Mark A. Stern

Monotonicity techniques developed to study the small scale structure of singularities of nonlinear pdes have been sharpened to give large scale information about eigenfunctions with small eigenvalues.
These improved monotonicity estimates can then return to nonlinear pde
to provide better understanding of their singularities.
In this talk, I discuss various applications of monotonicity estimates to eigenvalue problems arising in topology and number theory and conclude with an application to a toy problem concerning singularities arising in Yang Mills theory.

Monotonicity, Betti numbers, and small eigenvalues - AMS Special Session on Singer-Hopf Conjecture in Geometry and Topology · March 19, 2023 - March 19, 2023 Invited Talk AMS, U. Georgia, Atlanta

In this talk we survey the application of monotonicity methods to bound Betti numbers and the dimensions of Laplace Beltrami eigenspaces with small eigenvalue on negatively curved spaces.

Nahm transform for ALF spaces - U Florida Geometry Seminar · October 25, 2022 - October 25, 2022 Invited Talk U Florida Mathematics Dept. , U Florida

The goal of this talk is to explain aspects of the isometry between the moduli space of Yang-Mills instantons on ALF gravitational instantons of type A_{k-1} and the space of solutions to a system of odes on line segments.
This is joint work with Sergey Cherkis and Andres Larrain Hubach.

Nahm transform and ALF spaces - Duke Geometry/Topolgy Seminar · September 19, 2022 - September 19, 2022 Lecture Duke,

In this talk we consider the moduli space of Yang-Mills instantons on the family of hyperkahler 4 manifolds known as multi-center TaubNUT spaces. We describe the Nahm transform for flat manifolds. Then we sketch its extension to the above hyperkahler family, where it defines an isometry between the moduli space of instantons on the multi-center TaubNUT and the moduli space of solutions of a rococo system of ordinary differential equations. This is joint work with Sergey Cherkis and Andres Larrain Hubach

Nahm Transform Yoga - Geometry and physics of ALX metrics in gauge theory · July 29, 2022 - July 29, 2022 Invited Talk AIM, AIM

In this talk, we introduce many new variations of Nahm transforms, including transforms from Sp(k) connections to Sp(j) connections.

Bubbling threshholds for Yang Mills and harmonic maps in fat negatively curved spaces - Geometry, Analysis, and Quantum Physics of Monopoles · February 5, 2021 - February 5, 2021 Invited Talk BIRS, Banff, Alberta, Canada

I discuss the effect of negative curvature and large girth on singularity formation in Yang Mills and harmonic maps.

A coarse Cheeger inequality for 1-forms - UF Topology Seminar · January 14, 2020 - January 14, 2020 Invited Talk University of Florida,

Abstract: "Motivated by questions about the spectral geometry of
hyperbolic manifolds arising in number theory, we seek a lower bound for
the first nonzero eigenvalue of the Laplace Beltrami operator on 1-forms
on hyperbolic manifolds. We prove an analog of Cheeger's inequality.
Joint work with Michael Lipnowski."

A Coarse Cheeger Inequality for one Forms - Geometry and Topology Seminar · March 18, 2019 - March 18, 2019 Invited Talk Duke,

Monotonicity and Betti Number Bounds - Geometry Seminar · October 23, 2018 - October 23, 2018 Invited Talk Harvard,

Nahm Transform for ALF spaces - Geometry Seminar · March 6, 2018 Invited Talk SUNY Stonybrook,

Monotonicity and Betti Number Bounds - Geometry Seminar · February 15, 2018 Invited Talk CUNY Graduate Seminar,

Monotonicity and Betti number bounds - The Analysis of Gauge-Theoretic Moduli Spaces · August 29, 2017 - August 29, 2017 Invited Talk Banff International research Station, BIRS

A talk about my joint work with Luca DiCerbo on new estimates of Betti numbers

Instantons on ALF Spaces - Gauge Theory and Special Geometry · July 25, 2017 - July 25, 2017 Invited Talk Mathematical Congress of the Americas, Montreal

This talk was a report on my continuing joint work with Sergey Cherkis on the Nahm transform for instantons on ALF spaces

Spectral geometry of the 1 form Laplacian on hyperbolic manifolds. - Special Session on Geometry and Topology of Ball Quotients and Related Topics · May 6, 2017 - May 6, 2017 Invited Talk AMS Eastern Sectional Meeting, Hunter College, NYC

In this talk I discuss my recent work with Michael Lipnowski on estimates for eigenvalues of Laplace Beltrami operators motivated by conjectures of alegari et al.

Instantons on ALF spaces - City University NY · December 6, 2016 - December 6, 2016 Invited Talk CUNY ,

Instantons on ALF spaces - Waterloo Geometry seminar · December 2, 2016 - December 2, 2016 Invited Talk University of Waterloo,

In search of the second derivative - ICTP Elementary Notions Seminar · October 28, 2016 - October 28, 2016 Invited Talk ICTP, Trieste, Italy

This talk was written for an audience of mathematics students from underdeveloped countries.
Assuming only an undergraduate mathematics background, I introduced major questions and ideas of Yang Mills theory.

Instantons on ALF spaces - ICTP Geometry Seminar · October 24, 2016 - October 24, 2016 Invited Talk ICTP, Trieste, Italy

I discuss recent progress in collaboration with Sergey Cherkis and Andres Larrain-Hubach
on establishing Cherkis's conjectured Nahm transform for Yang Mills instantons on ALF gravitational instantons.

Instantons on ALF spaces - Harvard Geometry Seminar · October 11, 2016 - October 11, 2016 Invited Talk Harvard University,

Decay and Moduli of Yang Mills Instantons · July 21, 2015 Invited Talk Newton Institute,

Introduction to nonlinear harmonic forms · March 30, 2015 Invited Talk U. Maryland,

Introduction to nonlinear harmonic forms - Texas Geometry and Topology Conference · February 28, 2015 Invited Talk University of Houston,

Introduction to nonlinear harmonic forms · December 9, 2014 Invited Talk Chinese University, Hong Kong,

Introduction to nonlinear harmonic forms - Conference on Hodge Theory and L2-cohomology · November 22, 2014 Invited Talk Johns Hopkins,

Instanton Decay - Conference on Heterotic Strings and (0,2) QFT · April 2014 Invited Talk Texas A&M,

Symmetry, singularity, and decay of Yang Mills fields · March 20, 2014 Invited Talk Texas A&M,

Stability, Hodge theory, and Grassmann embeddings · May 21, 2013 Lecture Irvine

Hodge theory, Yang Mills, and Stability · January 28, 2013 Lecture U. of Arizona, Tucson

Asymptotic Hodge Theory of Vector Bundles · December 18, 2011 Lecture Sanya, Hainan, China

Stability, dynamics, and the quantum Hodge theory of vector bundles · November 30, 2010 Lecture Duke Mathematics

The Quantum Hodge Theory of Vector Bundles · March 12, 2010 Lecture CUNY Geometric Analysis 2010 Bubbling Phenomena and Non-Compactness

Geometry of Stable Yang-Mills connections · October 20, 2009 Lecture Graduate Center CUNY

Geometry of Minimal Yang-Mills connections · April 6, 2009 Lecture Warsaw

Geometry of Stable Yang-Mills connections · April 5, 2009 Lecture Texas A&M University

Minimal Energy Yang-Mills Connections and Interpolating Instantons · October 28, 2008 Lecture Mathematical Sciences Research Institute

Outreach & Engaged Scholarship

Judge - 2018 Yau Science Award · December 12, 2018 - December 15, 2018 Community Outreach Tsinghua University , Beijing Panelist in China's version of Mathematics Intel competition.

Judge - Dogrun-Yau Science Award · December 2016 - December 2016 Community Outreach Tsinghua University , Beijing Panelist in China's version of Mathematics Intel competition.

Judge - 2015 Yau Math Award · December 2015 - December 2015 Community Outreach Tsinghua University , Beijing Panelist in China's version of Mathematics Intel competition.

Judge - 2014 Yau Math Award · December 2014 - December 2014 Community Outreach Tsinghua University , Beijing Panelist in China's version of Mathematics Intel competition.

Judge - Hang Lung Mathematics Award · December 2014 - December 2014 Community Outreach City University Hong Kong , Hong Kong flag

Asia Panelist in Hong Kong's version of Mathematics Intel competition.

Judge - 2011 Yau Math Award · December 2011 Community Outreach Tsinghua University , Sanya, Hainan, China Panelist in China's version of Mathematics Intel competition.