Philip Engel
Assistant professor in mathematics
Department of Mathematics, Statistics, and Computer Science
University of Illinois in Chicago (UIC)
Chicago, Illinois
Office: SEO 535
Email: pengel@uic.edu
I am currently an assistant professor at UIC. Before that, I was a Bonn Junior Fellow at the University of Bonn for one year, an assistant professor at University of Georgia, and a postdoc at Harvard. I love talking about math and meeting new people, so please drop by my office if you feel like chatting. Thanks for visiting!
Here is a link to information on the course I'm currently teaching.
My research is in algebraic geometry and Hodge theory, especially degenerations of surfaces. I’m also fascinated by tilings and have thought a bit about Hurwitz theory.
Assistant professor in mathematics
Department of Mathematics, Statistics, and Computer Science
University of Illinois in Chicago (UIC)
Chicago, Illinois
Office: SEO 535
Email: pengel@uic.edu
I am currently an assistant professor at UIC. Before that, I was a Bonn Junior Fellow at the University of Bonn for one year, an assistant professor at University of Georgia, and a postdoc at Harvard. I love talking about math and meeting new people, so please drop by my office if you feel like chatting. Thanks for visiting!
Here is a link to information on the course I'm currently teaching.
My research is in algebraic geometry and Hodge theory, especially degenerations of surfaces. I’m also fascinated by tilings and have thought a bit about Hurwitz theory.
Publications and Preprints

WEAVING QUASICRYSTALS (click for photos & description), a joint art project with
Victoria Manganiello, appearing
in the Bonn Math Library, Luchsinger Gallery at Greenwich Academy, and maybe coming to
your mathematics institute!

Exact enumeration of fullerenes (with Peter Smillie, and an appendix by Jan Goedgebeur).
 Accepted to Duke Math Journal 2024. [+]
A fullerene, or buckyball, is a trivalent graph on the sphere with only pentagonal and hexagonal faces. We give an exact formula for the number of fullerenes with a given number of vertices. In previous work, Peter Smillie and I gave a formula which counted each fullerene by a natural weighting factor. We correct this leading term to give an exact count with no weighting factor. As input, the complicated computation requires the enumeration of fullerenes up to 400 carbon atoms. Algorithmic enumeration has a long history in computer science and computational chemistry. We employ the buckygen program, codeveloped by Jan Goedgebeur, to provide this necessary input to the paper.

Periods of elliptic surfaces with p_g=q=1 (with Francois Greer and Abigail Ward).
 Accepted to Forum of Math: Sigma 2024. [+]
We prove that the period mapping is dominant for elliptic surfaces over an elliptic curve with 12 nodal fibers, and that its degree is larger than 1.

Complete moduli of Enriques surfaces in Horikawa's model (with Valery Alexeev, Zack Garza, and Luca Schaffler).
 2023.
[+]
We describe a compactification of the moduli space of Enriques surfaces with a degree 2 numerical polarization. These Enriques surfaces arise from taking a K3 double cover of P1 x P1, and then quotienting by the fixedpoint free lift of an involution of P1 x P1. The image of the ramification divisor is then a canonical choice of polarizing divisor on the Enriques surface. There is an associated KSBA compactification, and we show that its normalization is also a semitoroidal compactification. The semifan is given explicitly in terms of a "folding" operation on Coxeter diagrams. Just as Kulikov models are encoded by integralaffine structures on the twosphere, certain nice dlt models of Enriques degenerations are encoded by integralaffine structures on either the disk or RP2. We construct these dlt models and all the stable surfaces at the boundary of moduli.

Compact moduli of K3 surfaces (with Valery Alexeev).
 Annals of Mathematics, 198(2): 727789, 2023.
[+]
In this paper, we resolve a longstanding open problem concerning the compactification of moduli of K3 surfaces, by proving that there are (semi)toroidal modular compactifications for all degrees 2d. Our approach introduces the notion of a recognizable divisor: this is an ample divisor on the generic polarized K3 surface which is welldefined on Kulikov surfaces. We prove that the stable pair compactification associated to such a recognizable divisor is normalized by a semitoroidal compactification. Furthermore, we show that the rational curve divisor defined by the sum of the rational curves in the polarization class, is recognizable for all degrees 2d.
 Stable pair
compactification of moduli of K3 surfaces of degree 2
(with Valery Alexeev and Alan Thompson).
 Crelles Journal, Vol. 799: 156, 2023.
[+]
A stable pair compactification of the moduli space of K3 surfaces can be constructed by choosing canonically an ample divisor on each K3. For degree 2 K3s, the ramification divisor of the canonical double covering map is such a choice. We prove that up to normalization, this compactification is one of Looijenga’s semitoroidal compactifications. We explicitly identify the semifan and describe the surfaces which fiber of the boundary.

On the nonabelian Hodge locus I (with Salim Tayou).
 2023. [+]
We partially resolve conjectures of Deligne and Simpson concerning local systems on quasiprojective varieties that underlie a polarized variation of Hodge structure. For local systems of "compact type", we prove (1) a relative form of Deligne's finiteness theorem, for any family of quasiprojective varieties, and (2) algebraicity of the corresponding nonabelian Hodge locus.

Mixed mock modularity of special divisors (with Francois Greer and Salim Tayou).
 2023. [+]
A well known result of KudlaMillson proves that the generating function of special (or Heegner) divisors in an orthogonal Shimura variety is a cohomologyvalued modular form. Shimura varieties admit toroidal compactifications, depending on combinatorial data called a fan. We prove that the Zariski closures of special divisors in a toroidal compactification can be corrected by an explicit linear combination of boundary divisors depending on the fan, so that the modularity property of the generating function is restored.

The ManinMumford conjecture in genus 2 and rational curves
on K3 surfaces (with Raju Krishnamoorthy and Daniel Litt).
 2022. [+]
We explore the question of whether every point on a K3 surface is contained in a rational curve. Unlike the phenomenon observed by Bogomolov and Tschinkel for algebraic closures of finite fields, in characteristic zero, not all points on a Kummer K3 are contained in a rational curve whose double cover in the abelian surface has genus 2.

Mirror symmetric compactifications of moduli spaces of K3 surfaces
with a nonsymplectic involution (with Valery Alexeev).
 2022. [+]
Following up on earlier work with Alexeev and Han, we explicitly compute the semitoroidal compactifications of the 50 moduli spaces of K3 surfaces with a nonsymplectic involution for which the fixed locus contains a curve of genus at least 2. We do so by finding Kulikov models for all Type III degenerations in all 75 moduli spaces of K3 surfaces with nonsymplectic involution.

The flex divisor of a K3 surface (with Valery Alexeev).
 IMRN rnac089, 2022. [+]
This paper computes the degree of the flex divisor of a primitively polarized K3 surface, defined as the set of points for which there is a pencil in the linear system of the polarization whose base locus is exactly that point.

Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal
(with Valery Alexeev and Adrian Brunyate).
 Geometry & Topology, 26: 3525–3588, 2022. [+]
We study the stable pair compactification of elliptic K3 moduli associated to the section plus the sum of the singular fibers. We prove that the normalization of this modular compactification is in fact a toroidal compactification, and compute the fan explicitly. It ends up being a refinement of a natural Coxeter fan, which results from decomposing each Coxeter chamber into nine subchambers.

Compact moduli of K3 surfaces with a nonsymplectic automorphism (with Valery Alexeev and Changho Han).
 Transactions of the AMS, Series B 11, 2024. [+]
We study moduli spaces of K3 surfaces with a nonsymplectic automorphism fixing a curve of genus at least two. We show that such a moduli space has a canonically defined compactification by KSBA stable pairs. Furthermore, we prove that this compactification is a Looijenga semitoroidal compactification for some semifan.
 Smoothings and
rational double point adjacencies for cusp singularities (with Robert Friedman).
 Journal of Differential Geometry 118(1): 23100, 2021.
[+]
This paper explores smoothings of cusp singularities further. We show that the number of smoothing components is at least the number of deformation types of anticanonical pairs with the dual cycle. We also classify the adjacencies of a cusp singularity to a rational double point configuration.
 Hurwitz theory of elliptic orbifolds, I.
 Geometry & Topology 25: 229–274, 2021.
[+]
This paper generalizes my work with Peter Smillie to higher genera. How many triangulations of a genus g surface are there with a specified set of nonzero curvatures? It turns out the generating function is a quasimodular form. The idea goes back to Eskin and Okounkov who connected these types of question to Hurwitz theory. The key is to count branched covers of an elliptic orbifold, i.e. a finite quotient of an elliptic curve, with specified branch profile, using representation theory of symmetric groups.
 Hurwitz theory of elliptic orbifolds, II.
 2018. [+]
This paper makes some difficult technical computations which were necessary to achieve the full quasimodularity of the Hurwitz theory of elliptic orbifolds. The key step is to realize that the Hurwitz theory of the elliptic orbifold of order N is encoded by a certain tensor product of vertex operators on N copies of Fock space, multiplied the rotation operator which cyclically rotates the N tensor factors.
 Appendix to
Contribution of onecylinder squaretiled surfaces to MasurVeech volumes
(with Vincent Delecroix, Elise Goujard, Peter Zograf, and Anton Zorich).
 Asterisque 415: 223274, 2020.
[+]
In the moduli space of abelian differentials, there is an equidistributed (with respect to the MasurVeech volume form) subset formed by squaretiled surfaces. This paper studies which proportion of the volume is due to squaretiled surfaces formed from a fixed number of vertical or horizontal cylinders. It turns out that the onecylinder contribution is asymptotic to the inverse of the genus. My appendix computes this contribution to the volume using some simple Hurwitz theory.
 The number of
convex tilings of the sphere by triangles, squares, or hexagons (with Peter Smillie).
 Geometry & Topology 22: 28392864, 2018.
[+]
A buckyball is a spherical carbon molecule whose rings have length 6 or less. We give a formula for the number of such combinatorial arrangements with a fixed number of carbon atoms. It turns out that these numbers form the Fourier coefficients of a modular form. The key is to use a theorem of Thurston’s which identifies these triangulations with lattice points in a Hermitian hyperbolic lattice, then integrate the Siegel theta correspondence for the corresponding symmetric space.

A proof of Looijenga's conjecture via integralaffine geometry.
 Journal of Differential Geometry 109(3): 467495, 2018.
[+]
A cusp singularity is a surface singularity whose minimal resolution is a cycle of rational curves. These singularities come in dual pairs. In 1981, Looijenga proved that if a cusp singularity is smoothable, then the dual cycle appears as the anticanonical divisor on a rational surface. He conjectured the converse. This paper gives a proof of Looijenga's cusp conjecture using surgeries on integralaffine surfaces.
Selected Conference Talks
 A panorama of moduli spaces, Goethe University, Frankfurt am Main, Spring 2024
 Moduli, Ktrivial Varieties, and Related Topics, IBSCCG (Korea), Spring 2024
 Recent Developments in Hodge Theory, IMSA, Spring 2021, Lectures I & II
 MSRI thematic semester on flat surfaces, Fall 2019
 Teichmuller Theory, Hyperbolicity and Dynamics at IMPA, Summer 2019
 Casa Matematica Oaxaca, Flat Surfaces and Dynamics on Moduli Space, II, Summer 2019
 Reptile Conference, Spring 2019, organized with Nikon Kurnosov
 Workshop on Modularity in GW Theory, Michigan, Spring 2019
 Workshop on Holomorphic Differentials, Stonybrook, Spring 2019
 Character Varieties and TQFT, University of Auckland, Fall 2018
 Flat Surfaces and Algebraic Curves, Oberwolfach, Fall 2018
 Algebraic and Combinatorial Aspects of Tropical Geometry (AMS Sectional), Ohio State, Spring 2018
 AGNUBC miniconference, Fall 2017
 Poisson Geometry and Stacks, Fields Institute, Summer 2017
 Conference on moduli spaces, mirror symmetry, and enumerative geometry, ICTP, Summer 2016
 Robert Friedman’s birthday, Perspectives on Complex Algebraic Geometry, Spring 2015
 IAS thematic year on topology of algebraic varieties, Fall 2014
Notes for Teaching
 Notes on 4manifolds (Spring 2021)
Links
 My Hypocrite Reader article on BWV 849 by Bach.
 My Hypocrite Reader article on selfreference.
 My Hypocrite Reader article on the integers.
 Some of my math friends: Salim Tayou, Peter Smillie, Raju Krishnamoorthy, Michael McBreen, Francois Greer, Eduard Duryev (who made the cool background image).