SOUTHEASTERN SECTIONAL SAMPLER

Jonathan D. Hauenstein

Numerical Algebraic Geometry and Opti-mizationConvex programming aims to minimize a convex objec-tive function over a convex set, called the feasible set. Forexample, linear programmingminimizes a linear functionover a polytope (intersection of finitely many linear half-spaces as in Figure 1(a)) while semidefinite programmingminimizes a linear function over a spectrahedron (inter-section of the cone of positive semidefinite matrices witha linear space as in Figure 1(b)).

(a) (b)

Figure 1. Example of (a) a polytope and (b) aspectrahedron.

When the feasible set has a nonempty interior, a stan-dard approach for solving convex programs are interiorpointmethods. Conversely, when the feasible set is empty,the program is said to be infeasible and the traditionalFarkas’ lemma is a standard approach for verifying infea-sibility. For example, every infeasible linear program canbe verified using the traditional Farkas’ lemma. However,there are so-called weakly infeasible semidefinite pro-grams where this is not the case. To illustrate, considerthe following semidefinite program:

(1)minimize 𝑥11

subject to [ 𝑥11 11 0 ] ⪰ 0

where 𝐴 ⪰ 0 means that 𝐴 is a positive semidefinitematrix. Since the determinant of the matrix in (1) is−1, the program (1) is clearly infeasible. Moreover, (1) isweakly infeasible since the corresponding alternative viathe traditional Farkas’ lemma is also infeasible, i.e., theredoes not exist 𝑦 ∈ ℝ2 such that

[ 0 𝑦1𝑦1 𝑦2 ] ⪰ 02 ⋅ 𝑦1 + 0 ⋅ 𝑦2 = −1.

Jonathan D. Hauenstein is associate professor of applied andcomputational mathematics and statistics at the University ofNotre Dame. His email address is hauenstein@nd.edu.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1739

One numerical challenge in identifying weakly infeasi-ble semidefinite programs is that perturbations can bestrongly infeasible or strictly feasible. For example,

minimize 𝑥11

subject to [ 𝑥11 11 𝜖 ] ⪰ 0

is strongly infeasible for 𝜖 < 0 and strictly feasible for𝜖 > 0. Liu and Pataki [3] showed that many commonly-used software packages in semidefinite programminghave difficulty identifying weakly infeasible semidefiniteprograms when the reason for infeasibility is not triviallyobvious. Suchmessy instanceswere obtainedbyobscuringtheir structure via row operations and rotations. Thus, achange of perspective was needed for identifying weaklyinfeasible semidefinite programs.

Using the lens of numerical algebraic geometry[1,4], the mathematical foundation of traditional interiorpoint methods is to numerically track a solution pathof a homotopy from a point in the interior of thefeasible set to an optimizer. With this viewpoint, weaklyinfeasible semidefinite programs can be identified [2]using the following three techniques from numericalalgebraic geometry: projective space for compactifyinginfinite length solution paths, adaptive precision pathtracking for navigating through ill-conditioned areas, andendgames for accurately computing singular endpoints.

To illustrate, we consider the following convex programmodified from (1):

(2)minimize 𝜆

subject to [ 𝑥11 +𝜆 11 𝜆 ] ⪰ 0.

The corresponding optimal value is easily observed tobe 𝜆∗ = 0, but this is actually an infimum that is notattained as a minimum, a condition that is equivalentto (1) being weakly infeasible. Therefore, optimizers to (2)are “at infinity” meaning that a solution path defined bytraditional interior point methods will have infinite lengthand approach an asymptote as represented in Figure 2(a).Compactification using projective space yields a finitelength path that can be efficiently tracked as representedin Figure 2(b).

Complex analysis enters the scene to accurately com-pute the endpoint. The winding number (also called thecycle number) of the endpoint for the path displayed inFigure 2(b) is 2, meaning that the path over the complexnumbers locally behaves like the complex square rootfunction. Hence, the Cauchy integral theorem can be usedto compute the endpoint of this path by integrating alonga closed loop as shown in Figure 3. Due to periodicity, nu-merical integration by the trapezoid rule is exponentiallyconvergent [5]. Such a procedure for computing the end-point is called the Cauchy endgame. Since any endpointwith winding number larger than 1 is necessarily singular,ill-conditioning that necessarily arises near the endpoint

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SOUTHEASTERN SECTIONAL SAMPLER

(a) (b)

Figure 2. (a) A plot of paths at a given (red) point mayhave infinite length with limiting asymptotecorresponding with 𝜆∗ = 0. (b) Compactificationusing projective space yields a finite-length path thatcan be efficiently tracked.

(a) (b)

Figure 3. To compute the endpoint of a path as inFigure 2(b), one uses the Cauchy integral theoremand integrates along a closed loop like the one withwinding number 2 with real (a) and imaginary (b)parts pictured here.

can be controlled using adaptive precision path trackingmethods.

This viewpoint for identifying weakly infeasible semi-definite programs using numerical algebraic geometryand the software package Bertini [1] along with severalother interactions of numerical algebraic geometry andoptimization will be discussed in Arkansas.

References[1] D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W.

Wampler, Numerically Solving Polynomial Systems withBertini, volume 25 of Software, Environments, and Tools. So-ciety for Industrial and Applied Mathematics, Philadelphia,PA, 2013. MR3155500

[2] J. D. Hauenstein, A. C. Liddell Jr., and Y. Zhang, Nu-merical algebraic geometry and semidefinite programming.Preprint available at dx.doi.org/10.7274/R0D798G4.

[3] M. Liu and G. Pataki, Exact duals and short certifi-cates of infeasibility and weak infeasibility in conic linearprogramming. Math. Program., 167(2), 435–480, 2018.MR3755739

[4] A. J. Sommese and C. W. Wampler, The Numerical Solutionof Systems of Polynomials Arising in Engineering and Science.World Scientific, Hackensack, NJ, 2005. MR2160078

[5] L. N. Trefethen and J. A. C. Weideman, The exponentiallyconvergent trapezoidal rule. SIAM Rev., 56(3), 385–458,2014. MR3245858

Photo CreditsAll figures by Jonathan D. Hauenstein.Photo of Jonathan D. Hauenstein by Matt Cashore, courtesy of

the University of Notre Dame.

Jonathan D. Hauen-stein

ABOUT THE AUTHOR

Jonathan D. Hauenstein has beenhonored with a DARPA YoungFaculty Award, Sloan Research Fel-lowship, Army Young InvestigatorAward, and Office of Naval Re-search Young Investigator Award.Outside of mathematics, he enjoysspending time with his wife andfour daughters.

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SOUTHEASTERN SECTIONAL SAMPLER

Kathryn Mann

Group Actions, Geometry and RigidityClassical representation theory is concerned with rep-resentations of discrete groups into Lie groups. Intopological or smooth dynamics we are concerned withrepresentations of discrete groups into the group of home-omorphisms or diffeomorphisms of a manifold 𝑀, and thebehavior of these representations under perturbation ordeformation. Our recent result says that rigidity can ariseonly in certain geometric ways.

RigidityA representation 𝜌 of a discrete group Γ into a topologicalgroup𝐺 is rigid, loosely speaking, if it has no non-obviousdeformations. “Obviousdeformations” arise by conjugacy:if 𝜌 ∶ Γ → 𝐺 is a representation and 𝑔𝑡 a path based atthe identity in 𝐺, then 𝛾 ↦ 𝑔𝑡𝜌(𝛾)𝑔−1𝑡 gives a continuouspath of representations starting at 𝜌. Thus, one way toformalize the notion of rigidity is to define 𝜌 to be rigidif it is an isolated point in the space of representations upto conjugacy, Hom(Γ,𝐺)/𝐺.

What is remarkable is that such examples exist atall. Perhaps the most famous rigidity result—and thefirst theorem that I remember being truly astounded byas a graduate student—is Mostow rigidity. In geometriclanguage, it says that a compact manifold of dimensionat least 3 admits at most one hyperbolic structure.In representation-theoretic language, it states that theinclusion Γ → SO(𝑛, 1) of a co-compact lattice Γ into theLie group SO(𝑛, 1), for 𝑛 ≥ 3, is rigid in the sense above.

Mostow rigidity completely fails in dimension 2; infact a genus 𝑔 compact surface has a much stud-ied (6𝑔 − 6)-dimensional moduli space of hyperbolicstructures. My talk is about how to recover rigidity bypassing to the nonlinear, dynamical setting of groups ofhomeomorphisms.

Geometry and Group ActionsThe story begins with hyperbolic structures on surfaces. IfΣ𝑔 is a surface of genus 𝑔 ≥ 2, equipped with a hyperbolicstructure, then the universal cover Σ̃𝑔 can be identifiedwith the hyperbolic plane and 𝜋1(Σ𝑔) with a subgroup ofthe isometry group SO(2, 1) ≅ PSL(2,ℝ). The hyperbolicplane has a natural compactification—in the Poincaré diskmodel depicted in Figure 1, the compactification adds thecircle at the boundary of the open disk—and the actionof PSL(2,ℝ) by hyperbolic isometries of the disc extendsto an action on 𝑆1 = ℝ∪{∞} by Möbius transformations.This is an example of what we call a “geometric” action

Kathryn Mann is the Manning Assistant Professor at Brown Uni-versity. Her email address is mann@math.brown.edu.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1738

Figure 1. The Poincaré disc model of ℍ2, tiled byfundamental domains for a genus 2 surface.Hyperbolic isometries extend to the boundary andprovide an example of a geometric action.

of 𝜋1(Σ𝑔) on the circle. More generally, we say an actionof a discrete group Γ on a manifold 𝑀 is geometric ifthe action Γ → Homeo(𝑀) factors through an embeddingΓ → 𝐺 → Homeo(𝑀), where 𝐺 is a connected Lie groupacting transitively on 𝑀, and Γ ⊂ 𝐺 a co-compact lattice.

It is not difficult to classify all geometric actionsof groups on the circle, they are virtually all surfacegroups, embedded into copies of PSL(2,ℝ) and its centralextensions byfinite cyclic groups. In earlierwork, I showedthat these geometric examples were all rigid—they areisolated points in the moduli space of representations ofa surface group into Homeo(𝑆1). Alternate, independentproofs have since been proposed by S. Matsumoto and J.Bowden.

Hidden Lie GroupsRecently, Maxime Wolff and I proved the remarkableconverse: if 𝜌 ∶ 𝜋1(Σ𝑔) → Homeo(𝑆1) is rigid, then 𝜌 isgeometric.1 In other words, an underlying geometricstructure is the only source of dynamical rigidity forsurface groups acting by homeomorphisms on the circle.

This result is much more difficult than the original“geometric implies rigid” direction. In that first direction,one is given a geometric representation 𝜌—which can bewritten down completely explicitly—and one just needs toshow that it is stable under perturbation. For the converse,

1Technically, this holds after passing to a Hausdorff quotient ofthe representation space.

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one starts with a completely mysterious representation,save for the knowledge that whatever it is, it can’t bedeformed. From there, the goal is to conjure up an ambientLie group.

The proof uses classical dynamical tools such as the ro-tation number of Poincaré, and various refinements of ourown invention, but alsomapping class groups and surfacetopology, a combination theorem for actions admittingMarkov-partition-like structures due to Matsumoto, andperspectives borrowed from Calegari, Ghys, and others.While the end result has turned into a cohesive narrative,the process felt like a three-year ordeal of “hit it witheverything we’ve got.” Fortunately, the philosophy of theproof—constructing geometry from rigidity—can be com-municated quite easily in a simplified setting that avoidsall the technical nightmare. That’s the version you’ll seein my talk.

Photo CreditsFigure by Kathryn Mann.Photo of Kathryn Mann by Jake Paleczny.

Kathryn Mann

ABOUT THE AUTHOR

Kathryn Mann works in geometrictopology and low-dimensional dy-namics, studying moduli spaces ofgroup actions on manifolds. Sheis the 2019 recipient of the AWM-Birman Research Prize in Topologyand Geometry.

6 Notices of the AMS Volume 65, Number 10