Benjy Firester

Benjy Firester

Math PhD student at MIT advised by Toby Colding and Tristan Collins

Levinson Fellow, MathWorks Fellow

AB/AM in Mathematics from Harvard (2023 Summa Cum Laude)

Recipient of the Mumford Prize and Friends Prize

Email: benjyfir at mit dot edu

arXiv ORCID Google Scholar
Benjy Firester presenting at Stanford

Research

Area-minimizing capillary cones (with R. Tsiamis and Y. Wang) [arXiv]
We construct non-flat minimal capillary cones with bi-orthogonal symmetry groups for any dimension and contact angle. These cones interpolate between rescalings of a singular solution to the one-phase problem and the free-boundary cone obtained by halving a Lawson cone along a hyperplane of symmetry. The existence and uniqueness of such cones is proved by solving a nonlinear free boundary equation parametrized by the contact angle and obtaining monotonicity properties for the solutions. The constructed cones are minimizing in ambient dimension 8 or higher, for appropriate contact angles, demonstrating that the regularity theory for minimizing capillary hypersurfaces can have singularities in codimension 7 and completing the capillary regularity theory for contact angles near \(\pi/2\). We further develop the connection between capillary hypersurfaces and solutions of the one-phase problem, consequently producing new examples of singular minimizing free boundaries for the Alt–Caffarelli functional.
Stability inequalities for one-phase cones (with R. Tsiamis and Y. Wang) [arXiv]
We obtain strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem. We prove that in dimension 7 and above, cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable. As a consequence, we obtain a bound on the first eigenvalue and the decay rates of Jacobi fields, with applications to the generic regularity of the one-phase problem.
Homogeneous optimal transport maps between oblique cones (with T. C. Collins and F. Tong) [arXiv]
We construct homogeneous optimal transport maps for the quadratic cost between convex cones with homogeneous, possibly degenerate, densities when the cones satisfy an obliqueness condition. The existence of such maps plays a central role in the boundary regularity theory for optimal transport maps between convex domains. Our results are also relevant for the existence of complete Calabi–Yau metrics on certain quasi-projective varieties.
On a general class of free boundary Monge–Ampère equations (with T. C. Collins) [arXiv]
We solve a general class of free boundary Monge–Ampère equations given by \(\det D^2 u = \lambda \frac{f(-u)}{g(u^\star)\, h(\nabla u)} \chi_{\{u < 0\}}\) in \(\mathbb{R}^n\), with \(\nabla u(\mathbb{R}^n) = P\) where \(P\) is a bounded convex set containing the origin, and \(h > 0\) on \(P\). Applications include optimal transport with degenerate densities, Monge–Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler–Ricci solitons on toric Fano manifolds.
Uniqueness of cylindrical tangent cones \(C_{p,q} \times \mathbb{R}\) (with R. Tsiamis and Y. Wang) [arXiv] To appear in Calc. Var. & PDE
We show the uniqueness of the cylindrical tangent cone \(C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}\) for area-minimizing hypersurfaces in \(\mathbb{R}^9\), completing the uniqueness of all tangent cones of the form \(C_{p,q} \times \mathbb{R}\) proved by Simon for dimensions at least 10 and Székelyhidi for the Simons cone.
Cohomogeneity two Ricci solitons with sub-Euclidean volume (with R. Tsiamis) [arXiv] [Journal] J. Geom. Anal. 35, 407 (2025).
We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with volume collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge–Ampère equation for the conformal factor coupled with ODEs. We obtain explicit complete expanding solitons as well as abstract existence results for shrinking and steady solitons with boundary. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics. We also classify local solutions of this Monge–Ampère equation to prove rigidity for these solitons.
Complete Calabi–Yau metrics from smoothing Calabi–Yau complete intersections [arXiv] [Journal] Geom. Dedicata (2024) 218:46
We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection \(V_0\) that is a Calabi–Yau cone, extending the work of Székelyhidi (2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by \(\mathbb{C} \times V_0\). This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Impact Hamiltonian systems and polygonal billiards (with L. Becker, S. Elliott, S. Gonen Cohen, M. Pnueli, V. Rom-Kedar) [arXiv] [Journal] Hamiltonian Systems: Dynamics, Analysis, Applications. Mathematical Sciences Research Institute Publications. Cambridge University Press; 2024:29–66.
The dynamics of a beam held on a horizontal frame by springs and bouncing off a step is described by a separable two degrees of freedom Hamiltonian system with impacts that respect, point wise, the separability symmetry. The energy in each degree of freedom is preserved, and the motion along each level set is conjugated, via action angle coordinates, to a geodesic flow on a flat two-dimensional surface in the four dimensional phase space. Yet, for a range of energies, these surfaces are not the simple Liouville–Arnold tori—these are tori of genus two, thus the motion on them is not conjugated to simple rotations. Namely, even though energy is not transferred between the two degrees of freedom, the impact system is quasi-integrable and is not of the Liouville–Arnold type. In fact, for each level set in this range, the motion is conjugated to the well studied and highly non-trivial dynamics of directional motion in L-shaped billiards, where the billiard area and shape as well as the direction of motion vary continuously on iso-energetic level sets.

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Geometric Analysis Reading Seminar

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