Matthew Badger | Department of Mathematics | University of Connecticut
Matthew Badger

Matthew Badger
University of Connecticut
Department of Mathematics
341 Mansfield Road, U-1009
Storrs, CT 06269-1009

Office: Monteith 326
Email: firstname.lastname _at_ uconn.edu

Spring 2024 Office Hours
M By Appointment
Tu 10:45-11:45
W 1:30-2:30
Th By Appointment
F By Appointment

Office hours are held in
Monteith 326

Geometry of Sets and Measures

I am an Associate Professor of Mathematics. I study the geometry of sets and measures in Euclidean space and general metric spaces using a mixture of geometric measure theory, harmonic analysis and quasiconformal analysis.
My recent work has focused on the question of how to construct good parameterizations of connected sets by Lipschitz and Hölder continuous maps.

harmonic measure of a subset of the spherea quasicirclea 1-rectifiable measure

Quick Links: [Curriculum Vitae | Teaching | Research]

Analysis at UConn

Analysis and Probability Seminar This is a research seminar featuring speakers from around the world. Organized by Sean Li and myself. In Spring 2024, we will hold the seminar on Tuesdays at 3:30.

PDE and Differential Geometry Seminar Another research seminar featuring latest advances in the field. Mondays at 2:30.

Recent and Upcoming Events

Geometry of Measures and Free Boundaries 2024: A conference in honor of Tatiana Toro. July 22-26, 2024 at University of Washington in Seattle. Introductory Mini-Courses on July 20 and 21.

2023 Northeast Analysis Network: September 23-24 at University of Rochester. This is a regional conference series in collaboration with SUNY Albany, Syracuse University, and University of Rochester. (We last hosted the conference at UConn in 2019.)

Geometric and Harmonic Analysis: a Conference for Graduate Students: March 29-31, 2019.

Nonsmooth Analysis: a Workshop for Postdocs: November 9-11, 2017.

[Archive of Past Events]

Teaching

[Math Course Schedules: Current Semester]

Spring 2024

Math 3151 Analysis II

Course materials for students available on HuskyCT

Ph.D. Students

Lisa Naples, Ph.D. August 2020

I am happy to advise Ph.D. students who would like to carry out research at the interface of analysis and geometry.

I can support a UConn graduate student as a research assistant (no teaching duties) in Summer 2024, Fall 2024, and Summer 2025.

Research

A non-technical description of my research with Raanan Schul on rectifiable measures can be found here.

Here is a picture related to my "Harmonic polynomials..." and "Flat points..." papers. The zero sets of homogeneous harmonic polynomials in x,y,z of odd degree may separate space into two components (cross your eyes to see a stereographic picture):

Intersecting Varieties

500x4y-1000x2y3+100y5 -5(x4+y4)z+10(x2+y2)z3+2z5=0 intersecting the unit sphere

Grants and Fellowships

NSF DMS 2403968
Analysis Program. Conference Grant. Geometry of Measures and Free Boundaries 2024.
NSF DMS 2154047
Analysis Program. Standard Grant. 2022 – 2025
NSF DMS 1901256
Analysis Program. Conference Grant. Northeast Analysis Network. 2019 – 2023
NSF DMS 1650546
Analysis Program. CAREER Award. 2017 – 2022
NSF DMS 1500382
Analysis Program. Standard Grant. 2015 – 2018
NSF DMS 1203497
2012 NSF Mathematical Sciences Postdoctoral Research Fellowship

Publications and Preprints

[Statistics] Newest preprints/papers are listed first.

Example of the zero set of a degree 5 homogeneous caloric polynomial in R^{2+1} with two nodal domains. For increased visibility, we show the zero set intersected with the unit sphere

#27 On the number of nodal domains of homogeneous caloric polynomials
(arXiv:2401.07268)
(with Cole Jeznach)
[Click to Show/Hide Abstract]
Status: preprint, submitted.

A grid of 9 squares of 3 different side lengths

#26 Square packings and rectifiable doubling measures
(arXiv:2309.01283)
(with Raanan Schul)
[Click to Show/Hide Abstract]
Status: preprint, submitted.
#25 A practical guide to writing an NSF grant proposal (Published Version)
This article for the Early Career Section of the AMS Notices was written at the request of the editors.
Citation: M. Badger, A practical guide to writing an NSF grant proposal, Notices Amer. Math. Soc. 70 (2023), no. 7, 1089-1093.

Super rotation of Szulkin's variety

#24 Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem
(arXiv:2210.17531 | Published Version)
(with Max Engelstein and Tatiana Toro)
Dedicado a Carlos Kenig, un gran maestro y amigo en conmemoración de sus 70 años
[Click to Show/Hide Abstract]
Status: accepted, to appear in Vietnam J. Math., online first, doi:10.1007/s10013-023-00668-6
#23 Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs
(arXiv:2208.10288 | Published Version)
(with Sean McCurdy)
[Click to Show/Hide Abstract]
Citation: M. Badger, S. McCurdy, Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs, Illinois J. Math. 67 (2023), no. 2, 275-331.
#22 Lower bounds on Bourgain's constant for harmonic measure
(arXiv:2205.15101 | Published Version)
(with Alyssa Genschaw)
[Click to Show/Hide Abstract]
Citation: M. Badger, A. Genschaw, Lower bounds on Bourgain's constant for harmonic measure, Canad. J. Math., First View, 1--20. doi:10.4153/S0008414X2300069X
#21 Identifying 1-rectifiable measures in Carnot groups
(arXiv:2109.06753 | Published Version)
(with Sean Li and Scott Zimmerman)
[Click to Show/Hide Abstract]
Citation: M. Badger, S. Li, S. Zimmerman, Identifying 1-rectifiable measures in Carnot groups, Anal. Geom. Metr. Spaces 11 (2023), Paper no. 20230102, 40 pp.
#20 Hausdorff dimension of caloric measure
(arXiv:2108.12340)
(with Alyssa Genschaw)
[Click to Show/Hide Abstract]
Status: accepted, to appear in Amer. J. Math.
#19 Radon measures and Lipschitz graphs
(arXiv:2007.08503 | Published Version)
(with Lisa Naples)
[Click to Show/Hide Abstract]
Citation: M. Badger, L. Naples, Radon measures and Lipschitz graphs, Bull. London Math. Soc. 53 (2021), no. 3, 921-936. https://doi.org/10.1112/blms.12473
#18 Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems
(arXiv:2002.11878 | Published Version)
(with Sean McCurdy)
[Click to Show/Hide Abstract]
Note: While revising an earlier draft of the manuscript, we identified a mistake in Schul's 2007 proof of the necessary conditions in the traveling salesman theorem in infinite-dimensional Hilbert space. We show how to correct the error in a minimal way, leaving the outline of virtually all of the original proofs intact.
Citation: M. Badger, S. McCurdy, Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems, Illinois J. Math. 67 (2023), no. 2, 203-274.
#17 Hölder parameterization of iterated function systems and a self-affine phenomenon
(arXiv:1910.08850 | Published Version)
(with Vyron Vellis)
[Click to Show/Hide Abstract]
Citation: M. Badger, V. Vellis, Hölder parameterization of iterated function systems and a self-affine phenomenon, Anal. Geom. Metr. Spaces 9 (2021), no. 1, 90-119.
#16 Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data
(arXiv:1807.08002 | Published Version)
(with Max Engelstein and Tatiana Toro)
[Click to Show/Hide Abstract]
Citation: M. Badger, M. Engelstein, T. Toro, Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data, Rev. Mat. Iberoam. 36 (2020), no. 5, 1375–1408. https://doi.org/10.4171/rmi/1170
#15 Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem
(arXiv:1806.01197 | Published Version)
(with Lisa Naples and Vyron Vellis)
[Click to Show/Hide Abstract]
Citation: M. Badger, L. Naples, V. Vellis, Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem, Adv. Math. 349 (2019), 564-647. doi:10.1016/j.aim.2019.04.011
#14 Generalized rectifiability of measures and the identification problem
(arXiv:1803.10022 | Published Version)
[Click to Show/Hide Abstract]
Note: This survey paper is based on a talk at the Northeast Analysis Network Conference held in Syracuse, New York in September 2017.
Citation: M. Badger, Generalized rectifiability of measures and the identification problem, Complex Anal. Synerg. 5 (2019), 2.
#13 Geometry of measures in real dimensions via Hölder parameterizations
(arXiv:1706.07846 | Published Version)
(with Vyron Vellis)
[Click to Show/Hide Abstract]
Citation: M. Badger, V. Vellis, Geometry of measures in real dimensions via Hölder parameterizations, J. Geom. Anal. 29 (2019), no. 2, 1153-1192. doi:10.1007/s12220-018-0034-2
#12 Multiscale analysis of 1-rectifiable measures II: characterizations
(arXiv:1602.03823 | Published Version)
(with Raanan Schul)
[Click to Show/Hide Abstract]
Citation: M. Badger, R. Schul, Multiscale analysis of 1-rectifiable measures II: characterizations, Anal. Geom. Metr. Spaces 5 (2017), no. 1, 1-39.
Related: H. Martikainen and T. Orponen (arXiv:1604.04091) have constructed a finite measure in the plane with bounded density-normalized L2 Jones function and vanishing lower 1-density. This implies that our use of β** in Theorem D is sharp and answers a question we posed following Theorem E.
#11 Structure of sets which are well approximated by zero sets of harmonic polynomials
(arXiv:1509.03211 | Published Version)
(with Max Engelstein and Tatiana Toro)
[Click to Show/Hide Abstract]
Citation: M. Badger, M. Engelstein, T. Toro, Structure of sets which are well approximated by zero sets of harmonic polynomials, Anal. PDE 10 (2017), no. 6, 1455-1495.
#10 Rectifiability and elliptic measures on 1-sided NTA domains with Ahflors-David regular boundaries
(arXiv:1507.02039 | Published Version)
(with Murat Akman, Steve Hofmann, and José María Martell)
[Click to Show/Hide Abstract]
Citation: M. Akman, M. Badger, S. Hofmann, J.M. Martell, Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries, Trans. Amer. Math. Soc. 369 (2017), no. 8, 2017, 5711-5745.
#9 Two sufficient conditions for rectifiable measures
(arXiv:1412.8357 | Published Version)
(with Raanan Schul)
[Click to Show/Hide Abstract]
Citation: M. Badger, R. Schul, Two sufficient conditions for rectifiable measures, Proc. Amer. Math. Soc. 144 (2016), 2445-2454.
#8 Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets
(arXiv:1409.7851 | Published Version)
(with Stephen Lewis)
[Click to Show/Hide Abstract]
Note: The arXiv version of the paper has outdated numbering. The published version is open access and is the authoritative version. On the other hand, the arXiv version has an additional example related to harmonic polynomials that the referee asked us to cut, but which is still interesting! See section 9.3 in the arXiv version.
Citation: M. Badger, S. Lewis, Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets, Forum Math. Sigma 3 (2015), e24, 63 pp.
#7 Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps
(arXiv:1403.2991 | Published Version)
(with Jonas Azzam, and Tatiana Toro)
[Click to Show/Hide Abstract]
Citation: J. Azzam, M. Badger, T. Toro, Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps, Adv. Math. 275 (2015), 195-259.
#6 Multiscale analysis of 1-rectifiable measures: necessary conditions
(arXiv:1307.0804 | Published Version)
(with Raanan Schul)
[Click to Show/Hide Abstract]
Citation: M. Badger, R. Schul, Multiscale analysis of 1-rectifiable measures: necessary conditions, Math. Ann. 361 (2015), no. 3-4, 1055-1072.
#5 Beurling's criterion and extremal metrics for Fuglede modulus
(arXiv:1207.5277 | Published Version)
[Click to Show/Hide Abstract]
Citation: M. Badger, Beurling's criterion and extremal metrics for Fuglede modulus, Ann. Acad. Sci. Fenn. Math. 38 (2013), 677-689.
#4 Quasisymmetry and rectifiability of quasispheres
(arXiv:1201.1581 | Published Version)
(with James T. Gill, Steffen Rohde, and Tatiana Toro)
[Click to Show/Hide Abstract]
Citation: M. Badger, J.T. Gill, S. Rohde, T. Toro, Quasisymmetry and rectifiability of quasispheres, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1413-1431.
#3 Flat points in zero sets of harmonic polynomials and harmonic measure from two sides
(arXiv:1109.1427 | Published Version)
[Click to Show/Hide Abstract]
Citation: M. Badger, Flat points in zero sets of harmonic polynomials and harmonic measure from two sides, J. London Math. Soc. 87 (2013), no. 1, 111-137.
#2 Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited
(arXiv:1003.4547 | Published Version)
[Click to Show/Hide Abstract]
Citation: M. Badger, Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited, Math. Z. 270 (2012), no. 1-2, 241-262.
#1 Harmonic polynomials and tangent measures of harmonic measure
(arXiv:0910.2591 | Published Version)
[Click to Show/Hide Abstract]
Citation: M. Badger, Harmonic polynomials and tangent measures of harmonic measure, Rev. Mat. Iberoam. 27 (2011), no. 3, 841-870.

Dissertation

PhD Thesis: Harmonic Polynomials and Free Boundary Regularity for Harmonic Measure from Two Sides. Defended on May 5, 2011.

Slides

Selected slides from research talks and colloquiua, in reverse chronological order:

3d Brownian motion and sets of dimension 2.99999 99999 99999
with Brownian motion / harmonic measure demos: Demo 1, Demo 2, Demo 3
Colloquium. University of Tennessee. February 2024
Square Packings and Rectifiable Doubling Measures
Harmonic Analysis Seminar. Universite Paris-Saclay. Orsay, France. December 2023
A Practical Guide to Writing an NSF Grant Proposal (NAN Edition)
2023 Northeast Analysis Network Meeting in Rochester, NY
Random curves and sets of dimension 2.99999 99999 99999
with Brownian motion / harmonic measure demos: Demo 1, Demo 2, Demo 3
UConn Mathematics REU Summer Talk Series
Updates on Traveling Salesman in Banach Spaces
AMS Special Session on Nonsmooth Analysis in Metric Spaces. Cincinnati meeting. April 2021.
Rectifiability of Measures: the Identification Problem
AMS Special Session in GMT and PDE. Joint Mathemathematics Meetings. Denver, CO 2020
Hölder parameterizations of Bedford-McMullen carpets and connected IFS
AMS Special Session. Analysis and Probability on Metric Spaces and Fractals. Madison, WI. September 2019
Open Problems about Curves, Sets, and Measures (Version 2)
PCMI Research Program Seminar. July 2018. Updated version of Talk at ORAM 2018.
Open Problems about Curves, Sets, and Measures
8th Ohio River Analysis Meeting, Lexington, March 2018.
Geometry of Radon measures via Hölder parameterizations
Geometric Measure Theory, Warwick, July 2017.
Structure theorems for Radon measures
Analysis on Metric Spaces, Pittsburgh, March 2017.
Singular Points for Two-Phase Free Boundary Problems for Harmonic Measure
SIAM Minisymposium on New Trends in Elliptic PDE, December 2015.
What is Nonsmooth Analysis?
An introductory colloquium (joint presentation with Vasileios Chousionis) for the UConn Special Semester in Nonsmooth Analysis. September 2015.

[Additional Slides]

Miscellaneous

Division Algebras over the Real Numbers
Not intended for publication. These are expository notes that I wrote as an undergraduate. Since they have been cited, I'm reposting them so that they are accessible.
Brownian Motion Demo
HTML 5 simulation of Brownian motion exiting a domain.
Bee Sting Bee
North American history in Ontario County, NY
Matthew Ward <link to>
Fiction and non-fiction by Connecticut writer Matthew Ward.
Date of Freshest Content: February 25, 2024.