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51. Group actions in geometric and arithmetic combinatorics

Author

Wheeler, James W. E. and Rudnev, Misha

Abstract

This project seeks to further and combine results and tools from the theory of growth in finite groups with state of the art methods of arithmetic and geometric combinatorics. This is a modern area of research at the crossroads of pure mathematics, with connections to computer science and coding and complexity theory, unified by the general theme of pseudorandomness. A significant progress in this area began in the 2000s after the foundational work of Helfgott, followed by Bourgain, Gamburd, Sarnak, and others. The growth phenomenon appears to be inherently connected with the renowned Sum-Product conjecture of Erdos and Szemerédi, towards which ˝ there has been a lot of progress in the past 15 years. More specifically the project has two main connected threads, first, the project considers specific families of groups, such as those of upper-triangular matrices, to uncover and categorise the structures therein that pose obstruction to growth and establish quantitative estimates for growth in their absence. The nature of these obstructions much depends on the field, where the matrix elements come from: analysing various scenarios to this effect is a specific novel feature of this project. Partially this scope of questions furthers the earlier results by Breuillard, Green and Tao, Gill and Helfgot, Murphy and Petridis and others. Growth in groups, and especially the concept of energy arising in its study are immediately related to geometric incidence theory estimates, arising in connection of these groups' actions on homogeneous spaces. This constitutes the other thread of the project, focusing on Möbius hyperbolae (which have connections to the two dimensional special linear group). The project improves on earlier results due to Bourgain, Solymosi and Tardos, Shkredov and others by using a special set of tools both from growth in groups and geometric incidence theory.

Published
2022

53. Incidences of M\'obius transformations in $\mathbb F_p$

Author

Warren, Audie and Wheeler, James

Subjects
Mathematics - Combinatorics
Abstract

We develop the methods used by Rudnev and Wheeler to prove an incidence theorem between arbitrary sets of M\"{o}bius transformations and point sets in $\mathbb F_p^2$. We also note some asymmetric incidence results, and give applications of these results to various problems in additive combinatorics and discrete geometry., Comment: 12 pages

Published
2021
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56. Incidence bounds with M\'obius hyperbolae in positive characteristic

Author

Rudnev, Misha and Wheeler, James

Subjects
Mathematics - Combinatorics
Abstract

We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates $H$ of the hyperbola $xy=\lambda\neq 0$, over a field of asymptotically large positive characteristic $p$. They improve recent bounds by Shkredov, which are based on using explicit incidence estimates in the early terminated procedure of repeated applications of the Cauchy-Schwarz inequality, underlying many qualitative results related to growth and expansion in groups. The improvement -- both quantitative, plus we are able to deal with a general $H$, rather than a Cartesian product -- is mostly due to a non-trivial "intermediate" bound on the number of $k$-rich M\"obius hyperbolae in positive characteristic. In addition, we make an observation that a certain energy-type quantity in the context of $H$ can be bounded via the $L^2$-moment of the Minkowski distance in $H$ and can therefore fetch the corresponding estimates apropos of the Erd\H{o}s distinct distance problem., Comment: 17 pages

Published
2021

64. Effects of Empagliflozin on Fluid Overload, Weight, and Blood Pressure in CKD

Author

Mayne, Kaitlin J., Staplin, Natalie, Keane, David F., Wanner, Christoph, Brenner, Susanne, Cejka, Vladimir, Stegbauer, Johannes, Judge, Parminder K., Preiss, David, Emberson, Jonathan, Trinca, Daniele, Dayanandan, Rejive, Lee, Ryonfa, Nolan, John, Omata, Akiko, Green, Jennifer B., Cherney, David Z.I., Hooi, Lai Seong, Pontremoli, Roberto, Tuttle, Katherine R., Lees, Jennifer S., Mark, Patrick B., Davies, Simon J., Hauske, Sibylle J., Steubl, Dominik, Brückmann, Martina, Landray, Martin J., Baigent, Colin, Haynes, Richard, Herrington, William G., Baigent, Colin, Landray, Martin J., Wanner, Christoph, Herrington, William G., Haynes, Richard, Green, Jennifer B., Hauske, Sibylle J., Brueckmann, Martina, Hopley, Mark, von-Eynatten, Maximillian, George, Jyothis, Brenner, Susanne, Cheung, Alfred K., Preiss, David, Liu, Zhi-Hong, Li, Jing, Hooi, Laiseong, Liu, Wen, Kadowaki, Takashi, Nangaku, Masaomi, Levin, Adeera, Cherney, David, Pontremoli, Roberto, Maggioni, Aldo P., Staplin, Natalie, Emberson, Jonathan, Hantel, Stefan, Goto, Shinya, Deo, Rajat, Tuttle, Katherine R., Hill, Michael, Judge, Parminder, Mayne, Kaitlin J., Ng, Sarah Y.A., Rossello, Xavier, Sammons, Emily, Zhu, Doreen, Sandercock, Peter, Bilous, Rudolf, Herzog, Charles, Whelton, Paul, Wittes, Janet, Bennett, Derrick, Achiri, Patricia, Ambrose, Chrissie, Badin, Cristina, Barton, Jill, Brown, Richard, Burke, Andy, Butler, Sebastian, Dayanandan, Rejive, Donaldson, Pia, Dykas, Robert, Fletcher, Lucy, Frederick, Kate, Kingston, Hannah, Gray, Mo, Harding, Emily, Hashimoto, Akiko, Howie, Lyn, Hurley, Susan, Lee, Ryonfa, Luker, Nik, Murphy, Kevin, Nakahara, Mariko, Nolan, John, Nunn, Michelle, Mulligan, Sorcha, Omata, Akiko, Pickworth, Sandra, Qiao, YanRu, Shah, Shraddha, Taylor, Karen, Timadjer, Alison, Willett, Monique, Wincott, Liz, Yan, Qin, Yu, Hui, Bowman, Louise, Chen, Fang, Clarke, Robert, Goonasekera, Michelle, Haynes, Richard, Herrington, William G., Judge, Parminder, Karsan, Waseem, Mafham, Marion, Mayne, Kaitlin J., Ng, Sarah Y. A., Preiss, David, Reith, Christina, Sammons, Emily, Zayed, Mohammed, Zhu, Doreen, Ellison, Ritva, Moys, Rowan, Stevens, Will, Verdel, Kevin, Wallendszus, Karl, Bowler, Chris, Brewer, Anna, Measor, Andy, Cui, Guanguo, Daniels, Charles, Field, Angela, Goodenough, Bob, Lawson, Ashley, Mostefai, Youcef, Radhakrishnan, Dheeptha, Syed, Samee, Xia, Shuang, Adewuyi-Dalton, Ruth, Arnold, Thomas, Beneat, Anne-Marie, Bhatt, Anoushka, Bird, Chloe, Breach, Andrew, Brown, Laura, Caple, Mark, Chavagnon, Tatyana, Chung, Karen, Clark, Sarah, Condurache, Luminita, Eichstadt, Katarzyna, Obrero, Marta Espino, Forest, Scarlett, French, Helen, Goodwin, Nick, Gordon, Andrew, Gordon, Joanne, Guest, Cat, Harding, Tina, Hill, Michael, Hozak, Michal, Lacey, Matthew, MacLean, David, Messinger, Louise, Moffat, Stewart, Radley, Martin, Shenton, Claire, Tipper, Sarah, Tyler, Jon, Weaving, Lesley, Wheeler, James, Williams, Elissa, Williams, Tim, Woodhouse, Hamish, Chamberlain, Angela, Chambers, Jo, Davies, Joanne, Donaldson, Denise, Faria-Shayler, Pati, Fleming-Brown, Denise, Ingell, Jennifer, Knott, Carol, Liew, Anna, Lochhead, Helen, Meek, Juliette, Rodriguez-Bachiller, Isabel, Wilson, Andrea, Zettergren, Patrick, AitSadi, Rach, Barton, Ian, Baxter, Alex, Bu, Yonghong, Danel, Lukasz, Grotjahn, Sonja, Kurien, Rijo, Lay, Michael, Maskill, Archie, Murawska, Aleksandra, Raff, Rachel, Young, Allen, Baigent, Colin, Haynes, Richard, Herrington, William G., Landray, Martin J., Preiss, David, Emberson, Jonathan, Sardell, Rebecca, Staplin, Natalie, Wanner, Christoph, Brenner, Susanne, Cejka, Vladimir, Fajardo-Moser, Marcela, Hartner, Christian, Poehler, Doris, Renner, Janina, Scheidemantel, Franziska, Haynes, Richard, Preiss, David, Herrington, William G., Judge, Parminder, Zhu, Doreen, Ng, Sarah Y. A., Mayne, Kaitlin J., Badin, Cristina, Chambers, Jo, Davies, Joanne, Donaldson, Denise, Gray, Mo, Harding, Emily, Ingell, Jenny, Qiao, Yanru, Shah, Shraddha, Wilson, Andrea, Zettergren, Patrick, Wanner, Christoph, Brenner, Susanne, Cejka, Vladimir, Ghavampour, Sharang, Knoppe, Anja, Schmidt-Gurtler, Hans, Dumann, Hubert, Merscher, Sybille, Patecki, Margret, Schlieper, Georg Rainer, Torp, Anke, Weber, Bianca, Zietz, Maja, Sitter, Thomas, Fuessl, Louise, Krappe, Julia, Loutan, Jerome, Vielhauer, Volker, Andriaccio, Luciano, Maurer, Magdalena, Winkelmann, Bernhard, Dursch, Martin, Seifert, Linda, Tenbusch, Linda, Weinmann-Menke, Julia, Boedecker, Simone, KaluzaSchilling, Wiebke, Kraus, Daniel, Krieger, Carina, Schmude, Margit, Schreiber, Anne, Eckrich, Ewelina, Tschope, Diethelm, Arbi, Abdulwahab, Lee-Barkey, Young, Stratmann, Bernd, Prib, Natalie, Rolfsmeier, Sina, Schneider, Irina, Rump, Lars, Stegbauer, Johannes, Pötz, Christine, Schemmelmann, Mara, Schmidt, Claudia, Haller, Hermann, Kaufeld, Jessica, Menne, Jan, Bahlmann-Kroll, Elisabeth, Bergner, Angela, Haynes, Richard, Herrington, William G., Zhu, Doreen, Gavrila, Madita, Lafferty, Kathryn, Rabara, Ria, Ruse, Sally, Weetman, Maria, Byrne, Cath, Jesky, Mark, Cowley, Alison, McHaffie, Emma, Waterfall, Holly, Taylor, Jo, Bough, Laura, Phillips, Thomas, Goodwin, Barbara Winter-, Frankel, Andrew, Tomlinson, James, Alegata, Marlon, Almasarwah, Rashid, Apostolidi, Anthoula, Vourvou, Maria, Walters, Thomas, Ugni, Shiva, Gunda, Smita, Oluyombo, Rotimi, Brindle, Vicki, Coutts, Ping, Fuller, Tracy, Nadar, Evelyn, Wong, Christopher, Goldsmith, Christopher, Barnes, Sherald, Bennett, Ann, Burston, Claire, Hope, Samantha, Hunt, Nicola, Kurian, Lini, Fish, Richard, Farrugia, Daniela, Lee, Judy, Sadler, Emma, Turner, Hannah, Clarke, Helen, Carnall, Victoria, Benyon, Sarah, Blake, Caroline, Estcourt, Stephanie, Piper, Jane, Morgan, Neal, Hutchinson, Carolyn, McKinley, Teresa, Doulton, Tim, Delaney, Michael, Montasser, Mahmoud, Hansen, Jenny, Loader, David, Moon, Angela, Morris, Frances, Fraser, Donald, Ali, Mohammad Alhadj, Griffin, Sian, Latif, Farah, Witczak, Justyna, Wonnacott, Alexa, Jeffers, Lynda, Webley, Yvette, Bell, Samira, Cosgrove, Leanne, Craik, Rachel, Murray, Shona, Khwaja, Arif, Jackson, Yvonne, Mbuyisa, Angeline, Sellars, Rachel, Lewington, Andrew, Baker, Richard, Dorey, Suzannah, Tobin, Kay, Wheatley, Rosalyn, Patel, Rajan, Mark, Patrick, Rankin, Alastair, Sullivan, Michael, Forsyth, Kirsty, and McDougall, Rowan

Published
2024
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66. Yang-Mills sources in biconformal gravity

Author

Muhwezi, Davis W. and Wheeler, James T.

Subjects
High Energy Physics - Theory, General Relativity and Quantum Cosmology
Abstract

Biconformal gravity, based on gauging of the conformal group to 2n dimensions, reproduces n-dim scale-covariant general relativity on the co-tangent bundle in any dimension. We generalize this result to include Yang-Mills matter sources formulated as SU\left(N\right) gauge theories with a twisted action on the full 2n-dimensional biconformal space. We show that the coupling of the sources to gravity does not stop the reduction of effective dimension 2n\rightarrow n of the gravity theory, and instead forces the Yang-Mills source to reduce to ordinary n-dimensional Yang-Mills theory on the gravitating cotangent bundle, with the usual Yang-Mills energy tensor as gravitational source. The results apply as well to gravity with sources on K\"ahler manifolds and in double field theories., Comment: 67 pages including appendices and references Clarified discussion of Kahler and Killing forms; added discussion of alternative Weyl gravity sources

Published
2020
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68. Growth in Some Finite Three-Dimensional Matrix Groups

Author

Murphy, Brendan and Wheeler, James

Subjects
Mathematics - Combinatorics
Abstract

We study the growth of product sets in some finite three-dimensional matrix groups. In particular, we prove two results about the group of $2\times 2$ upper triangular matrices over arbitrary finite fields: a product set estimate using techniques from multiplicative combinatorics, and an energy estimate using incidence geometry. The energy method gives better quantitative results, but only applies to small sets. We also prove an energy result for the Heisenberg group., Comment: 20 pages

Published
2020

69. Flatly Foliated Relativity

Author

Bray, Hubert, Hamm, Benjamin, Hirsch, Sven, Wheeler, James, and Zhang, Yiyue

Subjects
General Relativity and Quantum Cosmology, Mathematics - Analysis of PDEs, Mathematics - Differential Geometry
Abstract

Flatly Foliated Relativity (FFR) is a new theory which conceptually lies between Special Relativity (SR) and General Relativity (GR), in which spacetime is foliated by flat Euclidean spaces. While GR is based on the idea that "matter curves spacetime", FFR is based on the idea that "matter curves spacetime, but not space". This idea, inspired by the observed spatial flatness of our local universe, is realized by considering the same action as used in GR, but restricting it only to metrics which are foliated by flat spatial slices. FFR can be thought of as describing gravity without gravitational waves. In FFR, a positive cosmological constant implies several interesting properties which do not follow in GR: the metric equations are elliptic on each euclidean slice, there exists a unique vacuum solution among those spherically symmetric at infinity, and there exists a geometric way to define the arrow of time. Furthermore, as gravitational waves do not exist in FFR, there are simple analogs to the positive mass theorem and Penrose-type inequalities. Importantly, given that gravitational waves have a negligible effect on the curvature of spacetime, and that the universe appears to be locally flat, FFR may be a good approximation of GR. Moreover, FFR still admits many notable features of GR including the big bang, an accelerating expansion of the universe, and the Schwarzschild spacetime. Lastly, FFR is already known to have an existence theory for some simplified cases, which provokes an interesting discussion regarding the possibility of a more general existence theory, which may be relevant to understanding existence of solutions to GR., Comment: 29 pages, 2 figures

Published
2019

81. Notes

Author

Wheeler, James Scott

82. Index

Author

Wheeler, James Scott

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