---
OA_place: repository
OA_type: green
_id: '21931'
abstract:
- lang: eng
text: In 1873, James C. Maxwell conjectured that the electric field generated by
n point charges in generic position has at most (n-1)^2 isolated zeroes. The first
(nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and
Shapiro, who also posed two additional interesting conjectures. In this article,
we give the best upper bound known to date on the number of zeroes of the electric
field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov,
and Shapiro that the number of equilibria cannot exceed those of the distance
function defined by the unit point charges. Finally, we note that it is quite
possible that Maxwell's quadratic upper bound is not tight, so it is prudent to
find lower bounds. Hence, we also explore examples and construct configurations
of charges achieving the highest ratios of the number of electric field zeroes
by point charges found to this day.
article_number: e70163
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Goncalo
full_name: Oliveira, Goncalo
id: 58abbde8-f455-11eb-a497-98c8fd71b905
last_name: Oliveira
citation:
ama: Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic
potential. Proceedings of the London Mathematical Society. 2026;132(5).
doi:10.1112/plms.70163
apa: Edelsbrunner, H., Fillmore, C. D., & Oliveira, G. (2026). Counting equilibria
of the electrostatic potential. Proceedings of the London Mathematical Society.
Wiley. https://doi.org/10.1112/plms.70163
chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting
Equilibria of the Electrostatic Potential.” Proceedings of the London Mathematical
Society. Wiley, 2026. https://doi.org/10.1112/plms.70163.
ieee: H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of
the electrostatic potential,” Proceedings of the London Mathematical Society,
vol. 132, no. 5. Wiley, 2026.
ista: Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the
electrostatic potential. Proceedings of the London Mathematical Society. 132(5),
e70163.
mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.”
Proceedings of the London Mathematical Society, vol. 132, no. 5, e70163,
Wiley, 2026, doi:10.1112/plms.70163.
short: H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical
Society 132 (2026).
corr_author: '1'
date_created: 2026-05-31T22:02:13Z
date_published: 2026-05-01T00:00:00Z
date_updated: 2026-06-02T09:24:18Z
day: '01'
department:
- _id: HeEd
- _id: TaHa
doi: 10.1112/plms.70163
external_id:
arxiv:
- '2501.05315'
intvolume: ' 132'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2501.05315
month: '05'
oa: 1
oa_version: Preprint
publication: Proceedings of the London Mathematical Society
publication_identifier:
eissn:
- 1460-244X
issn:
- 0024-6115
publication_status: published
publisher: Wiley
quality_controlled: '1'
related_material:
record:
- id: '21050'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Counting equilibria of the electrostatic potential
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '17292'
abstract:
- lang: eng
text: The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkähler
4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of
the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian
isotopy classes of Lagrangians, and the Thomas-Yau conjecture on longtime existence
of the Lagrangian mean curvature ow. We also make observations concerning closed
geodesics, curve shortening flow and minimal surfaces.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jason D.
full_name: Lotay, Jason D.
last_name: Lotay
- first_name: Goncalo
full_name: Oliveira, Goncalo
id: 58abbde8-f455-11eb-a497-98c8fd71b905
last_name: Oliveira
citation:
ama: Lotay JD, Oliveira G. Special Lagrangians, Lagrangian mean curvature flow and
the Gibbons-Hawking ansatz. Journal of Differential Geometry. 2024;126(3):1121-1184.
doi:10.4310/jdg/1717348872
apa: Lotay, J. D., & Oliveira, G. (2024). Special Lagrangians, Lagrangian mean
curvature flow and the Gibbons-Hawking ansatz. Journal of Differential Geometry.
International Press. https://doi.org/10.4310/jdg/1717348872
chicago: Lotay, Jason D., and Goncalo Oliveira. “Special Lagrangians, Lagrangian
Mean Curvature Flow and the Gibbons-Hawking Ansatz.” Journal of Differential
Geometry. International Press, 2024. https://doi.org/10.4310/jdg/1717348872.
ieee: J. D. Lotay and G. Oliveira, “Special Lagrangians, Lagrangian mean curvature
flow and the Gibbons-Hawking ansatz,” Journal of Differential Geometry,
vol. 126, no. 3. International Press, pp. 1121–1184, 2024.
ista: Lotay JD, Oliveira G. 2024. Special Lagrangians, Lagrangian mean curvature
flow and the Gibbons-Hawking ansatz. Journal of Differential Geometry. 126(3),
1121–1184.
mla: Lotay, Jason D., and Goncalo Oliveira. “Special Lagrangians, Lagrangian Mean
Curvature Flow and the Gibbons-Hawking Ansatz.” Journal of Differential Geometry,
vol. 126, no. 3, International Press, 2024, pp. 1121–84, doi:10.4310/jdg/1717348872.
short: J.D. Lotay, G. Oliveira, Journal of Differential Geometry 126 (2024) 1121–1184.
corr_author: '1'
date_created: 2024-07-22T07:45:31Z
date_published: 2024-03-01T00:00:00Z
date_updated: 2025-09-08T08:27:51Z
day: '01'
department:
- _id: TaHa
doi: 10.4310/jdg/1717348872
external_id:
arxiv:
- '2002.10391'
isi:
- '001271790200007'
intvolume: ' 126'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2002.10391
month: '03'
oa: 1
oa_version: Preprint
page: 1121-1184
publication: Journal of Differential Geometry
publication_identifier:
issn:
- 0022-040X
publication_status: published
publisher: International Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking
ansatz
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 126
year: '2024'
...
---
_id: '12329'
abstract:
- lang: eng
text: In this article, we develop two independent and new approaches to model epidemic
spread in a network. Contrary to the most studied models, those developed here
allow for contacts with different probabilities of transmitting the disease (transmissibilities).
We then examine each of these models using some mean field type approximations.
The first model looks at the late-stage effects of an epidemic outbreak and allows
for the computation of the probability that a given vertex was infected. This
computation is based on a mean field approximation and only depends on the number
of contacts and their transmissibilities. This approach shares many similarities
with percolation models in networks. The second model we develop is a dynamic
model which we analyze using a mean field approximation which highly reduces the
dimensionality of the system. In particular, the original system which individually
analyses each vertex of the network is reduced to one with as many equations as
different transmissibilities. Perhaps the greatest contribution of this article
is the observation that, in both these models, the existence and size of an epidemic
outbreak are linked to the properties of a matrix which we call the R-matrix.
This is a generalization of the basic reproduction number which more precisely
characterizes the main routes of infection.
acknowledgement: Gonçalo Oliveira is supported by the NOMIS Foundation, Fundação Serrapilheira
1812-27395, by CNPq grants 428959/2018-0 and 307475/2018-2, and by FAPERJ through
the grant Jovem Cientista do Nosso Estado E-26/202.793/2019.
article_number: '468'
article_processing_charge: No
article_type: original
author:
- first_name: Arturo
full_name: Gómez, Arturo
last_name: Gómez
- first_name: Goncalo
full_name: Oliveira, Goncalo
id: 58abbde8-f455-11eb-a497-98c8fd71b905
last_name: Oliveira
citation:
ama: Gómez A, Oliveira G. New approaches to epidemic modeling on networks. Scientific
Reports. 2023;13. doi:10.1038/s41598-022-19827-9
apa: Gómez, A., & Oliveira, G. (2023). New approaches to epidemic modeling on
networks. Scientific Reports. Springer Nature. https://doi.org/10.1038/s41598-022-19827-9
chicago: Gómez, Arturo, and Goncalo Oliveira. “New Approaches to Epidemic Modeling
on Networks.” Scientific Reports. Springer Nature, 2023. https://doi.org/10.1038/s41598-022-19827-9.
ieee: A. Gómez and G. Oliveira, “New approaches to epidemic modeling on networks,”
Scientific Reports, vol. 13. Springer Nature, 2023.
ista: Gómez A, Oliveira G. 2023. New approaches to epidemic modeling on networks.
Scientific Reports. 13, 468.
mla: Gómez, Arturo, and Goncalo Oliveira. “New Approaches to Epidemic Modeling on
Networks.” Scientific Reports, vol. 13, 468, Springer Nature, 2023, doi:10.1038/s41598-022-19827-9.
short: A. Gómez, G. Oliveira, Scientific Reports 13 (2023).
corr_author: '1'
date_created: 2023-01-22T23:00:55Z
date_published: 2023-01-10T00:00:00Z
date_updated: 2024-10-09T21:03:29Z
day: '10'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1038/s41598-022-19827-9
external_id:
isi:
- '001003345000051'
file:
- access_level: open_access
checksum: a8b83739f4a951e83e0b2a778f03b327
content_type: application/pdf
creator: dernst
date_created: 2023-01-23T07:53:23Z
date_updated: 2023-01-23T07:53:23Z
file_id: '12336'
file_name: 2023_ScientificReports_Gomez.pdf
file_size: 2167792
relation: main_file
success: 1
file_date_updated: 2023-01-23T07:53:23Z
has_accepted_license: '1'
intvolume: ' 13'
isi: 1
language:
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month: '01'
oa: 1
oa_version: Published Version
publication: Scientific Reports
publication_identifier:
eissn:
- 2045-2322
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: New approaches to epidemic modeling on networks
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 13
year: '2023'
...