Number theory and Algebra Seminar at Asu (2026-27)

Title: Frobenius Descent and Perfectoid Recovery in Mixed Characteristic 

Abstract: A natural problem in mixed characteristics is to determine where Frobenius-compatible structure actually lives. For several determinantal-type singularities, I will explain that quotient-level Frobenius descent is exceptional: beyond the low-rank range, the standard quotient presentation is too rigid to support it. I will then discuss how the missing structure is recovered after passing to the right enlargement — geometrically in the Hankel setting, and tower-theoretically via canonical perfectoid constructions. The talk is based on joint work with Sankhaneel Bisui and Rajat Kumar Mishra.  

Title: Toward PNT Part II 

Abstract: We explore some more of Chebyshev’s work on the distribution of primes as well as some of Riemann’s deep contributions to the theory. 

Title: Componentwise linear ideals from sums 

Abstract: Ideals with linear resolutions arise naturally across commutative algebra and combinatorics. For example, the Cohen-Macaulay property was a crucial ingredient in the solution of the Upper Bound Conjecture, and it is well-known that a simplicial complex is Cohen-Macaulay if and only if the Stanley-Reisner ideal of its Alexander dual has a linear resolution. Componentwise linear ideals, introduced by Herzog and Hibi, generalize the notion of ideals with linear resolutions: an ideal I in a polynomial ring S is componentwise linear if the ideal generated by each component of I has a linear resolution.

In this talk, after introducing componentwise linear ideals and reviewing several of their characterizations, we will explain some necessary and sufficient conditions for the sum of two componentwise linear ideals to be componentwise linear. A complete answer in dimension two helps us to establish the equivalence between componentwise linear monomial ideals and ideals having linear quotients in k[x,y]. When the dimension is arbitrary, we describe a method to build a componentwise linear ideal from a given collection of componentwise linear monomial ideals satisfying some mild compatibility conditions, using only sum and squarefree translations. This is a joint work with Prof. Hailong Dao. 

Title: Graded Ehrhart theory of unimodular zonotopes 

Abstract: There is a well-studied connection between Hilbert series of toric varieties (algebraic geometry) and Ehrhart theory (combinatorics), which gives beautiful geometric meaning to concrete identities of generating functions associated to polytopes. Recently, Reiner and Rhoades stated conjectural q-analogues of the main theorems in Ehrhart theory, and Cavey provided a connection to the geometry of blown-up toric varieties. While Reiner and Rhoades's conjectures are false in general, we prove that they are true for the class of multiplicity-free subtoric varieties of a product of projective lines, which correspond to unimodular zonotopes. We prove our results using an alternate geometric model, known as an arrangement Schubert variety, which plays a central role in recent developments in matroid theory. Based on joint work with Ethan Partida.

Title: Principal jets of monomial ideals 

Abstract: The principal component of the jet scheme is the closure of the set of jets supported over the smooth points of the base scheme. It represents a large portion of the jet scheme and it can be relatively well behaved, as evidenced by a few case studies in the literature. We give a complete description of principal jets of monomial ideals, including their generators and irreducible decomposition. We also show how the Hilbert series and Betti numbers of the principal jets of a monomial ideal relate to the same invariants of the original ideal. 

Title: The Prime Number Theorem Part I  

Abstract: We will review some of the early work of Gauss and Chebyshev on the Prime Number Theorem. This will be the first of two talks on the subject.  

Title: On Arakawa lifting 

Abstract: By modular forms one would usually have holomorphic one in mind. Arakawa lifting is a theta lifting construction of non-holomorphic but real analytic automorphic forms on the indefinite symplectic group Sp(1,1) or real hyperbolic space of dimension four. The notion of the theta lifting is a generalization of theta functions, whose representation theoretic formulation is known as theta correspondence or Howe 

correspondence. The aim of this talk is to report a recent progress on arithmetic results on Arakawa lifts such as their automorphic L-functions in relation with their norms called the Petersson norm. 

Title: Irreducible modules of the degenerate affine Hecke algebra 

Abstract: The irreducible representations of the complex reflection group G(ℓ, 1, n) are indexed by ℓ-partitions; this is an ℓ-tuple of classic partitions but with n total boxes. One of the main objects of study for this talk will be the degenerate affine Hecke algebra of type G(ℓ, 1, n) studied in [1], which we denote by Hℓ,n. Such an algebra has also been studied by Dezelee [2] where it is called generalized graded Hecke algebra.

In this talk we aim to define the irreducible Hℓ,n-modules Sλ\µ which are indexed by twoℓ-partitions λ, µ and prove that for a certain subalgebra u of Hℓ,n each u-diagonalizable Hℓ,n-module can be obtained as one of the mentioned above. This result has an application on the rational Cherednik algebra of type G(ℓ, 1, n) since possesses a subalgebra isomorphic to Hℓ,n [3].

References

[1] Ram, A. and Shepler, A.V., Classification of graded Hecke algebras for complex reflection groups, Comentarii Mathematicii Helvetici, Vol. 78, 308-334.

[2] Dezelee, C. Generalized graded Hecke algebra for complex reflection group of type G(r, 1, n)

[3] Fishel, S., Griffeth, S., Manosalva, E. Unitary representations of the Cherednik algebra; V ∗-homology Math.Z. 299 (2021), no. 3-4, 2215-2255.

Title: Differential Operators on Spaces of Automorphic Forms 

Abstract: We will explore some of the recent work of Bombieri-Garrett on perturbations of linear operators, and how this relates to vanishing of L-functions.

Title: Selmer Groups and the Weak Mordell-Weil Theorem 

Abstract: For an elliptic curve E defined over a number field K, the Mordell-Weil theorem states that the group E(K) of K-rational points on the elliptic curve is finitely generated. The first step in proving this theorem is to prove the weak Mordell-Weil theorem, which states that the quotient group E(K)/mE(K) is finite for any positive integer m. Furthermore, the goal of finding generators for E(K) can be reduced to finding generators for any E(K)/mE(K).

We will discuss how E(K)/mE(K) can be computed using Selmer groups and group cohomology, and work through an explicit example computing E(Q)/2E(Q) for an elliptic curve defined over the rational numbers. 

Title: Number Theoretic Methods in Differential Geometry

Abstract: Geodesics are an important concept in differential geometry. In non-Euclidean model geometries, it is possible to have closed geodesics, and naturally a question arises. Does the Geodesic intersect itself before it completes its path? Such a geodesic is called non-simple. In this talk, we will explore a construction of real hyperbolic manifolds all of whose closed geodesics are simple. Almost all of the techniques used are number theoretic in nature, utilizing quaternion algebras formed over number fields or (characteristic 0) local fields. Time permitting, we will also discuss current developments in translating this project to complex hyperbolic space.

Title: Implicitization and Syzygies

Abstract: Given a rational map between projective spaces, a natural question that arises is how to study the (closed) image as a subvariety. As the coordinates are given parametrically, a natural question that arises is how to obtain the corresponding implicit system. This so-called implicitization problem has been studied to great length by geometers, algebraists and, in recent years, the geometric modeling community, for its applications to computer-aided design. In this talk, we discuss the connections of this geometric problem to the algebraic notion of syzygies. We discuss the history of the problem, as well as its modern treatment and applications, and end with a variety of open-ended questions for future research.