Michael Goldberg Professor Department Head
Office: 4114 French Hall West Phone: 513-556-4052 Fax: 513-556-3417 E-mail: Michael . Goldberg @ uc . edu	Mailing address: Dept. of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025

Research Summary
I use techniques from Fourier analysis to study partial differential equations.
(I also use partial differential equations as an excuse to do Fourier analysis!)

My current projects involve linear dispersive estimates for the Schrödinger,
wave and/or Dirac equations with a short-range potential, plus some more
classical Fourier Transform bounds.

The 14^th Ohio River Analysis Meeting will take place March 29-30, 2025 at the University of Cincinnati!
Homepages for previous meetings: 2024 2023, 2022, 2021, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011.

These slides from some of my past talks discuss specific problems and results.
The work is currently supported by the Simons Foundation under grant #635369.

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Teaching

Courses taught at UC:

• Math 1062 - Calculus II [Spring 2020]
• Math 2063 - Multivariable Calculus [Fall 2016] [Spring 2019] [Spring 2021]
• Math 2073 - Ordinary Differential Equations [Fall 2017] [Fall 2018]
• Math 2074 - Dynamical Systems [Spring 2015]
• Math 6003 - Abstract Linear Algebra [Fall 2022] [Fall 2023] [Fall 2024]
• Math 6007 - PDEs and Fourier Analysis [Spring 2016]
• Math 7004 - Topology
• Math 7005 - Ordinary Differential Equations [Fall 2016] [Fall 2019]
• Math 7006 - Partial Differential Equations [Spring 2015] [Spring 2018]
• Math 8003 - Functional Analysis [Fall 2015]
• Math 251 - Calculus I
• Math 252 - Calculus II
• Math 253 - Calculus III
• Math 627-629 - Partial Differential Equations

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Publications and Preprints

Due to copyright restrictions, some papers hosted on this site appear in their (unrevised) preprint version.
For access to the peer-reviewed version of record, please click on the journal title to visit the publisher's homepage.

• Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials (with M. B. Erdogan and W. Green), preprint 2023.  [pdf]
• Spectral multipliers and wave propagation for Hamiltonians with a scalar potential (With M. Beceanu), J. Funct. Anal. 286 (2024), no. 6, Article 110300.  [pdf]
• Counterexamples to L^p boundedness of wave operators for classical and higher order Schröodinger operators (with M. B. Erdogan and W. Green) J. Funct. Anal. 285 (2023), no. 5, Article 110008.  [pdf]
• On the L^p Boundedness of the Wave Operators for Fourth Order Schrödinger Operators (with W. Green), Trans. Amer. Math. Soc. 374 (2021), 4075-4092.  [pdf]
• Time Integrable Weighted Dispersive Estimates for the Fourth Order Schrödinger Equation in Three Dimensions (with W. Green), Bull. London Math. Soc. 54 (2022), no. 2, 428-448.  [pdf]
• Strichartz Estimates for the Schrödinger Equation with a Measure-Valued Potential (with M. B. Erdogan and W. Green), Proc. Amer. Math. Soc. Ser. B, 8 (2021), 336-348.  [pdf]
• Restrictions of Higher Derivatives of the Fourier Transform (with D. Stolyarov), Trans. Amer. Math. Soc. Ser. B 7 (2020), 46-96.  [pdf]
• The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates (with M. B. Erdogan and W. Green), J. Spectr. Theory 11 (2021), no. 3, 935-979.  [pdf]
• Limiting Absorption Principle and Strichartz Estimates for Dirac Operators in Two and Higher Dimensions (with M. B. Erdogan and W. Green), Comm. Math. Phys. 367 (2019), no. 1, 241-263.  [pdf]
• On the L^p Boundedness of Wave Operators for Two-Dimensional Schrödinger Operators with Threshold Obstructions (with M. B. Erdogan and W. Green), J. Funct. Anal. 274 (2018), no. 7, 2139-2161.  [pdf]
• On the L^p Boundedness of Wave Operators for Four-Dimensional Schrödinger Operators with a Threshold Eigenvalue (with W. Green), Ann. Henri Poincaré 18 (2017), no. 4, 1269-1288.  [pdf]
• The L^p Boundedness of Wave Operators for Schrödinger Operators with Threshold Singularities (with W. Green), Adv. Math. 303 (2016), 360-389.  [pdf]
• The Helmholtz Equation with L^p Data and Bochner-Riesz Multipliers. Math. Res. Lett. 23 (2016), no. 6, 1665-1679.  [pdf]
• Dispersive Estimates for Higher Dimensional Schrödinger Operators with Threshold Eigenvalues I: The Odd Dimensional Case (with W. Green). J. Funct. Anal., 269 (2015), no. 3, 633-682.  [pdf]
• Dispersive Estimates for Higher Dimensional Schrödinger Operators with Threshold Eigenvalues II: The Even Dimensional Case (with W. Green), J. Spectr. Theory 7 (2017), no. 1, 33-86.  [pdf]
• Dispersive Estimates for Four Dimensional Schrödinger and Wave Equations with Obstructions at Zero Energy (with M. B. Erdogan and W. Green). Comm. PDE, 39 (2014), no. 10, 1936-1964.  [pdf]
• The Klein-Gordon Equation on Z² and the Quantum Harmonic Lattice (with V. Borovyk). J. Math. Pures Appl. (9) 107 (2017), no. 6, 667-696.  [pdf]
• Strichartz Estimates and Maximal Operators for the Wave Equation in R³ (with M. Beceanu). J. Funct. Anal. 266 (2014), no. 3, 1476-1510.  [pdf]
• Dispersive Estimates for Schrödinger Operators with Measure-Valued Potentials in R³. Indiana Univ. Math. J. 61 (2012), no. 6, 2123-2141.  [pdf]
• Schrödinger Dispersive Estimates for a Scaling-Critical Class of Potentials (with M. Beceanu), Comm. Math. Phys. 314 (2012), no. 2, 471-481.  [pdf] 
• A Dispersive Bound for Three-Dimensional Schrödinger Operators with Zero Energy Eigenvalues, Comm. PDE 35 (2010), 1610-1634.  [pdf] 
• Strichartz Estimates for Schrödinger Operators with a Non-Smooth Magnetic Potential, Discrete Contin. Dyn. Syst. 31 (2011), no. 1, 109-118.   [pdf] 
• Strichartz Estimates for the Schrödinger Equation with Time-Periodic L^n/2 Potentials, J. Funct. Anal. 256 (2009), 718-746.  [dvi]  [ps]  [pdf] 
• Strichartz and Smoothing Estimates for Schrödinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions (with M. B. Erdogan and W. Schlag), Forum Math. 21 (2009), no. 4, 687-722.   [dvi]  [ps]  [pdf] 
• Strichartz and Smoothing Estimates for Schrödinger Operators with Large Magnetic Potentials in R³ (with M. B. Erdogan and W. Schlag), J. Eur. Math. Soc. 10 (2008), no. 2, 507-531.   [dvi]  [pdf] 
• Transport in the One-Dimensional Schrödinger Equation, Proc. Amer. Math. Soc. 135 (2007), 3171-3179.   [pdf] 
• Counterexamples of Strichartz Inequalities for Schrödinger Equations with Repulsive Potentials (with L. Vega and N. Visciglia), Intl. Math. Res. Not. 2006 (2006), Article ID 13927, 16pp.   [dvi]  [pdf] 
• A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions (with M. Visan), Comm. Math. Phys. 266 (2006), no. 1, 211-238.  [dvi]  [pdf] 
• Dispersive Bounds for the Three-Dimensional Schrödinger Equation with Almost Critical Potentials, Geom. and Funct. Anal. 16 (2006), no. 3, 517-536.  [dvi]  [ps]  [pdf]
• Dispersive Estimates for the Three-Dimensional Schrödinger Equation with Rough Potentials, Amer. J. Math. 128 (2006) 731-750.   [dvi]  [ps]  [pdf]
• A Limiting Absorption Principle for the Three-Dimensional Schrödinger Equation with L^p Potentials (with W. Schlag), Intl. Math. Res. Not. 2004:75 (2004), 4049-4071.  [dvi]  [ps]  [pdf]
• Dispersive Estimates for Schrödinger Operators in Dimensions One and Three (with W. Schlag), Comm. Math. Phys. 251 (2004), no. 1, 157-178.  [dvi]  [ps]  [pdf]
• Matrix A_p Weights via Maximal Functions, Pac. J. Math. 211 (2003), 201-220.  [pdf]
• Asymptotic Properties of the Vector Carleson Embedding Theorem, Proc. Amer. Math. Soc. 130 (2002), 529-531.  [pdf]
• Vector A₂ Weights and a Hardy-Littlewood Maximal Function (with M. Christ), Trans. Amer. Math. Soc. 353 (2001), 1995-2002.  [pdf]

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Education
AB., Mathematics, Princeton University, 1997
Ph.D., Mathematics, University of California, Berkeley, 2002

Here is my full Curriculum Vitae .

Fun Stuff

U.S. National Weather Service	Geyser Observation and Study Association
Current weather watches and warnings from the NWS	Old Faithful Webcam from the National Park Service in Yellowstone

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