Research

1. S. Park, Explicit expressions for the gamma vector leading to connections to upper/lower bounds and structural properties

We give explicit formulas for the gamma vector directly in terms of the input polynomial which extends to arbitrary polynomials. More specifically, we can express them as a linear combination of the coefficients of the starting polynomial (using Catalan numbers and binomial coefficients) or in terms of the derivative of a simple quotient of the input polynomial. The first formula suggests relations to common noncrossing partition/Coxeter group structures in existing gamma positivty examples. The second one shows that the gamma vector measures the difference between local and global information when the input polynomial is the h-polynomial of a simplicial complex. It can also be used to relate signs/inequalities of the gamma vector to upper/lower bounds for the coefficients of the input polynomial. The expression in terms of a derivative and relation to the bounds complements and extends initial observations made by Gal when he defined the gamma vector. Finally, we use the form taken by sums involved while making these estimates and connections between matroid-related examples and intersection numbers to connect these properties of the gamma vector to algebraic structures. For example, this includes characteristic classes involved in log concavity and Schur positivity problems.

2. S. Park, Symmetries and intrinsic vs. extrinsic properties of M_{0, n}bar
   Our motivation is the following question: How much of the combinatorial structure of the moduli space of stable rational curves with n marked points is “intrinsic” to (the geometry of) the space itself? We view this from the lens of the natural permutation action on the points. In fact, it is known that this action does *not* extend to other wonderful compactifications of the complement of the A_{n – 2} hyperplane arrangement. We determine the differences in intersection patterns of parallel faces of associahedra and permutohedra inducing this rigidity property. As a consequence, we show that this is reflected in most of terms of degree at least 2 in the Chow ring. On the other hand, we consider degree 1 elements from the perspective of S_n-invariance (e.g. suitable Lefschetz elements) and how it connects to recent positivity results. In particular, we show that the log concave sequences arising from degree 1 Hodge-Riemann relations using S_n-invariant elements of the Picard group have a special (recursive) structure. They take the form of polynomials in (quantum) Littlewood-Richardson coefficients multiplied by terms such as partition components, factorials, and multinomial coefficients. Finally, we connect higher degree Hodge-Riemann relations (of other rings) to the geometry of this moduli space via Toeplitz matrices.

3. S. Park, Matroidal Cayley-Bacharach and independence/dependence of geometric properties of matroids
   We study the relationship between “geometric” properties of matroids and the matroidal Cayley-Bacharach property of degree a MCB(a) defined by Levinson and Ullery. From the perspective of matroid polytopes/generalized permutohedra, we see from the case of nestohedra that the MCB(a) property can have a natural description in terms of properties of polytopes while *not* being a combinatorial invariant of polytopes. On the other hand, there seem to be a close relationship between combinatorial properties in the case of paving matroids (which are conjecturally almost all matroids of a given rank) and supersolvable line/hyperplane arrangements. The paving matroids involve a relationship between the degree a and the Chow rings of matroids. Using supersolvable line arrangements, we find a family of matroids other than the case of representable matroids where the MCB(a) property measures the failure of a set of points to impose independent conditions on the space of hypersurface of a given degree. In general, MCB(a) for supersolvable hyperplane arrangements has a recursive property from MCB(b) for lower degrees b and covers of appropriate subarrangements.

4. S. Park, Anti-Ramsey theory problems, lattice point counts on polytopes, and Hodge structures on the cohomology of toric varieties
   By "anti--Ramsey theory problems", we mean the number of edge colorings of graphs such that a specified subgraphs are *not* monochromatic. We find families of graphs and subgraphs such that this number is determined by a lattice point count. The idea is to combine a reinterpretation of simplicial chromatic polynomials and connections between h-vectors and lattice point counts of polytopes. Note that this follows up on our earlier work expressing simplicial chromatic polynomials in terms of h-vectors of auxiliary simplicial complexes. As a result, we obtain a family of “anti-Ramsey” questions addressed using geometric/structural methods.

5. S. Park, Matroids satisfying the matroidal Cayley--Bacharach property and ranks of covering flats
   We first show that there are no nontrivial bounds on ranks of proper flats that cover the underlying set of a matroid satisfying the matroidal analogue of the Cayley-Bacharach property. This gives a negative answer to a question in recent work of Levinson and Ullery. Next, we look at the matroidal Cayley-Bacharach property from the point of view of polytopes associated with matroids that are studied. We consider a (generic) class of matroids where the matroidal Cayley-Bacharach property depends on a collection of set-theoretic properties depending on the ranks of the flats of the matroids arising from the polytopes.

6. S. Park, Simplicial chromatic polynomials as Hilbert series of Stanley--Reisner rings
   This project started with our initial observation that Euler characteristic-like invariants of ordered configuration spaces of distinct points on a manifold and can be altered to obtain chromatic polynomials of graphs. This means only preventing the coordinates corresponding to adjacent vertices from being equal to each other. As it turns out, properties of this modified configuration space such as this one were studied earlier by Eastwood and Huggett and there is a higher-dimensional version of this connection arising from simplicial complexes in work of Cooper-de Silva-Sazdanovic. This polynomial (the simplicial chromatic polynomial) is uniquely determined up to normalization by a deletion-contraction type relation. While they study the polynomial from a topological point of view, we find an explicit combinatorial interpretation for a large class of initial simplicial complexes. More specifically, we find that they arise from the Hilbert series of Stanley-Reisner rings associated to auxiliary simplicial complexes. In addition, *any* simplicial complex can be set to be the auxiliary simplicial complex of *some* simplicial complex.
   Note that these polynomials are closely related to characteristic polynomials of diagonal/hypergraph linear subspace arrangements (or their associated polymatroids). Since they are determined by h-vectors of auxiliary simplicial complexes, we found some connections between these simplicial chromatic polynomials and other questions involving log concavity, symmetries between a polynomial and its reciprocal polynomial, and cyclotomic polynomials along the way.

7. S. Park, Graph coloring-related properties of (generating functions of) Hodge--Deligne polynomials
   We were considering some connections between Euler characteristic-like invariants (e.g. Hodge-Deligne polynomials) of configuration spaces and chromatic polynomials. It turns out that there is a connection between colorings of *directed* graphs and Hodge-Deligne polynomials as well. We take a look at what this means and how it relates to existing structures between Hodge numbers (e.g. birational invariants) and properties of configuration spaces.

8. S. Park, Characterizing cubic hypersurfaces via projective geometry
   Under certain numerical/generic conditions, we show that cubic hypersurfaces are characterized by a projective geometry construction. This uses a cut and paste relation (in the Grothendieck of varieties) of Galkin and Shinder matching pairs of points with an incidence correspondence involving the third point of intersection and the line spanned by the first two points (filtering out instances where the line is contained in the given variety). Weakening these conditions extends the possibilities to complete intersections of two quadric hypersurfaces or two quartic hypersurfaces. As a special case, we find generic hypersurfaces of a given degree satisfying this cut and paste relation must be cubic hypersurfaces.

9. S. Park, Motivic limits for Fano varieties of k-planes, The Quarterly Journal of Mathematics, haac012 (2022)
   We show that "most'' of certain properties of Fano varieties of k-planes (k-planes contained in a given projective variety) are determined by symmetric products of points on the given variety, Grassmannians of appropriate dimensions, and incidence correspondences of points in linear subspaces. Examples of properties in question are those compatible with cut and paste constructions such as Poincare polynomials, Euler characteristics, and Hodge-Deligne polynomials. The main idea is to construct an approximate/motivic limit version of a relation of Galkin and Shinder in the Grothendieck ring of varieties. This means building a correspondence between points and incidence correspondences coming from the intersection of a variety in projective space with a linear subspace of complementary dimension and filtering out loci where this map is not a bijection (which includes terms from Fano varieties of k-planes).

10. S. Park, Decomposability and Mordell-Weil ranks of Jacobians using Picard numbers (not submitted)
    We study number field analogues of some questions of Ekedahl and Serre about the decomposability of Jacobians of curves C over number fields as a product of elliptic curves. The main case considered involves self-products E^g and we approach this question by studying the Picard numbers of self-products of the curves C involved under specialization to primes. This involves methods previously used by Costa, Elsenhans, and Jahnel to study those of K3 surfaces. As a result, we give bounds on the genus of such curves with respect to initial arithmetic invariants (e.g. norms of primes related to reduction properties or heights) and obtain infinite families where the reduction modulo a prime is maximal or minimal when such decompositions exist. In addition, we rule out cases where the curves have a large automorphism group. Finally, we show that Picard numbers of self-products of curves can also be used to study jumps of Mordell-Weil ranks via results of Ulmer on Mordell-Weil ranks of Jacobians over function fields and endomorphism rings.

11. L. Chua, B. Gunby, S. Park, and A. Yuan, Proof of a conjecture of Guy on class numbers, International Journal of Number Theory, 11 (2015), pages 1345 - 1355.
    We resolve a conjecture of Guy on a congruence between class numbers of quadratic fields Q(sqrt(\pm p)) and continued fraction expansions of \sqrt(p). The tools used were some algebraic number theory, results of Zagier connecting these class numbers with the continued fraction expansions, Jacobi symbols, and Dedekind sums. While this question is apparently about class numbers, it is interesting to note that the main ideas used are combinatorial arguments rather than the structure of the class group.

12. L. Chua, S. Park, and G. Smith, Bounded gaps between primes in special sequences, Proceedings of the American Mathematical Society, 143 (2015), pages 4597 - 4611.
    In Maynard’s work on bounded gaps between primes, it was shown that any subset of the primes which is “well-distributed” in arithmetic progressions contains many primes which are “close together”. We adapt his method to show that there are bounded gaps between sequences of the form [bn], where b is an irrational number of finite type.

13. S. Park, Arithmetic properties of generalized Fibonacci sequences
    We consider a generalization of the Fibonacci sequence which shares some arithmetic properties with the original sequence. This includes a resolution to some conjectures of Chen, Moll, and Sagan on periodicity, d-adic valuations, and the behavior of an analogue of the Riemann zeta function. Also, we give an algebraic description of the periodicity property considered and study how it is distributed.

14. S. Park, Discriminators of quadratic polynomials
    For polynomials f and positive integers n, we study the discriminator D_f(n), which is the smallest number m such that. f(1),…,f(n) are distinct mod m. This was first defined in the context of computing square roots of a long sequence of numbers for a computer simulation. While this quantity has been studied for certain classes of polynomials, it is very complicated in general. We focus on polynomials of the form f(x) = x(dx – 1) where this problem is more tractable and extend results of Sun for d = 2, 3 where D_f(n) = d^([log_d n] + 1) to d = 2^r for positive integers r. Afterwards, we also study cases where d = p^r for other primes p (e.g. using bounds) and observe using computational methods that discriminator values are concentrated around prime powers even after increasing the size of the prime p or power r. This gives a potential method for generating prime numbers using discriminators of polynomials.