Talk till you are stuck, Spring 2025

    Description of the seminar

The seminar will focus on Algebraic Geometry, and PhD students will present their research or recent thoughts until they reach questions they are currently grappling with. While the seminar is primarily AG-centered, it is very much open to number theorists and algebraic topologists due to the many connections between these fields. But essentially anyone with "AG-questions" and an "AG-audience" in mind is welcome to present.

The format is of 1 hour and 30/15 minutes: the first hour for a proper introduction and presentation of known or new results, and the remaining time for discussion. During the discussion, the speaker and audience can ask questions, share suggestions, and brainstorm ideas. Speakers should aim to distill the "I am stuck with" part of their talk into something the audience can engage with during the seminar.

Hopefully, this will be a fun way to foster collaboration in the Department or just learn more about each other's research.

    Logistics Info

The seminar will be held on Thursdays from 4:10pm to 5:30pm in Room 622 in the Math Department at Columbia.

Please email me at mp3947 at columbia dot edu if you are interested in giving a talk and/or you want to be added to the mailing list.

    Schedule

1. January 30th
           Organizational meeting


2. February 6th
           Speaker: Sofia Wood
           Title: Invariants of compactifications of the universal Jacobian over the moduli space of curves
           Abstract: We discuss some invariants of compactified Jacobians over the moduli space of curves, and some related questions.


3. February 13th
           Speaker: Morena Porzio
           Title: Stable birationality of Symmetric powers Sym^n_X of cubic hypersurfaces X for small n
           Abstract: Today we will discuss the stable birational type of Sym^n_X where X is a hypersurface. For cubic surfaces this is well understood for big n: however the arguments involved break down in higher dimensions and for small n. Ad hoc arguments have been developed to tackle this issue but they still leave some "leftovers" even in the surface case. Today we will go over the known results, see what the leftovers n's are and discuss what can and cannot work in these cases.


4. February 27th
           Speaker: Morena Porzio
           Title: When is the set of degree n points dense on a surface?
           Abstract: Faltings' Theorem describes completely when a curve C over a number field can have (resp. has) dense k-points: however, already for surfaces there are many open questions. After a brief overview on that matter, we will focus on a broader question: given a variety X when is the set of degree n points Zariski dense? For curves the situation is fairly understood (Viray--Vogt), while for surfaces it's work in progress. In a joint work with Berg, Fu, Gazaki, Rawson, and Vogt, we worked out the case of the product of curves with low genus, but "some cases" can still be better understood.


5. March 13th
           Speaker: Matthew Hase-Liu
           Title: A geometric perspective on an analytic-number-theoretic approach to understanding the geometric properties of a moduli-theoretic analogue of a generalization of the classic Waring's problem from analytic number theory.
           Abstract: The second-most interesting moduli space is the moduli space of curves on a hypersurface. I will explain how to analyze the geometry of this space (including irreducibility, dimension, and quality of singularities) by geometrizing tools from analytic number theory. For example, I will show that the Fano scheme of lines on any sufficiently low-degree smooth hypersurface has at worst terminal singularities. Such tools have been successful in approaching Waring's problem and Manin's conjecture, which are in some sense the number-theoretic analogues of these geometric questions.


6. March 27th
           Speaker: Nicolás Vilches
           Title: Explicit deformation theory
           Abstract: Deformation theory is a powerful tool that gives us a local description of moduli problems. There is a catch: getting explicit descriptions tends to be tricky. We will survey various approaches one can take towards this goal in various concrete examples.


7. April 3rd: No seminar today.


8. April 10th
           Speaker: Alex Scheffelin
           Title: Cohomological Dimension of Regular Local Rings
           Abstract: Local Cohomology is an algebraic invariant associated to modules which plays the role of cohomology with support in a closed subset in algebraic geometry. We will first define local cohomology, explain its relations to cohomology of schemes, highlighting the relationship between cohomology on open subsets of projective space and local cohomology. We will then define cohomological dimension, and introduce a problem for regular local rings which has implications for cohomology of open subsets of projective space.


9. May 1st
           Speaker: Kuan-Wen Chen
           Title: Hodge Numbers of Hypersurfaces
           Abstract: Given a smooth hypersurface in a smooth projective variety over complex numbers, a natural question is how to compute its Betti and Hodge numbers from the data of the ambient variety. When the hypersurface is defined by an ample line bundle, the standard approach involves the Lefschetz hyperplane theorem and the computation of Hirzebruch genera. In this talk, we explore a method for the case where the hypersurface is not ample and apply it to a concrete example—Schoen’s Calabi–Yau manifolds. We will also discuss some open questions arising from this result.