TATERS

The geproci property is a recent development in the world of geometry. We call a set of points Z ⊆ P3(k) an (a, b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011 and nondegenerate non-grids have been known only since 2018. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in my thesis, a procedure was known for creating specific nondegenerate non-grid (a, b)-geproci sets for 4 ≤ a ≤ b, but it was not known what other examples there can be. Furthermore, before the work in my thesis, almost all examples of geproci sets that were known were contained in unions of b disjoint lines (known as half grids) and there was no known way to generate new examples of non-half grids. Here, I will discuss how to use geometry in positive characteristics to find new methods of producing geproci half grids and non-half grids.

A classical way to study singularities of complex algebraic varieties is to look at fundamental group of a small nonsingular open set around the singularity.  In complex algebraic geometry, klt singularities are a centrally important class of singularities and a great deal of effort has been expended recently in studying their associated local fundamental groups.

In characteristic p > 0 commutative algebra, the F-signature measures how close a strongly F-regular ring is from being non-singular and has been used to bound the size of the local étale fundamental group. Here F-regular singularities are a characteristic p > 0 analog of klt singularities. In this talk, using the perfectoidization of Bhatt–Scholze, we will introduce a mixed characteristic analog of F-signature and Hilbert–Kunz multiplicity.

As an application, we show it can be used to provide an explicit upper bound on the size of the étale fundamental group of the regular locus of a BCM-regular singularity (related to results of Xu, Braun, Zhuang, Carvajal–Rojas, Tucker, Bhatt–Gabber–Olsson, and others in characteristic zero and characteristic p). BCM-regular singularities, as introduced by Pérez-R.G. and Ma and myself, can be thought of as a mixed characteristic analog of klt and F-regular singularities from characteristic zero or p > 0 respectively. This is joint work with Hanlin Cai, Seungsu Lee, Linquan Ma and Kevin Tucker.

A hyperplane arrangement is a union of codimension one linear spaces.  These simple objects provide fertile ground for interactions between combinatorics, algebra, algebraic geometry, topology, and group actions.  The combinatorics of an arrangement is encoded by the pattern of intersections among the hyperplanes, called its intersection lattice.  On the other hand, a key algebraic object is the module of vector fields tangent to the arrangement, called the module of logarithmic derivations, which was introduced by Saito in 1980 to study the singularities of hypersurfaces.  An enduring mystery in the theory of hyperplane arrangements is which algebraic properties of the module of logarithmic derivations can be determined from the intersection lattice, and which properties depend fundamentally on the geometry (i.e. the exact hyperplanes).  At the center of this mystery is Terao's conjecture, which proposes that the algebraic property of freeness can be determined only from the intersection lattice.  In this talk we explain how rigidity of planar frameworks (dating back to Maxwell) connects the geometry of a projective line arrangement with extremal syzygies of its module of derivations.  This is joint work with Jessica Sidman and Will Traves.

I’d like to tell a story about the numbers 1, 8/7, and 2 (in that order).  This story is fundamentally about combinatorics and linear algebra, framed in terms of probability, featuring a little bit of algebraic geometry and electrical engineering.  It is a small sliver of the work for which June Huh was awarded a Fields Medal.

Contact topology arose historically in classical mechanics and optics, but has developed into an independent field of mathematics over the last century. Contact structures arise only on odd-dimensional manifolds and can do so in various ways, particularly as the boundary of suitably "convex" manifolds equipped with a complex or symplectic structure. Not every contact manifold arises this way; those that do are called "fillable". On the other hand, every contact structure on a 3-dimensional manifold can be obtained by a process called contact surgery, starting from a suitable knot or link in the 3-sphere. I will describe a geometric-topological characterization of those knots that give rise to fillable contact structures under contact surgery, which was obtained in joint work with Bülent Tosun. As time permits we will explore further consequences and applications.

An asymmetric random variable X is said to be symmetrization resistant if every independent random variable Y that produces a symmetric sum X+Y has a greater variance than that of X. Asymmetric Bernoulli random variables were shown to be symmetrization resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. Proving symmetrization resistance appears to be difficult: little is known about other asymmetric distributions. We introduce the notion of entropic symmetrization resistance which is the same as symmetrization resistance except that the entropy (rather than variance) of Y must exceed that of X. We show that Bernoulli random variables exhibit entropic symmetrization resistance exactly when they exhibit symmetrization resistance. We also extend the underlying entropy and variance inequalities to the hypercube. Finally, we explore the possibility of extensions to non-Bernoulli random variables. This talk is based on joint work with Mokshay Madiman.

In joint work with Tài  Huy Hà and Susan Morey we introduced the notion of initially regular sequences on $R/I$, where $I$ is any homogeneous ideal in a polynomial ring $R$. We will discuss this notion, and show how we can construct certain types of (initially) regular sequences on $R/I$ that give effective bounds on the depth of $R/I$. Moreover, we will discuss when these sequences remain (initially) regular sequences on $R/I^t$ and give lower bounds on ${\rm depth} R/I^t$ for $t\ge 2$.

The pursuit of consilience between general relativity and quantum mechanics has driven research in physics for nearly a century. The approaches generally favor one theory over the other as a starting point, and then work to bridge the conceptual gap through various constructions. One such family of approaches, called topological quantum field theories, start with a quantum field theory and attempt to restrict calculations to those feasible on a dynamic metric (essentially "making room" for general relativity). These theories integrate diverse mathematical disciplines including geometric topology, category theory, and representation theory. In this presentation I give motivation from the physics perspective and provide a an overview of the framework and machinery of a particular topological quantum field theory called Dijkgraaf-Witten Theory. In the end I demonstrate how two different interpretations of this theory yield Mednykh's formula; a well known result which links the fundamental group of a surface to the representations of finite groups. 

Voting arises as a natural result of the desire to make collective decisions. Votes can be cast in multiple ways, most of which can be thought of as a partial ordering on the set of candidates. The votes can also be tallied in multiple ways. Different voting methods can result in different outcomes for the same election, even without voters changing their opinions. Voting theory is a way to study these ballot collection and tallying procedures, including the properties held by such procedures.

In Approval Voting, each voter gives a single point to each candidate in the set of candidates of which they approve, and the candidate with the most points wins the election. Recently, there has been increased interest in Satisfaction Approval Voting and Quadratic Voting, in which each voter evenly spreads out a single point among the candidates of which they approve. Satisfaction Approval Voting, Quadratic Voting, and Approval Voting can then be described by each voter submitting a vector of 0’s and 1’s, and scaling that vector to be a unit vector using the p-norm (these examples correspond to p = 1, 2, ∞ respectively). This suggests a family of voting procedures we are calling p-Norm Approval Voting. In this talk, we will examine some desirable properties in voting procedures and how they do, or don’t, apply in the p-Norm Approval Voting scenario. (Hari Nathan, Michael Orrison, Katharine Shultis, Jessica Sorrells)

Automata are a useful tool for producing tractable groups with surprising properties. Some particularly nice automata are what are called bireversible automata but these turn out to be few and far between. In this talk we will give an introduction to automata groups and discuss some results on producing lamplighter groups with bireversible automata. This is based on my joint work with Benjamin Steinberg.

This talk will explore the utility of data-driven path metrics for the clustering and visualization of high-dimensional data. These metrics are defined by solving an optimal path problem in a proximity graph, and are characterized by a parameter harmonizing density-based and geometric features. First, we will discuss theoretical properties of these metrics and implications for clustering: in particular, spectral clustering with path metrics leads to strong theoretical guarantees on estimating number of clusters and cluster accuracy. Furthermore, as the sample size converges to infinity, the eigenvalues and eigenvectors of the discrete path metric graph Laplacian converge to those of a continuum operator; this operator generates a diffusion which is accelerated in regions of high data density, allowing for the rapid exploration of elongated data structures. Secondly, we will discuss dimension reduction with path metrics. Despite the allure of visually striking results from dimension reduction algorithms, can we be sure the perceived patterns are genuinely intrinsic to the data? We will see how designing algorithms which incorporate data-driven path metrics can lead to desirable and understandable properties for data visualization, and specifically focus on their application to the analysis of single cell RNA sequence data. Finally, we will discuss generalizations of these metrics to simplex paths, and their utility for multi-manifold clustering.

The Curry-Howard-Lambek correspondence and “proofs-as-programs paradigm" establishes a deep connection between programming languages, intuitionistic logic, and category theory. Proof assistants like Lean, which have found increasing use by practicing mathematicians, are based on this connection. Mathematicians, however, reason classically, freely using the law of the excluded middle. Where does classical logic sit in this perspective? A theorem of Griffin in 1990 showed that classical logic is related in the proofs-as-programs paradigm to control flow operators and continuation-passing style in programming. In this talk, I will convey the flavor of the proofs-as-programs paradigm and of continuations and control flow in programming, toward understanding Griffin’s result. I will then look at continuations from a more mathematical angle as continuation monads in cartesian closed categories, noting some mathematically interesting examples. My goal for this talk will be to motivate the idea that continuation monads, originally a notion from computer science, may be relevant in areas not obviously related to logic and computer science.

Aperiodic Wang protosets - sets of square Wang tiles that tile the plane but do not admit any periodic tilings - have been studied since the 1960s, when the first such example consisting of over 20,000 tiles was discovered. In 2015, Jeandel and Rao discovered, through an exhaustive computer search, an aperiodic Wang tile protoset consisting of 11 tiles, and further, they proved that no Wang tile protoset with 10 or fewer tiles can be aperiodic, so their order-11 aperiodic protoset is of minimal order. In 2019, S. Labbé presented a Markov partition for the Jeandel-Rao aperiodic Wang protoset. This construction is remarkable. The ingredients are a geometric partition P of a 2-dimension torus T into polygonal shapes, each of which has a label from {0,1,2,…,11}, along with a simple Z^2 action R on this torus. Here is why this construction is remarkable: If you pick any point p in T and consider its orbit in T under the action R, keeping track of which atom of P the points of the orbit fall, the corresponding two-dimensional “word” corresponds directly to a valid tiling formed from the Jeandel-Rao protoset. This dynamical systems approach to encoding Wang tilings is very novel and allows us to use the tools of dynamical systems in analyzing spaces of tilings. In this talk, we will give a gentle introduction to this topic and discuss some computer experiments that show how other well-known aperiodic Wang tile protosets seem to also have Markov partitions that encode tilings.