Adelaide, Adelaide SA 5005, Australia
5 shared publications - Michael K. Murray
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
4 shared publications - Raymond F. Vozzo
School of Mathematical Sciences, University of Adelaide, SA, Australia
4 shared publications - Richard J. Szabo
Heriot-Watt University, Edinburgh, UK
4 shared publications - Jouko Mickelsson
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
2 shared publications - Marcos Orseli
Department of Mathematics, University of Auckland, 1010 Auckland, New Zealand
2 shared publications
Latest Publications
- 17 December 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 379 (3) , 1853-1882
Equivariant index on toric contact manifolds
We compute the equivariant index of the twisted horizontal Dolbeault operator on compact toric contact manifolds of Reeb type. The operator is elliptic transverse to the Reeb foliation and its equivariant index defines a distribution on the torus. Using the good cone condition, we show that the symbol localises to the closed Reeb orbits corresponding to the edges of the moment cone and obtain an Atiyah-Bott-Lefschetz type formula for the index. For the horizontal Dolbeault operator, we obtain an expression for the index as a sum over the lattice points of the moment cone, by applying an adaptation of the Lawrence-Varchenko polytope decomposition to rational polyhedral cones.
- 17 December 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 379 (3) , 1853-1882
Equivariant index on toric contact manifolds
We compute the equivariant index of the twisted horizontal Dolbeault operator on compact toric contact manifolds of Reeb type. The operator is elliptic transverse to the Reeb foliation and its equivariant index defines a distribution on the torus. Using the good cone condition, we show that the symbol localises to the closed Reeb orbits corresponding to the edges of the moment cone and obtain an Atiyah-Bott-Lefschetz type formula for the index. For the horizontal Dolbeault operator, we obtain an expression for the index as a sum over the lattice points of the moment cone, by applying an adaptation of the Lawrence-Varchenko polytope decomposition to rational polyhedral cones.
- Open Access
Brieskorn spheres, cyclic group actions and the Milnor conjecture
In this paper we further develop the theory of equivariant Seiberg-Witten-Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain the following applications. First, we show that the knot concordance invariants $\theta^{(c)}$ defined by the first author satisfy $\theta^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever $c$ is a prime not dividing $ab$. Since $\theta^{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture of a similar flavour to the proofs using the Ozsv\'ath-Szab\'o $\tau$-invariant or Rasmussen $s$-invariant. Second, we prove that a free cyclic group action on a Brieskorn homology $3$-sphere $Y = \Sigma(a_1 , \dots , a_r)$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. This generalises to arbitrary $r$ the result of Anvari-Hambleton in the case $r=3$. Third, given a finite subgroup of the Seifert circle action on $Y = \Sigma(a_1 , \dots , a_r)$ of prime order $p$ acting non-freely on $Y$, we prove that if the rank of $HF_{red}^+(Y)$ is greater than $p$ times the rank of $HF_{red}^+(Y/\mathbb{Z}_p)$, then the $\mathbb{Z}_p$-action on $Y$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. We also prove a similar non-extension result for equivariant connected sums of Brieskorn homology spheres.Open AccessBrieskorn spheres, cyclic group actions and the Milnor conjecture
In this paper we further develop the theory of equivariant Seiberg-Witten-Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain the following applications. First, we show that the knot concordance invariants $\theta^{(c)}$ defined by the first author satisfy $\theta^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever $c$ is a prime not dividing $ab$. Since $\theta^{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture of a similar flavour to the proofs using the Ozsv\'ath-Szab\'o $\tau$-invariant or Rasmussen $s$-invariant. Second, we prove that a free cyclic group action on a Brieskorn homology $3$-sphere $Y = \Sigma(a_1 , \dots , a_r)$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. This generalises to arbitrary $r$ the result of Anvari-Hambleton in the case $r=3$. Third, given a finite subgroup of the Seifert circle action on $Y = \Sigma(a_1 , \dots , a_r)$ of prime order $p$ acting non-freely on $Y$, we prove that if the rank of $HF_{red}^+(Y)$ is greater than $p$ times the rank of $HF_{red}^+(Y/\mathbb{Z}_p)$, then the $\mathbb{Z}_p$-action on $Y$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. We also prove a similar non-extension result for equivariant connected sums of Brieskorn homology spheres. - Open Access
- 18 March 2024
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 24 (1) , 493-554
Equivariant Seiberg–Witten–Floer cohomology
We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology 3-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of d-invariants d G;c .Y; s/ which are indexed by the group cohomology of G. These invariants satisfy a Frøyshov-type inequality under equivariant cobordisms. Lastly, we consider a variety of applications of these d-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding 4-manifolds, Nielsen realisation problems for 4-manifolds with boundary and obstructions to equivariant embeddings of 3-manifolds in 4-manifolds.
- 18 March 2024
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 24 (1) , 493-554
Open AccessEquivariant Seiberg–Witten–Floer cohomology
We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology 3-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of d-invariants d G;c .Y; s/ which are indexed by the group cohomology of G. These invariants satisfy a Frøyshov-type inequality under equivariant cobordisms. Lastly, we consider a variety of applications of these d-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding 4-manifolds, Nielsen realisation problems for 4-manifolds with boundary and obstructions to equivariant embeddings of 3-manifolds in 4-manifolds.
- Open Access
New invariants of involutions from Seiberg-Witten Floer theory
We study equivariant Seiberg-Witten Floer theory of rational homology $3$-spheres in the special case where the group action is given by an involution. The case of involutions deserves special attention because we can couple the involution to the charge conjugation symmetry of Seiberg-Witten theory. This leads to new Floer-theoretic invariants which we study and apply in a variety of applications. In particular, we construct a series of delta-invariants $δ^E_*, δ^R_*, δ^S_*$ which are the equivariant equivalents of the Ozsváth-Szabó $d$-invariant. The delta-invariants come in three types: equivariant, Real and spin depending on the type of the spin$^c$-structure involved. The delta-invariants satisfy many useful properties, including a Froyshov-type inequality for equivariant cobordisms. We compute the delta-invariants in a wide range of examples including: equivariant plumbings, branched double covers of knots and equivariant Dehn surgery. We also consider various applications including obstructions to extending involutions over bounding $4$-manifolds, non-smoothable involutions on $4$-manifolds with boundary, equivariant embeddings of $3$-manifolds in $4$-manifolds and non-orientable surfaces bounding knots.Open AccessNew invariants of involutions from Seiberg-Witten Floer theory
We study equivariant Seiberg-Witten Floer theory of rational homology $3$-spheres in the special case where the group action is given by an involution. The case of involutions deserves special attention because we can couple the involution to the charge conjugation symmetry of Seiberg-Witten theory. This leads to new Floer-theoretic invariants which we study and apply in a variety of applications. In particular, we construct a series of delta-invariants $δ^E_*, δ^R_*, δ^S_*$ which are the equivariant equivalents of the Ozsváth-Szabó $d$-invariant. The delta-invariants come in three types: equivariant, Real and spin depending on the type of the spin$^c$-structure involved. The delta-invariants satisfy many useful properties, including a Froyshov-type inequality for equivariant cobordisms. We compute the delta-invariants in a wide range of examples including: equivariant plumbings, branched double covers of knots and equivariant Dehn surgery. We also consider various applications including obstructions to extending involutions over bounding $4$-manifolds, non-smoothable involutions on $4$-manifolds with boundary, equivariant embeddings of $3$-manifolds in $4$-manifolds and non-orientable surfaces bounding knots. - Open Access
Equivariant Index on Toric Contact Manifolds
We compute the equivariant index of the twisted horizontal Dolbeault operator on compact toric contact manifolds of Reeb type. The operator is elliptic transverse to the Reeb foliation and its equivariant index defines a distribution on the torus. Using the good cone condition, we show that the symbol localises to the closed Reeb orbits corresponding to the edges of the moment cone and obtain an Atiyah-Bott-Lefschetz type formula for the index. For the horizontal Dolbeault operator, we obtain an expression for the index as a sum over the lattice points of the moment cone, by applying an adaptation of the Lawrence-Varchenko polytope decomposition to rational polyhedral cones.
Open AccessEquivariant Index on Toric Contact Manifolds
We compute the equivariant index of the twisted horizontal Dolbeault operator on compact toric contact manifolds of Reeb type. The operator is elliptic transverse to the Reeb foliation and its equivariant index defines a distribution on the torus. Using the good cone condition, we show that the symbol localises to the closed Reeb orbits corresponding to the edges of the moment cone and obtain an Atiyah-Bott-Lefschetz type formula for the index. For the horizontal Dolbeault operator, we obtain an expression for the index as a sum over the lattice points of the moment cone, by applying an adaptation of the Lawrence-Varchenko polytope decomposition to rational polyhedral cones.
- Open Access
Equivariant Seiberg-Witten-Floer cohomology
We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology $3$-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of $d$-invariants $d_{G,c}(Y,\mathfrak{s})$ which are indexed by the group cohomology of $G$. These invariants satisfy a Froyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these $d$-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding $4$-manifolds, Nielsen realisation problems for $4$-manifolds with boundary and obstructions to equivariant embeddings of $3$-manifolds in $4$-manifolds.Open AccessEquivariant Seiberg-Witten-Floer cohomology
We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology $3$-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of $d$-invariants $d_{G,c}(Y,\mathfrak{s})$ which are indexed by the group cohomology of $G$. These invariants satisfy a Froyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these $d$-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding $4$-manifolds, Nielsen realisation problems for $4$-manifolds with boundary and obstructions to equivariant embeddings of $3$-manifolds in $4$-manifolds. - 18 August 2020
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 378 (3) , 1843-1873
Sign Choices for Orientifolds
- 18 August 2020
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 378 (3) , 1843-1873
Sign Choices for Orientifolds
- Open Access
Sign choices for orientifolds
We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant $p$-gerbes with $p\geq-1$, which give rise to sign choices and are related by coboundary maps. We prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations. Concrete calculations for spherical and toroidal orientifolds are carried out, and in particular we exhibit a four-dimensional orientifold where not every sign choice is geometrically attainable. We elucidate how the $K$-theory groups associated with invariant $p$-gerbes for $p=-1,0,1$ interact with the coboundary maps. This allows us to interpret a notion of $K$-theory due to Gao and Hori as a special case of twisted $KR$-theory, which consequently implies the homotopy invariance and Fredholm module description of their construction.Open AccessSign choices for orientifolds
We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant $p$-gerbes with $p\geq-1$, which give rise to sign choices and are related by coboundary maps. We prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations. Concrete calculations for spherical and toroidal orientifolds are carried out, and in particular we exhibit a four-dimensional orientifold where not every sign choice is geometrically attainable. We elucidate how the $K$-theory groups associated with invariant $p$-gerbes for $p=-1,0,1$ interact with the coboundary maps. This allows us to interpret a notion of $K$-theory due to Gao and Hori as a special case of twisted $KR$-theory, which consequently implies the homotopy invariance and Fredholm module description of their construction. - Open Access
Real bundle gerbes, orientifolds and twisted KR-homology
We consider Real bundle gerbes on manifolds equipped with an involution and prove that they are classified by their Real Dixmier-Douady class in Grothendieck's equivariant sheaf cohomology. We show that the Grothendieck group of Real bundle gerbe modules is isomorphic to twisted KR-theory for a torsion Real Dixmier-Douady class. Using these modules as building blocks, we introduce geometric cycles for twisted KR-homology and prove that they generate a real-oriented generalised homology theory dual to twisted KR-theory for Real closed manifolds, and more generally for Real finite CW-complexes, for any Real Dixmier-Douady class. This is achieved by defining an explicit natural transformation to analytic twisted KR-homology and proving that it is an isomorphism. Our model both refines and extends previous results by Wang and Baum-Carey-Wang to the Real setting. Our constructions further provide a new framework for the classification of orientifolds in string theory, providing precise conditions for orientifold lifts of H-fluxes and for orientifold projections of open string states.
Open AccessReal bundle gerbes, orientifolds and twisted KR-homology
We consider Real bundle gerbes on manifolds equipped with an involution and prove that they are classified by their Real Dixmier-Douady class in Grothendieck's equivariant sheaf cohomology. We show that the Grothendieck group of Real bundle gerbe modules is isomorphic to twisted KR-theory for a torsion Real Dixmier-Douady class. Using these modules as building blocks, we introduce geometric cycles for twisted KR-homology and prove that they generate a real-oriented generalised homology theory dual to twisted KR-theory for Real closed manifolds, and more generally for Real finite CW-complexes, for any Real Dixmier-Douady class. This is achieved by defining an explicit natural transformation to analytic twisted KR-homology and proving that it is an isomorphism. Our model both refines and extends previous results by Wang and Baum-Carey-Wang to the Real setting. Our constructions further provide a new framework for the classification of orientifolds in string theory, providing precise conditions for orientifold lifts of H-fluxes and for orientifold projections of open string states.
- 1 January 2019
- Published by International Press of Boston in Advances in Theoretical and Mathematical Physics
- Vol. 23 (8) , 2093-2159
Real bundle gerbes, orientifolds and twisted $KR$-homology
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2019
- Published by International Press of Boston in Advances in Theoretical and Mathematical Physics
- Vol. 23 (8) , 2093-2159
Real bundle gerbes, orientifolds and twisted $KR$-homology
International Press of Boston - publishers of scholarly mathematical and scientific journals and books