Title: Cohen-Macaulay Modules on Curves and Lattice Cohomology
Date: January 14, 2025
Venue: Deformation of Complex Singularities and Related Topics
Abstract: A Noetherian local ring containing a field admits at least one module with depth equal to the Krull dimension of the ring, called a maximal Cohen-Macaulay (MCM) module. A local ring is Cohen-Macaulay if and only if it is a MCM module over itself, and more broadly it is of interest to classify the MCM modules over a given ring. In particular, a ring is said to be of finite representation type if it admits only finitely many indecomposable MCM modules up to isomorphism; we will discuss how to characterize this property in the one-dimensional case using the lattice cohomology of the corresponding curve singularity. Based on joint work with András Némethi.
Title: Ideal-Theoretic Study of the Milnor Fibration
Date: June 5, 2024
Venue: International Congress on Complex Geometry, Singularities and Dynamics: In honor of José Seade
Abstract: If a holomorphic function germ has an isolated critical point, the topology of its Milnor fiber can be recovered from the ideal generated by its partial derivatives in the local power series ring of the ambient space. We give a relative version of this result for families of holomorphic functions with non-isolated critical points.
Slides: 20240605_pepefest.pdf
Title: Derived Categories III: Exact Triangles and Triangulated Categories
Date: May 27, 2024
Venue: Alfréd Rényi Institute of Mathematics Junior Singularities Seminar
Notes: The third of a series of talks on derived categories, the details of which can be found on my teaching page. This talk covered exact triangles in the homotopy category and derived category, triangulated categories, and the existence of long exact sequences arising from left or right derived functors. A digression on adjoint functors was also included.
Title: Derived Categories II: Derived Category and Functors
Date: May 13, 2024
Venue: Alfréd Rényi Institute of Mathematics Junior Singularities Seminar
Notes: The second of a series of talks on derived categories, the details of which can be found on my teaching page. This talk introduced the derived category and the constructions of the (right and left) derived functors of left- and right-exact functors on the underlying category.
Title: Derived Categories I: Preliminaries
Date: May 6, 2024
Venue: Alfréd Rényi Institute of Mathematics Junior Singularities Seminar
Notes: The first in a series of talks introducing the basic idea behind the derived category and some of its applications in topology; for the details, see my teaching page. This talk gave an overview of abelian categories, various flavors of additive functor, (co)chain complexes, and the homotopy category.
Title: Understanding Flatness Geometrically
Date: April 29, 2024
Venue: Alfréd Rényi Institute of Mathematics Junior Singularities Seminar
Notes: A talk on the geometric picture of flatness given by the definition in terms of normal cones, corresponding roughly to Section 2.3 and Subsection 2.4.1 of my thesis. The suggested materials to read in advance were Sections 5.1, 5.3, & 6.5 of David Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry and, optionally, Theorem 22.3 of Hideyuki Matsumura's Commutative Ring Theory.
Title: Ideal-Theoretic Study of Critical Points
Date: April 5, 2024
Venue: Alfréd Rényi Institute of Mathematics Algebraic Geometry and Differential Topology Seminar
Abstract: The local behavior of a holomorphic function at a critical point is described by the Milnor fibration. We explain how the ideal generated by the function's partial derivatives in the local power series ring can be used to obtain information about this behavior, and in particular its variation in families, for arbitrary-dimensional singularities.
Video: https://video.renyi.hu/video/ideal-theoretic-study-of-critical-points-777
Notes: Another talk on the material covered in the preprint at arXiv:2212.12807 and in my thesis, with a particular emphasis on expositing the idea of non-reduced structure.
Title: Milnor Fiber Consistency via Flatness
Date: December 13, 2023
Venue: Thesis Defense
Abstract: What can we say about the topology of the level sets of a homogeneous polynomial, or of a holomorphic function locally near a critical point? As it turns out, a lot less than you'd hope! We will discuss a new way of approaching the problem using machinery from algebraic (and complex-analytic) geometry.
Notes: This was my thesis defense; the thesis in question can be found here.
Title: Milnor Fiber Consistency via Flatness
Date: November 7, 2023
Venue: IberoSing International Workshop 2023: Mirror Symmetry & Hodge Ideals
Abstract: The Milnor fibration gives a well-defined notion of the smooth local fiber of a holomorphic function at a critical point. Milnor's work in the isolated case suggests that this fiber's topology should be controlled by the scheme-theoretic invariants of the critical locus; we give results which demonstrate that this is true in a relative sense. Specifically, we show that the local smooth fiber varies nicely in families where the embedded critical locus satisfies certain algebraic consistency requirements and discuss various implications.
Poster: 20231107_iberosing_ii.pdf
Notes: Corresponds to the preprint at arXiv:2212.12807; more or less identical to the poster presented at 115AM in Jaca.
Title: Why Does Normalization Resolve Singularities in Codimension 1?
Date: October 18, 2023
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: We review the definition of the normalization of an algebraic variety, discuss the geometric intuition for it, and explain why normalizing removes codimension-1 singularities. This talk is intended to be accessible to people taking first-semester algebraic geometry.
Notes: I didn't put it in the abstract, but the answer (or one answer, anyway) to the titular question turns out to be a statement about normal cones! See my thesis, in particular Subsection 2.4.4, for more details.
Title: Milnor Fiber Consistency via Flatness
Date: July 1, 2023
Venue: The Tenth Congress of Romanian Mathematicians
Abstract: The Milnor fibration gives a well-defined notion of the smooth local fiber of a holomorphic function at a critical point. Milnor's work in the isolated case suggests that this fiber's topology should be controlled by the scheme-theoretic invariants of the critical locus; we give results which demonstrate that this is true in a relative sense. Specifically, we show that the local smooth fiber varies nicely in families where the embedded critical locus satisfies certain algebraic consistency requirements and discuss implications for homogeneous polynomials and other special cases.
Slides: 20230701_crm.pdf
Notes: Corresponds to the preprint at arXiv:2212.12807.
Title: Milnor Fiber Consistency via Flatness
Date: June 22, 2023
Venue: Singularities and Applications Research School
Abstract: The Milnor fibration gives a well-defined notion of the smooth local fiber of a holomorphic function at a critical point. Milnor's work in the isolated case suggests that this fiber's topology should be controlled by the scheme-theoretic invariants of the critical locus; we give results which demonstrate that this is true in a relative sense. Specifically, we show that the local smooth fiber varies nicely in families where the embedded critical locus satisfies certain algebraic consistency requirements and discuss implications for homogeneous polynomials and other special cases.
Notes: Corresponds to the preprint at arXiv:2212.12807.
Title: Milnor Fiber Consistency via Flatness
Date: June 16, 2023
Venue: 115AM: Algebraic and Topological Interplay of Algebraic Varieties
Abstract: The Milnor fibration gives a well-defined notion of the smooth local fiber of a holomorphic function at a critical point. Milnor's work in the isolated case suggests that this fiber's topology should be controlled by the scheme-theoretic invariants of the critical locus; we give results which demonstrate that this is true in a relative sense. Specifically, we show that the local smooth fiber varies nicely in families where the embedded critical locus satisfies certain algebraic consistency requirements and discuss implications for homogeneous polynomials and other special cases.
Poster: 20230616_115am.pdf
Notes: Corresponds to the preprint at arXiv:2212.12807.
Title: Milnor Fiber Consistency via Flatness
Date: March 13, 2023
Venue: Singularities in the Midwest VIII
Abstract: The Milnor fibration gives a well-defined notion of the smooth local fiber of a holomorphic function at a critical point. Milnor's work in the isolated case suggests that this fiber's topology should be controlled by the scheme-theoretic invariants of the critical locus; we give results which demonstrate that this is true in a relative sense. Specifically, we show that the local smooth fiber varies nicely in families where the embedded critical locus satisfies certain algebraic consistency requirements and discuss implications for homogeneous polynomials and other special cases.
Slides: 20230313_singsinmidwest.pdf
Notes: Similar to many of the earlier presentations on this list, with a correction to the main result. Corresponds to the preprint at arXiv:2212.12807.
Title: Normal Cones in Algebraic Geometry
Date: February 15, 2023
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: In this talk, we'll go over the definition of the normal cone of a closed subscheme, explore the geometric intuition behind it via a construction called the Rees algebra, and explain how it can be used to give geometric characterizations of apparently algebraic notions such as flatness and depth.
Title: Smooth Fibers, Critical Points, and Flatness
Date: October 28, 2022
Venue: Universidad Autónoma de Madrid Algebraic Geometry Seminar
Abstract: Given a family of homogeneous polynomials depending holomorphically on some parameters, we can ask when the fibers away from the origin remain consistent as we vary the parameters. We give a sufficient condition for this to occur - namely, the flatness of the polynomials' critical loci over the parameter space - by way of the Milnor fibration.
Notes: A talk covering the same material as the previous poster, with an emphasis on aspects that might interest algebraic geometers.
Title: Milnor Fiber Consistency via Flatness
Date: October 27, 2022
Venue: IberoSing International Workshop 2022
Abstract: The Milnor fiber of a holomorphic function defining an isolated singularity can be understood by perturbing the function slightly to one with only Morse critical points. For an arbitrary-dimensional singularity, perturbation is no longer guaranteed to preserve the Milnor fiber, making an analogous approach difficult. We present a new algebraic condition - the flatness of the critical locus over the parameter space - under which this problem does not arise, giving a new avenue for Milnor fiber computations.
Poster: 20221027_iberosing.pdf
Notes: Similar to some of the earlier presentations on this list, with the addition of some applications to homogeneous polynomials.
Title: Revenge of the Classical Topology
Date: October 7, 2022
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: The Zariski topology is pretty cool, but, if we're working over the complex numbers, we can also think about the classical topology we're used to from other areas of math. In this talk, we'll discuss analytification, the process of passing from an algebraic variety (or scheme) to the corresponding classical object (or complex-analytic space), and touch on various facts about the relationship between the two, such as Serre's GAGA principle. If time permits, we'll also talk a little about tools we can use to gain insight about varieties once we've analytified them, such as the theory of stratifications.
Title: Milnor Fiber Consistency for Deformations of Arbitrarily-Singular Hypersurfaces
Date: July 20, 2022
Venue: CIMPA Research School on Singularities and Applications
Abstract: The Milnor fiber of a holomorphic function defining an isolated singularity can be understood by perturbing the function slightly to one with only Morse critical points. For an arbitrary-dimensional singularity, perturbation is no longer guaranteed to preserve the Milnor fiber, making an analogous approach difficult. We present a new algebraic condition - the flatness of the critical locus over the parameter space - under which this problem does not arise, giving a new avenue for Milnor fiber computations.
Slides: 20220720_cimpasaocarlos.pdf
Video: https://youtu.be/yMAeAk3PEO8?t=108
Notes: An in-person version of the previous talk, now with a better example usage of the theorem.
Title: Milnor Fiber Consistency via Flatness
Date: June 1, 2022
Venue: Iberoamerican Webminar of Young Researchers in Singularity Theory
Abstract: Given a holomorphic family of function germs defining hypersurface singularities, we can ask whether the Milnor fiber varies consistently; in the isolated case, it is well-known that the answer is always yes (in the sense that the family defines a fibration above the complement of the discriminant), and this allows us to obtain a distinguished basis of vanishing cycles for a singularity by perturbing it slightly. In the non-isolated case, this is not always true, and there has long been interest in finding conditions under which this kind of consistency does occur. We give a powerful algebraic condition which is sufficient for this purpose - namely, that the analogous statement will hold so long as the critical locus of the family, considered as an analytic scheme, is flat over the parameter space.
Slides: 20220601_iberoamerican.pdf
Video: https://drive.google.com/file/d/17Swfe-6vWSq0sk8J7l-dBirx6UZJju_G/view
Notes: Basically the same as the previous talk, with the addition of a counterexample due to Bobadilla and some embarrassing technical issues. Watch at your own risk!
Title: Consistency of Milnor Fibers for Deformations of Arbitrary-Dimensional Hypersurface Singularities
Date: May 2, 2022
Venue: University of Wisconsin-Madison Topology and Singularities Seminar
Abstract: Given a holomorphic family of function germs defining hypersurface singularities, we can ask whether the Milnor fiber varies consistently; in the isolated case, it is well-known that the answer is always yes (in the sense that the family defines a fibration above the complement of the discriminant), and this allows us to obtain a distinguished basis of vanishing cycles for a singularity by perturbing it slightly. In the non-isolated case, this is not always true, and there has long been interest in finding conditions under which this kind of consistency does occur. We give a powerful algebraic condition which is sufficient (and possibly necessary) for this purpose - namely, that the analogous statement will hold so long as the critical locus of the family, considered as an analytic scheme, is flat over the parameter space.
Slides: 20220502_uwmadisonsingularities.pdf
Video: https://uwmadison.app.box.com/s/i7kcnfd992qxdky2bat053l3qjjaa8s7/file/953658629482
Notes: It later turned out that the "possibly necessary" part of the abstract was a tad optimistic.
Title: Geometric Intuitions for Flatness
Date: April 7, 2022
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Flatness is often described as the correct characterization of what it means to have "a nicely varying family of things" in the setting of algebraic geometry. In this talk, which is intended to be pretty low-key, I'll dig a little bit more into what that means, and discuss some theorems and examples that refine and clarify this intuition in various ways.
Title: An Introduction to the Deformation Theory of Complete Intersection Singularities
Date: March 18, 2021
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Essentially what it says in the title; I'll give a fairly laid-back overview of some of the basic definitions and results about deformations of complete intersection singularities, including the Kodaira-Spencer map and the existence of versal deformations in the isolated case. If time permits, I'll discuss Morsification of isolated singularities. Very little background will be assumed.
Slides: GAGS_CIS.pdf
Notes: Important clarification not mentioned on the slides: The Milnor fiber is well-defined only in certain cases!
Title: The Topology of Milnor Fibers
Date: December 15, 2020
Venue: Specialty Exam
Slides: specialty_exam.pdf
Notes: The original abstract of this talk is lost to time, but it is mainly a roundup of various results on the topology of the Milnor fibration. Also, I was not aware of this at the time, but it should be noted that the quantity discussed at the start of Section IV is actually a case of a known concept, the multiplicity of a prime ideal in a module. This can be found in Section 3.6 of David Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry.
Title: Embrace the Singularity: An Introduction to Stratified Morse Theory
Date: April 15, 2020
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Early on in the semester, Colin told us a bit about Morse Theory, and how it lets us get a handle on the (classical) topology of smooth complex varieties. As we all know, however, not everything in life goes smoothly, and so too in algebraic geometry. Singular varieties, when given the classical topology, are not manifolds, but they can be described in terms of manifolds by means of something called a Whitney stratification. This allows us to develop a version of Morse Theory that applies to singular spaces (and also, with a bit of work, to smooth spaces that fail to be nice in other ways, like non-compact manifolds!), called Stratified Morse Theory. After going through the appropriate definitions and briefly reviewing the results of classical Morse Theory, we'll discuss the so-called Main Theorem of Stratified Morse Theory and survey some of its consequences.
Slides: GAGS_SMT.pdf
Notes: "Colin" here is Colin Crowley. The slides have some blank spaces for pictures I drew during the talk, which didn't end up getting saved anywhere.
Title: Tropicalization Blues
Date: November 13, 2019
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Tropicalization turns algebro-geometric objects into piecewise linear ones which can then be studied through the lens of combinatorics. In this talk, I will introduce the basic construction, then discuss some of the recent efforts to generalize and improve upon it, touching upon the Giansiracusa tropicalization and developing gazing wistfully in the direction of the machinery of ordered blueprints necessary for the Lorscheid tropicalization.
Notes: Turns out I didn't do so hot estimating the timing on this one, so we didn't get anywhere close to the ordered blueprint stuff. This is a framework due to Oliver Lorscheid which essentially lets one formalize various notions of tropicalization as base changes in an appropriate category; if you'd like to learn more, I wrote this summary for Dan Corey's Introduction to Tropical Geometry course in Spring 2020.
Title: Kindergarten GAGA
Date: April 10, 2019
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Join me in regressing to an infantile state as we discuss Serre's 1956 paper Algebraic geometry and analytic geometry, widely considered to be the most influential work ever authored by a mathematician under the age of five. We will define the notion of an analytic space, construct the analytic space associated to any algebraic variety over C, and examine the relationships between the two, including the equivalence between coherent algebraic sheaves and coherent analytic sheaves in the projective case.
Notes: This abstract is not entirely serious. In fact, Serre was over five years old when he wrote GAGA.