Disclaimer: For many of my presentations I have included the notes I used
when giving the talk. These notes are extremely raw and unpolished. They
almost surely contain many errors, typos, and mistakes; not to mention things that only
make sense to me. So when reading them be catious and keep their context in mind.
Research Presentations
Asymptotic Syzygies for Products of Projective Space Algebra Seminar, University of Kentucky - Lexington, KY - 4/24/19
I will discuss results describing the asymptotic syzygies of
products of projective space, in the vein of the explicit
methods of Ein, Erman, and Lazarsfeld’s non-vanishing results
on projective space.
Semi-Ample Asymptotic Syzygies Commutative Algebra Seminar, University of Minnesota - Minneapolis, MN - 4/11/19
I will discuss the asymptotic non-vanishing of syzygies for
products of projective spaces, generalizing the monomial methods
of Ein-Erman-Lazarsfeld. This provides the first example of how
the asymptotic syzygies of a smooth projective variety whose
embedding line bundle grows in a semi-ample fashion behave in
nuanced and previously unseen ways.
Asymptotic Syzygies for Products of Projective Space (Poster) 2019 AWM Research Symposium, Rice University - Houston, TX - 4/6/19
I will discuss results describing the asymptotic syzygies of
products of projective space, in the vein of the explicit
methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on
projective space.
Asymptotic Syzygies for Products of Projective Space KUMUNUkr, University of Nebraska - Lincoln, NE - 3/30/19
I will discuss results describing the asymptotic syzygies of
products of projective space, in the vein of the explicit
methods of Ein, Erman, and Lazarsfeld’s non-vanishing results
on projective space.
Asymptotic Syzygies for Products of Projective Space AMS Spring Southeastern Sectional, Auburn Univeristy - Auburn, AL - 3/16/19
I will discuss results describing the asymptotic syzygies of
products of projective space, in the vein of the explicit
methods of Ein, Erman, and Lazarsfeld’s non-vanishing results
on projective space.
Asymptotic Syzygies for Products of Projective Space AMS Special Session, Joint Math Meetings - Baltimore, MD - 1/17/19
I will discuss results describing the asymptotic syzygies of
products of projective space, in the vein of the explicit
methods of Ein, Erman, and Lazarsfeld’s non-vanishing results
on \(\mathbb{P}^n\).
Covering Abelian Varieties and Effective Bertini Algebra and Algebraic Geometry Seminar, University of Wisconsin - Madison, WI - 11/3/18
I will discuss recent work showing that every abelian variety
is covered by a Jacobian whose dimension is bounded. This is
joint with Wanlin Li.
Asymptotic Syzygies for Products of Projective Space AMS Fall Central Sectional, University of Michigan - Ann Arbor, MI - 10/20/18
I will discuss results describing the asymptotic syzygies of
products of projective space, in the vein of the explicit
methods of Ein, Erman, and Lazarsfeld’s non-vanishing results
on \(\mathbb{P}^{n}\).
Asymptotic Syzygies in the Semi-Ample Setting (Poster) AGNES Poster Session, Brown University - Providence, RI - 9/22/18
In recent years numerous conjectures have been made describing
the asymptotic Betti numbers of a projective variety $X\subset
\mathbb{P}^{r}$ as the embedding line bundle becomes more ample.
I will present recent work attempting to generalize these
conjectures to the case when the embedding line bundle becomes
more semi-ample. (Recall a line bundle is semi-ample if a
sufficiently large multiple is base point free.) In particular,
I will discuss how the monomial methods of Ein, Erman, and
Lazarsfeld for proving non-vanishing results on $\mathbb{P}^{n}$
can be extended to prove non-vanishing results for products of
projective space.
Covering Abelian Varieties and Effective Bertini Algebraic Geometry and Number Theory Seminar, Rice University - Houston, TX - 9/18/18
Until recently the Betti tables of \(\mathbb{P}^{2}\) under the
d’uple Veronese embedding were only known for \(d<5\). I will
discuss a project that synthesizes a number of new examples by
using numerical linear algebra and large-scale distributed
computing. Using this data we have found evidence supporting
several existing, and new, conjectures on the syzygies of
\(\mathbb{P}^2\). This is joint work with Daniel Erman, Steve
Goldstein, and Jay Yang.
Asymptotic Syzygies in the Semi-Ample Setting Algebra and Algebraic Geometry Seminar, University of Wisconsin - Madison, WI - 2/9/18
In recent years numerous conjectures have been made describing
the asymptotic Betti numbers of a projective variety $X\subset
\mathbb{P}^{r}$ as the embedding line bundle becomes more ample.
I will present recent work attempting to generalize these
conjectures to the case when the embedding line bundle becomes
more semi-ample. (Recall a line bundle is semi-ample if a
sufficiently large multiple is base point free.) In particular,
I will discuss how the monomial methods of Ein, Erman, and
Lazarsfeld for proving non-vanishing results on $\mathbb{P}^{n}$
can be extended to prove non-vanishing results for products of
projective space.
Asymptotic Syzygies in the Semi-Ample Setting (Poster) AWM Poster Session, Joint Math Meetings - San Diego, CA - 1/12/18
In recent years numerous conjectures have been made describing
the asymptotic Betti numbers of a projective variety $X\subset
\mathbb{P}^{r}$ as the embedding line bundle becomes more ample.
I will present recent work attempting to generalize these
conjectures to the case when the embedding line bundle becomes
more semi-ample. (Recall a line bundle is semi-ample if a
sufficiently large multiple is base point free.) In particular,
I will discuss how the monomial methods of Ein, Erman, and
Lazarsfeld for proving non-vanishing results on $\mathbb{P}^{n}$
can be extended to prove non-vanishing results for products of
projective space.
Asymptotic Syzygies in the Semi-Ample Setting Commutative Algebra Seminar, University of Michigan - Ann Arbor, MI - 12/7/17
In recent years numerous conjectures have been made describing
the asymptotic Betti numbers of a projective variety as the
embedding line bundle becomes more ample. I will discuss recent
work attempting to generalize these conjectures to the case when
the embedding line bundle becomes more semi-ample. (Recall a
line bundle is semi-ample if a sufficiently large multiple is
base point free.) In particular, I will discuss how the monomial
methods of Ein, Erman, and Lazarsfeld for proving non-vanishing
results on projective space can be extended to prove
non-vanishing results for products of projective space.
Asymptotic Syzygies in the Semi-Ample Setting Structures on Free Resolutions, Texas Tech University - Lubbock, TX - 10/27/17
In recent years numerous conjectures have been made describing the asymptotic Betti numbers of a projective variety $X\subset \mathbb{P}^r$ as the embedding line bundle becomes more ample. I will discuss recent work attempting to generalize these conjectures to the case when the embedding line bundle becomes more semi-ample. (Recall a line bundle is semi-ample if a sufficiently large multiple is base point free.) In particular, I will discuss how the monomial methods of Ein, Erman, and Lazarsfeld for proving non-vanishing results on $\mathbb{P}^n$ can be extended to prove non-vanishing results for products of projective space.
A Distributed Numerical Approach to Syzygies of P^2 - (Poster) Free Resolutions and Computations, University of California - Berkeley - 7/18/17
Until recently the Betti tables of $\mathbb{P}^{2}$ under the d'uple embedding were only known for $d<5$. I will present on a project that synthesizes a number of new examples by using numerical linear algebra and large-scale distributed computing. Using this data we have found evidence supporting several existing, and new, conjectures on the syzygies of $\mathbb{P}^2$. This is joint work with Daniel Erman, Steve Goldstein, and Jay Yang.
The Degree of SO(n) and Low-rank SDP Applied Algebra Seminar, University of Wisconsin - 3/31/17
The group of special orthogonal matrices has the structure of a variety. This talk will focus on how certain geometric properties of this variety are related to the convergence of a particular algorithm for solving a low-rank semidefinite programing (SDP) problem. In particular, we provide a closed formula for the degree of SO(n), and show that from this one can count the number of critical points of this low-rank SDP problem. Additionally, I will discuss our computational work exploring the real locus of SO(n).This is based upon joint work with Madeline Brandt, Taylor Brysiewicz, Robert Krone, and Elina Robeva.
A Probabilistic Approach to Noether Normalization - (Poster) RTG Lectures in Arithmetic Geometry, Rice University - 2/18/17
Classically, Noether normalization says that if \(X\) is an \(n\)-dimensional projective (resp. affine) variety over a field $\kk$ then their exists a finite morphism \(\phi:X\rightarrow\mathbb{P}^n_{\textbf{k}}\) (resp. \(\mathbb{A}^n_{\textbf{k}}\)). By studying the distribution of systems of parameters we show similar results hold when \(\textbf{k}\) is replaced by certain dimension one integral domains namely \(\mathbb{Z}\) or \(\textbf{F}_q[t]\).
Noether Normalization in Families - (Poster) Introductory Workshop: Combinatorial Algebraic Geometry, Fields Institute - 8/16/16
Classically, Noether normalization says that if \(X\) is an \(n\)-dimensional projective (resp. affine) variety over a field $\kk$ then their exists a finite morphism \(\phi:X\rightarrow\mathbb{P}^n_{\textbf{k}}\) (resp. \(\mathbb{A}^n_{\textbf{k}}\)). By studying the distribution of systems of parameters we show similar results hold when \(\textbf{k}\) is replaced by certain dimension one integral domains namely \(\mathbb{Z}\) or \(\textbf{F}_q[t]\).
Noether Normalization in Families - (Poster) Commutative Algebra and Its Interactions with Algebraic Geometry, University of Michigan - 7/8/16
Classically, Noether normalization says that if \(X\) is an \(n\)-dimensional projective (resp. affine) variety over a field $\kk$ then their exists a finite morphism \(\phi:X\rightarrow\mathbb{P}^n_{\textbf{k}}\) (resp. \(\mathbb{A}^n_{\textbf{k}}\)). By studying the distribution of systems of parameters we show similar results hold when \(\textbf{k}\) is replaced by certain dimension one integral domains namely \(\mathbb{Z}\) or \(\textbf{F}_q[t]\).
Noether Normalization in Families - (Poster) Midwest Commutative Algebra and Commutative Algebra Conference, University of Notre Dame - 5/17/16
Classically, Noether normalization says that if \(X\) is an \(n\)-dimensional projective (resp. affine) variety over a field $\kk$ then their exists a finite morphism \(\phi:X\rightarrow\mathbb{P}^n_{\textbf{k}}\) (resp. \(\mathbb{A}^n_{\textbf{k}}\)). By studying the distribution of systems of parameters we show similar results hold when \(\textbf{k}\) is replaced by certain dimension one integral domains namely \(\mathbb{Z}\) or \(\textbf{F}_q[t]\).
Noether Normalization in Families Algebraic Geometry Seminar, University of Wisconsin - 4/15/16
Classically, Noether normalization says that any projective (resp. affine) variety of dimension n over a field admits a finite surjective morphism to \(\mathbb{P}^n\) (resp. \(\mathbb{A}^n\)). I will discuss whether we can generalize such theorems to other bases like \(\mathbb{Z}\), \(\mathbb{C}[t]\), etc. This is based on joint work with Daniel Erman.
Probability of Systems of Parameters Speciality Exam, University of Wisconsin - 2/29/16
Given \(k\) homogenous polynomials in \(n+1\) variables what do we expect about the dimension of their zero set? Well a good guess would be that each polynomial imposes a dimension one condition and so their zero set should have dimension \(n-(k+1)\). In this talk I will discuss how, over a finite field, we may put this intuition to the test. This is based on joint work with Daniel Erman.
Betti Tables of Graph Curves Midwest Algebraic Geometry Graduate Conference, University of Illinois - Chicago - 4/11/15
Given a graph one may obtain a reducible algebraic curve by associating a \(\mathbb{P}^1\) to each vertex with two \(\mathbb{P}^1\)'s intersecting if there is an edge between the associated vertices. Such curve are called graph curves, or line arrangements, and were introduced by Bayer and Eisenbud in studying Greens conjecture. I will discuss how the combinatorics of the graph affect the Betti table of its associated curve. In particular, I will present formulas for the Betti table for all graph curves of genus zero and one. Additionally, I will give formulas for the graded Betti numbers for a class of curves of higher genus. This talk is based on joint work with Pete Vermeire, Evan Nash, Ben Perez, and Pin-Hung Kao.
Betti Tables of Graph Curves Algebraic Geometry Seminar, University of Wisconsin - 12/12/14
Given a graph one may obtain a reducible algebraic curve by associating a \(\mathbb{P}^1\) to each vertex with two \(\mathbb{P}^1\)'s intersecting if there is an edge between the associated vertices. Such curve are called graph curves, or line arrangements, and were introduced by Bayer and Eisenbud in studying Greens conjecture. I will discuss how the combinatorics of the graph affect the Betti table of its associated curve. In particular, I will present formulas for the Betti table for all graph curves of genus zero and one. Additionally, I will give formulas for the graded Betti numbers for a class of curves of higher genus. This talk is based on joint work with Pete Vermeire, Evan Nash, Ben Perez, and Pin-Hung Kao.
Expository Presentations
Intro to Moduli Spaces and Group Work AG Reading Group, University of Wisconsin - 3/8/19
I will try present and motivate the definition of fine and coarse moduli spaces via active learning and group work.
Kissing Conics AMS Student Chapter Seminar, University of Wisconsin - 10/2/18
Have you every wondered how you can easily tell when two plane conics kiss (i.e. are tangent to each other at a point)? If so this talk is for you, if not, well there will be donuts.
The Hilbert and Quot Schemes Examples in Algebraic Geometry (Reading Group), University of Wisconsin - 4/5/17
I will give a introduction to the Hilbert and Quot schemes outlining how one might go about constructing them.
The Many Faces of the Grassmannian Examples in Algebraic Geometry (Reading Group), University of Wisconsin - 2/1/17
There are many different ways one might "describe" a variety; one might give equations, one might glue it from other spaces, one might parameterize it, and so on. We will explore these various ways by looking at the Grassmannian.
Complexity in Algebraic Geometry Positivity in Algebraic Geometry (Reading Group), University of Wisconsin - 10/18/16
I will discuss some of the material presented in Section I.1.8 of Lazarsfeld's Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series. In particular, I will try and motivate the notion of the Castelnuovo — Mumford regularity of a coherent sheaf on projective space via the question: How can we measure the complexity of a projective variety?
Some Numbers Are Sometimes Bigger Than Others (Sometimes...) AMS Student Chapter Seminar, University of Wisconsin - 10/8/14
I will write down two numbers and show that one of them is larger than the other.
Vignettes in Algebraic Geometry Graduate Algebraic Geometry Seminar, University of Wisconsin - 9/14/16
Algebraic geometry is a massive forest, and it is often easy to become lost in the thicket of technical detail and seemingly endless abstraction. The goal of this talk is to take a step back out of these weeds, and return to our roots as algebraic geometers. By looking at three different classical problems we will explore various parts of algebraic geometry, and hopefully motivate the development of some of its larger machinery. Each problem will slowly build with no prerequisite assumed of the listener in the beginning.
A Crazy Way to Define Homology AMS Student Chapter Seminar, University of Wisconsin - 4/27/16
This talk will be like a costume party!! However, instead of pretending to be an astronaut I will pretend to be a topologist, and try and say something about the Dold-Thom theorem, which gives a connected between the homotopy groups and homology groups of connected CW complexes. So I guess this talk will be nothing like a costume party, but feel free to wear a costume if you want.
Divisors and Stuff - (3 Talks) Graduate Algebraic Geometry Seminar, University of Wisconsin - 2/21 - 3/9/16
The goal of my talks will be to prove Zariski Decomposition, which says that on a smooth surface divisors take a particularly nice form. In the first talk I will review the basics about divisors i.e. what they are, what we do with them, etc. In the second talk I will construct the intersection pairing on a smooth surface and begin proving Zariski Decomposition. Finally, the third talk will focus on completing the proof of Zariski Decomposition and explaining how it can be utilized and extended to higher dimensions.
Counting Categorically Graduate Number Theory Seminar, University of Wisconsin - 1/26/15
As number theorists we are often interested in counting various things: quadratic fields, quadratic forms, random permutations. This talk will not focus in any detail about these examples, but instead we will explore various ways we might approaching counting. In particular, I will try and convince you that keeping track of additional structure when counting often leads to more natural answers. This is based in part on Tom Leinster's The Euler characteristic of a category. Despite the title very little about category theory will be used, and the speaker will do their best to keep things down to Earth.
Dynamics, Covers, and Kneading Sequences AMS Student Chapter Seminar, University of Wisconsin - 10/21/15
Given a continuous map f:X—>X of topological spaces and a point x in X one can consider the set {x, f(x), f(f(x)), f(f(f(x))),…} i.e, the orbit of x under the map f. The study of such things even in simple cases, for example when X is the complex numbers and f is a (quadratic) polynomial, turns out to be quite complex (pun sort of intended). (It also gives rise to main source of pretty pictures mathematicians put on posters.) In this talk I want to show how the study of such orbits is related to the following question: How can one tell if a (ramified) covering of S^2 comes from a rational function? No background will be assumed and there will be pretty pictures to stare at.
The Ring - (2 Talks) Graduate Algebraic Geometry Seminar, University of Wisconsin - 9/23 - 9/30/15
The Grothendieck ring of varieties is an incredibly mysterious object that seems to capture a bunch of arithmetic, geometric, and topological data regarding algebraic varieties. We will explore some of these connections. For example, we will see how the Weil Conjectures are related to stable birational geometry. No background will be assumed and the speaker will try and keep things accessible to all.
The Important Questions Graduate Number Theory Seminar, University of Wisconsin - 9/15/15
Did the Universe come from nothing? Why are we moral? Where did we come from? According to some signs on Bascom Hill these are the important questions in life. Sadly the poor person who made these signs does not know what the really important questions are: What is David Zureick-Brown going to saying in his NTS talk? How many rational points are on the projective curve given (in affine coordinates) by: $$y^2 = x^6 + 8x^5 + 22x^4 + 22x^3 + 5x^2 + 6x + 1?$$ If you would like to be enlightened by the answers to these truly important questions come to my talk where everything will be illuminated... Or at least some of the background for Coleman and Chabauty's method for finding rational points on curves will be discussed. PS: The number of references to Elijah Wood will be bounded - just like the number of rational points on our curves.
Bertini Theorems in Various Settings (2 Talks) Graduate Algebraic Geometry Seminar, University of Wisconsin - Spring 2015
I will discuss how the classical Bertini Theorems fail over finite fields, and how similar results may be obtained in these settings. This will focus on Poonen's paper, Bertini Theorems Over Finite Fields.
Mean, Median, and Mode - Well Actually Just Median AMS Student Chapter Seminar, University of Wisconsin - 2/25/15
Given a finite set of numbers there are many different ways to measure the center of the set. Three of the more common are mean, median, mode familiar to us from high school. This talk will focus on the concept of the median, and why in many ways it's frigin' sweet. In particular, we will explore how we can extend the notion of a median to higher dimensions, and apply it to create more robust statistics. It will be awesome, and there will be donuts.
Rational Varieties and Rational Points Graduate Number Theory Seminar, University of Wisconsin - 2/17/15
A variety \(X\) over a field \(k\) is rational if it is birationally equivalent to \(\mathbb{P}^n\). The question of whether or not \(X\) is rational depends on the ground field in often subtle ways. In fact, it turns out that in some case this whether or not \(X\) is rational over \(k\) is equivalent to the existence of a smooth \(k\)-rational point. We will explore this connection by considering quadric and cubic hypersurfaces over various fields.
Topics in Resolutions of Singularities Graduate Algebraic Geometry Seminar, University of Wisconsin - 10/14
The study of resolutions of singularities dates back to the 19th century when many very geometric methods of resolving singularities were developed. We will explore some of these by looking at the case of curves. Given time I will also discuss how these methods do and do not generalize to higher dimensional varieties.
Intro to Complex Dynamics Graduate Number Theory Seminar, University of Wisconsin - 10/28/14
Given a polynomial \(f(z)\) with complex coefficients, we can ask for which complex numbers \(p\) is the set \(\{f(p), f(f(p)), f(f(f(p))),...\}\) bounded, that is to ask which complex numbers have bounded forward orbit under \(f(z)\)? Alternatively we can turn the question around and ask for a fixed complex number \(p\), for which (complex) polynomials is the forward orbit of \(p\) bounded? Finite? Periodic? These questions give the interesting fractal pictures many of you have probably seen. Amazingly many of the tools needed to approach these questions, arose well before computers allowed us to generate images like the one above. In this talk we will explore some of the basic tools and results of complex dynamics paying particular attention relations to number theory. The goal being to present some of the background material need for Laura DeMarco's talk later in the week. (Also getting to see a really cool area of mathematics!)
Hex on the Beach AMS Student Chapter Seminar, University of Wisconsin - 10/8/14
The game of Hex is a two player game played on a hexagonal grid attributed in part to John Nash. (This is the game he is playing in A Beautiful Mind.) Despite being relatively easy to pick up, and pretty hard to master, this game has surprising connections to some interesting mathematics. This talk will introduce the game of Hex, and then explore some of these connections. *As it is a lot more fun once you've actually played Hex feel free to join me at 3:00pm on the 9th floor to actually play a few games of Hex!*
Through various programs – especially math circles – I have given numerous outreach presentations.
A list of these can be found here. A list of presentations I gave as a undergraduate can be found
here.