Boston College Algebraic Geometry Seminar

In his shire theorem, Polya proves that the zeros of iterated derivatives of a rational function in the complex plane accumulate on the union of edges of the Voronoi diagram of the poles of this function. Recasting the local arguments of Polya into the language of translation surfaces, we prove a generalization describing the asymptotic distribution of the zeros of a meromorphic function on a compact Riemann surface under the iterations of a linear differential operator defined by meromorphic 1-form. The accumulation set of these zeros is the union of edges of a generalized Voronoi diagram defined jointly by the initial function and the singular flat metric on the Riemann surface induced by the differential. This process offers a completely novel approach to the practical problem of finding a flat geometric presentation (a polygon with identification of pairs of edges) of a translation surface defined in terms of algebraic or complex-analytic data. This is a joint work with Rikard Bogvad, Boris Shapiro and Sangsan Warakkagun.

Understanding the mathematical foundation of machine learning theory has become essential in this ever changing world of AI. In particular, neural networks with Rectified Linear Unit (ReLU) activations underlie almost every other machine learning problem. In this talk, I will develop a dictionary between feedforward ReLU neural networks and toric geometry. The central insight underlying this work is that the function computed by a feedforward ReLU neural network with no biases can be interpreted as the support function of a Q-Cartier divisor on a rational polyhedral fan. With the toric geometry framework, we have a complete classification of functions realizable by unbiased shallow ReLU neural networks by computing intersection numbers of the ReLU Cartier divisor and torus-invariant curves.

Let X be a smooth Fano variety over the complex numbers. A natural object to study is the moduli space of stable maps to X. In particular, one fundamental question concerning this moduli space is to determine its irreducible components and their dimensions, as such knowledge usually sheds light on the arithmetic and the enumerative geometry of X.

In this talk, I will discuss this question through the lens of Geometric Manin's Conjecture. I will present a classification result on the components of the space of genus one stable maps when X is a smooth cubic threefold. The proof proceeds by distinguishing the components via the notion of free curves, before invoking Mori's Bend-and-Break to inductively show their irreducibility.

A fundamental question in algebra is the classification of finitely generated projective modules (or algebraic vector bundles, if you like) over a ring. In the world of topology, vector bundles over a finite CW complex are classified up to finite ambiguity by their Chern classes, and it is natural to ask whether an analogous question is true for algebraic vector bundles, say over a smooth complex affine variety. The desired statement is indeed true in low ranks and dimensions, but begins to fail when examining rank two bundles over fourfolds. Using techniques drawn from motivic homotopy theory (I won't assume the audience knows anything about this!), we establish conditions under which there are only finitely many rank two algebraic vector bundles over a smooth affine fourfold. We provide many concrete examples, as well as some applications towards splitting questions or the classification of symplectic vector bundles. This is joint work with M. Opie and T. Syed.

Motivated by a topological proof of Manin’s conjecture over global function fields, Ellenberg and Venkatesh envisioned homological stability for the space of sections of Fano fibrations. In this talk we discuss this property in the context of weak approximation and establish such a stability for certain Fano fibrations. This is joint work with Yuri Tschinkel.

To a polynomial f over Q, one can associate two zeta functions, using two distinct norms on field extensions of Q. The Archimedean zeta function originates in Gelfand’s 1954 problem, and is closely related to the existence of fundamental solutions of linear PDEs. Its non-Archimedean cousin, the p-adic zeta function, was motivated by Borewicz-Schfarevich’s 1966 question on the rationality of generating functions counting solutions of f modulo powers of a prime p. A central theme is to understand what information about f is encoded in these zeta functions. 

In this talk, I will survey recent progress on this theme, by the means of mixed Hodge modules. On the Archimedean side, we resolve a question of Mustațǎ-Popa, showing that it detects whether the hypersurface f=0 has rational singularities. We also refine Bernstein’s classical bound on the order of poles, using the monodromy weight on the Milnor fibers of f, which in turn provides a negative answer to a long-standing question of Loeser  (1985). Finally, building on insights from the complex case, we establish the Strong Monodromy Conjecture for the p-adic zeta function of hyperplane arrangements and confirm the n/d conjecture of Budur-Mustațǎ-Teilter (2009).

This is based on several joint works with Dougal Davis and Andras Lörincz.