Mathematics and StatisticsDepartment of Mathematics and Statisticshttps://hdl.handle.net/2144/9982022-05-19T23:50:12Z2022-05-19T23:50:12ZSlopes of modular forms and reducible Galois representations: an oversight in the ghost conjecturePollack, RobertBergdall, Johnhttps://hdl.handle.net/2144/443552022-05-05T04:40:24ZSlopes of modular forms and reducible Galois representations: an oversight in the ghost conjecture
Pollack, Robert; Bergdall, John
The ghost conjecture, formulated by this article’s authors, predicts
the list of p-adic valuations of the non-zero ap-eigenvalues (“slopes”) for
overconvergent p-adic modular eigenforms in terms of the Newton polygon of
an easy-to-describe power series (the “ghost series”). The prediction is restricted
to eigenforms whose Galois representation modulo p is reducible on a
decomposition group at p. It has been discovered, however, that the conjecture
is not formulated correctly. Here we explain the issue and propose a salvage.
p-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou - IBuyukboduk, KazimPollack, RobertSasaki, Shuhttps://hdl.handle.net/2144/443452022-05-04T04:50:57Zp-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou - I
Buyukboduk, Kazim; Pollack, Robert; Sasaki, Shu
Let p be an odd prime. Given an imaginary quadratic field K = Q(sqrt(−D_K) where p splits with D_K > 3, and a p-ordinary newform f ∈ Sk(Γ0(N)) such that N verifies the Heegner hypothesis relative to K, we prove a p-adic Gross–Zagier formula for the critical slope p-stabilization of f (assuming that it is non-θ-critical). In the particular case when f = fA is the newform of weight 2 associated to an elliptic curve A that has good ordi- nary reduction at p, this allows us to verify a conjecture of Perrin-Riou. The p-adic Gross–Zagier formula we prove has applications also towards the Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one.
Functional Gaussian approximations in Hilbert spaces: the non-diffusive caseBourguin, SolesneCampese, SimonDang, Thanhhttps://hdl.handle.net/2144/443412022-05-05T07:49:40ZFunctional Gaussian approximations in Hilbert spaces: the non-diffusive case
Bourguin, Solesne; Campese, Simon; Dang, Thanh
We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting,
thus obtaining a) quantitative central limit theorems for approximation of arbitrary nondegenerate
Gaussian random elements taking values in a separable Hilbert space and b) fourth
moment bounds for approximating sequences with finite chaos expansion. Our results rely
on an infinite-dimensional version of Stein’s method of exchangeable pairs combined with the
so-called Gamma calculus. Two applications are included: Brownian approximation of Poisson
processes in Besov-Liouville spaces and a functional limit theorem for an edge-counting statistic
of a random geometric graph.
The Mutational signature comprehensive analysis toolkit (musicatk) for the discovery, prediction, and exploration of mutational signaturesChevalier, AaronYang, ShiyiKhurshid, ZainabSahelijo, NathanTong, TongHuggins, Jonathan H.Yajima, MasanaoCampbell, Joshua D.https://hdl.handle.net/2144/443282022-05-02T04:59:36Z2021-12-01T00:00:00ZThe Mutational signature comprehensive analysis toolkit (musicatk) for the discovery, prediction, and exploration of mutational signatures
Chevalier, Aaron; Yang, Shiyi; Khurshid, Zainab; Sahelijo, Nathan; Tong, Tong; Huggins, Jonathan H.; Yajima, Masanao; Campbell, Joshua D.
Mutational signatures are patterns of somatic alterations in the genome caused by carcinogenic exposures or aberrant cellular processes. To provide a comprehensive workflow for preprocessing, analysis, and visualization of mutational signatures, we created the Mutational Signature Comprehensive Analysis Toolkit (musicatk) package. musicatk enables users to select different schemas for counting mutation types and to easily combine count tables from different schemas. Multiple distinct methods are available to deconvolute signatures and exposures or to predict exposures in individual samples given a pre-existing set of signatures. Additional exploratory features include the ability to compare signatures to the Catalogue Of Somatic Mutations In Cancer (COSMIC) database, embed tumors in two dimensions with uniform manifold approximation and projection, cluster tumors into subgroups based on exposure frequencies, identify differentially active exposures between tumor subgroups, and plot exposure distributions across user-defined annotations such as tumor type. Overall, musicatk will enable users to gain novel insights into the patterns of mutational signatures observed in cancer cohorts. SIGNIFICANCE: The musicatk package empowers researchers to characterize mutational signatures and tumor heterogeneity with a comprehensive set of preprocessing utilities, discovery and prediction tools, and multiple functions for downstream analysis and visualization.
2021-12-01T00:00:00ZComparison of SG/AG mirror constructionsLin, Yu-ShenCheung, Man-Waihttps://hdl.handle.net/2144/443262022-05-02T04:59:34ZComparison of SG/AG mirror constructions
Lin, Yu-Shen; Cheung, Man-Wai
The Torelli theorem for ALH∗ gravitational instantonsLin, Yu-ShenCollins, Tristan C.Jacob, Adamhttps://hdl.handle.net/2144/443252022-05-02T04:59:33ZThe Torelli theorem for ALH∗ gravitational instantons
Lin, Yu-Shen; Collins, Tristan C.; Jacob, Adam
We give a short proof of the Torelli theorem for ALH∗ gravitational instantons, using the authors' previous construction of mirror special Lagrangian fibrations in del Pezzo surfaces and rational elliptic surfaces together with recent work of Sun-Zhang. In particular, this includes an identification of 10 diffeomorphism types of ALH∗b gravitational instantons.
Scattering diagrams from holomorphic discs in log Calabi-Yau surfacesLin, Yu-ShenBardwell-Evans, SamCheung, Man-Wai MandyHong, Hansolhttps://hdl.handle.net/2144/443242022-05-02T04:59:32ZScattering diagrams from holomorphic discs in log Calabi-Yau surfaces
Lin, Yu-Shen; Bardwell-Evans, Sam; Cheung, Man-Wai Mandy; Hong, Hansol
We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert and the canonical scattering diagrams of Gross-Hacking-Keel. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.
Some examples of family Floer mirrorsLin, Yu-ShenCheung, Man-Waihttps://hdl.handle.net/2144/443222022-05-02T04:59:31ZSome examples of family Floer mirrors
Lin, Yu-Shen; Cheung, Man-Wai
In this article, we give explicit calculations for the family Floer mirrors of some non-compact Calabi-
Yau surfaces. We compare it with the mirror construction of Gross-Hacking-Keel-Siebert for suitably
chosen log Calabi-Yau pairs and the rank two cluster varieties of nite type. In particular, the analytifications of the later two give partial compactifications of the family Floer mirrors that we computed.
The problem of perfect predictors in statistical spike train modelsFarhoodi, SahandEden, Uri T.https://hdl.handle.net/2144/442492022-04-20T04:33:53Z2021-08-23T00:00:00ZThe problem of perfect predictors in statistical spike train models
Farhoodi, Sahand; Eden, Uri T.
2021-08-23T00:00:00ZChildren’s use of reasoning by exclusion to track identities of occluded objectsCheng, ChenKibbe, Melissa M.https://hdl.handle.net/2144/441112022-04-01T04:57:24Z2021-08-01T00:00:00ZChildren’s use of reasoning by exclusion to track identities of occluded objects
Cheng, Chen; Kibbe, Melissa M.
Reasoning by exclusion allows us to infer properties of
unobserved objects from currently observed objects,
formalized by P or Q, not P, therefore Q. Previous work
suggested that, by age 3, children can use this kind of reasoning
to infer the location of a hidden object after learning that
another location is empty (e.g. Mody & Carey, 2016). In the
current study, we asked whether children could use reasoning
by exclusion to infer the identities of previously unobserved
occluded objects in a task that required them to track the
locations of multiple occluded objects. Forty-nine 4-7-yearolds
viewed animated arrays of virtual “cards” depicting
images which were then hidden by occluders. The occluders
then swapped locations during the maintenance period.
Children were asked to select which card was hidden in a
probed location. During the encoding period, we manipulated
whether children saw all the card faces (Face-up block) or all
but one of the card faces (Exclusion block), for which children
had to reason by exclusion to infer the target in half of the trials.
We found that all children succeeded in the Face-up block, but
only 6-year-olds succeed in the Exclusion block when they had
to deploy logical reasoning to identify a previously-unseen
hidden target. Our results suggest that children’s ability to
reason by exclusion to infer the identity of a hidden target while
tracking multiple objects and locations may undergo protracted
development.
2021-08-01T00:00:00ZKaehler geometry of quiver varieties and machine learningLau, Siu-cheongJeffreys, Georgehttps://hdl.handle.net/2144/441102022-04-01T04:57:11ZKaehler geometry of quiver varieties and machine learning
Lau, Siu-cheong; Jeffreys, George
We develop an algebro-geometric formulation for neural networks in machine
learning using the moduli space of framed quiver representations. We find natural Hermitian
metrics on the universal bundles over the moduli which are compatible with the GIT
quotient construction by the general linear group, and show that their Ricci curvatures
give a Kahler metric on the moduli. Moreover, we use toric moment maps to construct
activation functions, and prove the universal approximation theorem for the multi-variable
activation function constructed from the complex projective space.
Decontamination of ambient RNA in single-cell RNA-seq with DecontXYang, ShiyiCorbett, Sean E.Koga, YusukeWang, ZheJohnson, W. EvanYajima, MasanaoCampbell, Joshua D.https://hdl.handle.net/2144/439562022-03-04T04:48:22Z2020-03-05T00:00:00ZDecontamination of ambient RNA in single-cell RNA-seq with DecontX
Yang, Shiyi; Corbett, Sean E.; Koga, Yusuke; Wang, Zhe; Johnson, W. Evan; Yajima, Masanao; Campbell, Joshua D.
Droplet-based microfluidic devices have become widely used to perform single-cell RNA sequencing (scRNA-seq). However, ambient RNA present in the cell suspension can be aberrantly counted along with a cell's native mRNA and result in cross-contamination of transcripts between different cell populations. DecontX is a novel Bayesian method to estimate and remove contamination in individual cells. DecontX accurately predicts contamination levels in a mouse-human mixture dataset and removes aberrant expression of marker genes in PBMC datasets. We also compare the contamination levels between four different scRNA-seq protocols. Overall, DecontX can be incorporated into scRNA-seq workflows to improve downstream analyses.
2020-03-05T00:00:00ZPKC downregulation upon rapamycin treatment attenuates mitochondrial diseaseMartin-Perez, MiguelGrillo, Anthony S.Ito, Takashi K.Valente, Anthony S.Han, JeehaeEntwisle, Samuel W.Huang, Heather Z.Kim, DayaeYajima, MasanaoKaeberlein, MattVillén, Judithttps://hdl.handle.net/2144/439552022-03-04T04:48:08Z2020-12-01T00:00:00ZPKC downregulation upon rapamycin treatment attenuates mitochondrial disease
Martin-Perez, Miguel; Grillo, Anthony S.; Ito, Takashi K.; Valente, Anthony S.; Han, Jeehae; Entwisle, Samuel W.; Huang, Heather Z.; Kim, Dayae; Yajima, Masanao; Kaeberlein, Matt; Villén, Judit
Leigh syndrome is a fatal neurometabolic disorder caused by defects in mitochondrial function. Mechanistic target of rapamycin (mTOR) inhibition with rapamycin attenuates disease progression in a mouse model of Leigh syndrome (Ndufs4 knock-out (KO) mouse); however, the mechanism of rescue is unknown. Here we identify protein kinase C (PKC) downregulation as a key event mediating the beneficial effects of rapamycin treatment of Ndufs4 KO mice. Assessing the impact of rapamycin on the brain proteome and phosphoproteome of Ndufs4 KO mice, we find that rapamycin restores mitochondrial protein levels, inhibits signalling through both mTOR complexes and reduces the abundance and activity of multiple PKC isoforms. Administration of PKC inhibitors increases survival, delays neurological deficits, prevents hair loss and decreases inflammation in Ndufs4 KO mice. Thus, PKC may be a viable therapeutic target for treating severe mitochondrial disease.
Published in final edited form as:
Nat Metab. 2020 December ; 2(12): 1472–1481. doi:10.1038/s42255-020-00319-x.
2020-12-01T00:00:00ZSystems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial dataSalins, Michaelhttps://hdl.handle.net/2144/439412022-03-02T10:26:03Z2021-12-01T00:00:00ZSystems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial data
Salins, Michael
Large deviations principles characterize the exponential decay rates of the probabilities of rare events. Cerrai and Röckner (2004) proved that systems of stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over bounded sets of initial data.
This paper proves uniform large deviations results for a system of stochastic reaction–diffusion equations in a more general setting than Cerrai and Röckner. Furthermore, this paper identifies two common situations where the large deviations principle is uniform over unbounded sets of initial data, enabling the characterization of Freidlin–Wentzell exit time and exit shape asymptotics from unbounded sets.
2021-12-01T00:00:00ZExistence and uniqueness of global solutions to the stochastic heat equation with superlinear drift on an unbounded spatial domainSalins, Michaelhttps://hdl.handle.net/2144/439402022-03-02T10:29:20Z2021-12-28T00:00:00ZExistence and uniqueness of global solutions to the stochastic heat equation with superlinear drift on an unbounded spatial domain
Salins, Michael
We prove the existence and uniqueness of global solutions to the
semilinear stochastic heat equation on an unbounded spatial domain
with forcing terms that grow superlinearly and satisfy an Osgood condition R
1/|f(u)|du = +∞ along with additional restrictions. For example, consider the forcing f(u) = u log(e
e + |u|) log(log(e
e + |u|)). A
new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.
2021-12-28T00:00:00ZGlobal solutions for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativitySalins, Michaelhttps://hdl.handle.net/2144/439382022-03-02T04:41:46Z2022-01-01T00:00:00ZGlobal solutions for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity
Salins, Michael
A condition is identified that implies that solutions to the stochastic reaction-diffusion
equation ∂u
∂t = Au + f(u) + σ(u)W˙ on a bounded spatial domain never explode.
We consider the case where σ grows polynomially and f is polynomially dissipative,
meaning that f strongly forces solutions toward finite values. This result demonstrates
the role that the deterministic forcing term f plays in preventing explosion
2022-01-01T00:00:00ZChallenges and opportunities in high-dimensional variational inferenceDhaka, Akash KumarCatalina, AlejandroWelandawe, ManushiHuggins, Jonathan H.Riis Andersen, MichaelVehtari, Akihttps://hdl.handle.net/2144/438012022-02-09T19:21:58Z2021-12-06T00:00:00ZChallenges and opportunities in high-dimensional variational inference
Dhaka, Akash Kumar; Catalina, Alejandro; Welandawe, Manushi; Huggins, Jonathan H.; Riis Andersen, Michael; Vehtari, Aki
We explore the limitations of and best practices for using black-box variational inference to estimate posterior summaries of the model parameters. By taking an importance sampling perspective, we are able to explain and empirically demonstrate: 1) why the intuitions about the behavior of approximate families and divergences for low-dimensional posteriors fail for higher-dimensional posteriors, 2) how we can diagnose the pre-asymptotic reliability of variational inference in practice by examining the behavior of the density ratios (i.e., importance weights), 3) why the choice of variational objective is not as relevant for higher-dimensional posteriors, and 4) why, although flexible variational families can provide some benefits in higher dimensions, they also introduce additional optimization challenges. Based on these findings, for high-dimensional posteriors we recommend using the exclusive KL divergence that is most stable and easiest to optimize, and then focusing on improving the variational family or using model parameter transformations to make the posterior more similar to the approximating family. Our results also show that in low to moderate dimensions, heavy-tailed variational families and mass-covering divergences can increase the chances that the approximation can be improved by importance sampling.
2021-12-06T00:00:00ZDihedral Artin representations and CM fieldsRohrlich, David E.https://hdl.handle.net/2144/437862022-02-08T06:11:27Z2022-01-01T00:00:00ZDihedral Artin representations and CM fields
Rohrlich, David E.
For a fixed CM field K with maximal totally real subfield F, we
consider dihedral Artin representations of F induced from K. We prove that
a positive proportion of such representations have image D4.
2022-01-01T00:00:00ZQuaternionic Artin representations and nontraditional arithmetic statisticsRohrlich, David E.https://hdl.handle.net/2144/437852022-02-08T06:11:53Z2019-06-13T00:00:00ZQuaternionic Artin representations and nontraditional arithmetic statistics
Rohrlich, David E.
We classify and then attempt to count the real quadratic fields
(ordered by the size of the totally positive fundamental unit, as in Sarnak
[14], [15]) from which quaternionic Artin representations of minimal conductor
can be induced. Some of our results can be interpreted as criteria for a real
quadratic field to be contained in a Galois extension of Q with controlled
ramification and Galois group isomorphic to a generalized quaternion group.
2019-06-13T00:00:00ZAsymptotic approximation of a modified compressible Navier-Stokes systemGoh, RyanWelter, RolandWayne, C. Eugenehttps://hdl.handle.net/2144/437822022-02-08T10:31:09Z2022-01-01T00:00:00ZAsymptotic approximation of a modified compressible Navier-Stokes system
Goh, Ryan; Welter, Roland; Wayne, C. Eugene
We study the effects of localization on the long time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of jointly hyperbolic-parabolic equations in terms of Hermite functions in which $n^{th}$ order approximations can be computed for solutions with $n^{th}$ order moments. We then obtain existence of solutions to the mcNS system in weighted spaces and, based on the decay rates obtained for the various pieces of the solutions, determine the optimal choice of asymptotic approximation with respect to the various localization assumptions, which in certain cases can be evaluated explicitly in terms of Hermite functions.
2022-01-01T00:00:00Z