Thursday, December 8, 2022. Exley Science Center 113

Mathematics Colloquium

Joseph Silverman, Brown

Finite orbits of points on surfaces that admit three non-commuting involutions

Abstract: The classical Markoff-Hurwitz equation

 M : x^2 + y^2 + z^2 = axyz + b

 admits three non-commuting involutions coming from the three double covers M --> A^2. There has recently been considerable interest in studying the orbit structure of the (Z/pZ)-points of M under the action of the involutions. In this talk I will discuss some of this history, and then describe analogous results and conjectures on K3 surfaces W in P^1 x P^1 x P^1 given by the vanishing of a (2,2,2) form. Just as with the Markoff-Hurwitz surface, the three projections W --> P^1 x P^1 are double covers that induce three non-commuting involutions on W. Let G be the group of automorphisms of W generated by these involutions. We investigate the G-orbit structure of the points of W. In particular, we study G-orbital components of W(Z/pZ) and finite G-orbits in W(C). This nice blend of number theory, geometry, and dynamics requires no pre-requisites beyond an undergraduate algebra course. (This is joint work with Elena Fuchs, Matthew Litman, and Austin Tran.)

 

Wednesday, November 30, 2022

Topology Seminar

Genevieve Walsh, Tufts

Groups with planar boundaries

Abstract: Relatively hyperbolic groups generalize geometrically finite Kleinian groups acting on real hyperbolic space $H^3$.   The boundaries of relatively hyperbolic groups generalize the limit sets of geometrically finite Kleinian groups. We will define relatively hyperbolic groups and their boundaries, and give more examples. There are many interesting questions about these boundaries.   Since the boundary of $H^3$ is $S^2$,  the limit set of every Kleinian group is planar.  Can every relatively hyperbolic group with planar boundary be realized as a Kleinian group? The answer is no, and we will give illustrative examples to show the many ways this can fail. However, we will show that in the absence of cut points in the boundary, such groups have the property that all their peripheral subgroups are surface groups.  Under additional hypotheses, we outline a proof that relatively hyperbolic groups with planar boundaries can be realized as Kleinian groups. This is joint work with Chris Hruska.

 

Thursday, November 17, 2022

Mathematics Colloquium

Maggie Regan, Duke

Parameterized polynomial systems and their applications

Abstract: Many problems in computer vision and engineering can be formulated using a parameterized system of polynomials which must be solved for given instances of the parameters. Solutions and behavior over the real numbers are those that provide meaningful information for these applications. By using homotopy continuation within numerical algebraic geometry, one can solve these parameterized polynomial systems. First, I propose a new approach which uses locally adaptive methods and sparse matrix calculations to solve parameterized overdetermined systems in projective space. Examples are provided in 2D image reconstruction to compare the new methods with traditional approaches in numerical algebraic geometry. Second, I discuss new homotopy continuation methods for solving two minimal trifocal calibrated relative pose problems defined by point and line correspondences, which appear together, e.g., in urban scenes or observing curves. Experiments are shown using real and synthetic data to demonstrate that challenging scenes can be reconstructed where standard methods fail. Third, we present a new definition of monodromy action over the real numbers which helps describe behavior between real solutions. An application in kinematics is discussed to highlight the computational method and impact on calibration.  Finally, other projects stemming from real algebraic geometry are discussed, as well as future research questions.  

 

Wednesday, November 16, 2022

Topology Seminar

Mark Pengitore, University of Virginia

Residual finiteness of the mapping class group with respect to solvable covers

Abstract: In this talk, we will quantify residual finiteness of the mapping class group of finite type with respect to congruence quotients coming from characteristic finite index covers of the underlying surface with solvable deck group. We refer to these quotients as congruence quotients of solvable type. In particular, we construct an infinite sequence of mapping classes where the minimal congruence quotient of solvable type that detects one of these mapping classes has size which is super polynomial in word length.

 

Monday, November 14, 2022

Mathematics Colloquium

Maddie Weinstein, Stanford

Metric Algebraic Geometry

Abstract: A real algebraic variety is a set of points in real Euclidean space that satisfy a system of polynomial equations.  Metric algebraic geometry is the study of properties of real algebraic varieties that depend on a metric. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, and the reach of an algebraic variety. We also show applications to the study of the geometry of data with nonlinear models.

 

Tuesday, November 11, 2022

Logic Seminar

James Hanson, Univerity of Maryland

How bad could it be? The semilattice of definable sets in continuous logic

Abstract:  After giving an overview of the basic properties of definable sets in continuous logic, we will give a largely visual proof that any finite lattice is the partial order of definable sets in some superstable continuous first-order theory.

 

Thursday, November 10, 2022

Mathematics Colloquium

Iris Yoon, Univeristy of Oxford

How topology reveals structure in data

Abstract: We live in an exciting time where new data is generated at an exponential rate. Such data explosion necessitates the development of novel methods for studying large, noisy, and complex data. One interesting aspect of data is its shape and structure. In this talk, I will discuss recent developments in applied topology that studies the structure of data. In particular, I will show how constructions in topology, such as Dowker complexes and sheaf theory, and network science, such as hypergraphs, reveal interesting structures in data. I will discuss the mathematical challenges of extending these constructions to data science.

 

Wednesday, November 9, 2022

Topology Seminar

Giuseppe Martone, Yale

The Hilbert pressure metric for the space of finite area convex RP^2 surfaces

Abstract: Real convex projective structures on a surface S are a generalization of hyperbolic structures on S, and they come equipped with a natural Hilbert metric. The space of real convex projective structures is a prominent example of a higher rank Teichmuller space.

 In this talk, we construct a mapping class group invariant path-metric on the space of finite area real convex projective structures on S which is naturally associated to the Hilbert metric.  This construction is motivated by McMullen’s dynamical interpretation of the Weil-Petersson metric on the space of hyperbolic surfaces and by Bridgeman, Canary, Labourie, and Sambarino’s pressure metrics. The key dynamical ingredients in our construction come from the Thermodynamic Formalism of countable Markov shifts.

This talk is based on joint work with Harry Bray, Dick Canary and Nyima Kao.

 

Tuesday, November 8, 2022

Computer Science Seminar

David Kahn, Carnegie Mellon University

The quantum physicist's method in automatic amortized resource analysis

Abstract: This talk presents the quantum physicist's method in Automatic Amortized Resource Analysis (AARA).  AARA is a type system that can be used to automatically derive cost bounds for programs. The technique works by using types to locally define potential functions for the physicist's method of amortized analysis, where potential energy is metaphorically stored in data structures.

However, simple program patterns like branching and tree traversals can require non-local reasoning to derive good cost bounds, and this is incompatible with AARA's locally-defined potential functions. The quantum physicist's method is a framework that can construct the needed non-local potential functions out of local ones, upgrading AARA's use of the physicist's method. The resulting system has been implemented and can provide better cost bounds for real code with only moderate overhead.

 

Friday, November 4, 2022

Algebra Seminar

Andrew Obus, Baruch College

Fun with Mac Lane valuations

Abstract:  Mac Lane's technique of "inductive valuations" is over 85 years old, but has only recently been used to attack problems about arithmetic surfaces.  This first half of the talk will be an explicit, hands-on introduction to the theory, requiring no background beyond the definition of a discrete valuation.  We will then discuss how the theory helps us work with models of the projective line, and how the insights gained can be used to resolve certain types of singularities of arithmetic surfaces, as well as to prove conductor-discriminant inequalities for hyperelliptic curves.

 

Wednesday, November 2, 2022

Topology Seminar

Akshat Das, UConn

A three gap theorem for adele rings

Abstract: The classical three gap theorem for rotations on $\R/\Z$ was first proved in the late 1950's and since then it has been reproved numerous times and generalized in many ways. In order to understand problems in dynamics that are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on $\mathbb{R}/\mathbb{Z}$ to the setting of rotations on adelic tori.

In this talk, we begin with an introduction to the setting of an adelic torus. We then state and prove a natural generalization of the Three Gap Theorem for rotations on adelic tori, which is joint work with Alan Haynes. Our proof uses an adaptation of a lattice based approach to gaps problems in Diophantine approximation. We reformulate our problem as a problem about bounding a certain function on a space of lattices. We also give an exhaustive list of examples to prove that the bound we get is best possible.

 

Wednesday, October 26, 2022

Topology Seminar

Andrew Yarmola, Princeton

Canonical knots associated to curves on surfaces

Abstract: Let S be a surface of negative Euler characteristic and C an (oriented) closed curve on S. The set of tangents to C is a knot in the unit tangent bundle UT(S) and, drilling this knot, we obtain a 3-manifold M_C. Any invariant of M_C is automatically a mapping class group invariant of C. In this talk, we will discuss some parallels between these canonical knots in UT(S) and knots in S^3. In particular, we show that M_C uniquely determines C up to mapping classes and we will go over results that explain the behavior the hyperbolic volume of M_C. This behavior will be characterized in terms of topological and geometric properties of C. For example, when C is a filling pair of simple closed curves, we show that the volume is coarsely comparable to Weil-Petersson distance between strata in Teichmuller space. Lastly, we will explain algorithmic methods and tools for building such links and computing other invariants. This work is joint with Tommaso Cremaschi, Jacob Intrater,  and Jose Andres Rodriguez-Migueles.

 

Thursday, October 20, 2022

Mathematics Colloquium

Shelly Harvey, Rice

Knotting and linking in 4-dimensions

Abstract: Knots are circles embedded into Euclidean space.  Links are knots with multiple components.  The classification of links is essential for understanding the fundamental objects in low-dimensional topology:  3- and 4-dimensional manifolds since every 3- and 4-manifold can be represented by a weighted link.  When studying 3-manifolds, one considers isotopy as the relevant equivalence relation whereas when studying 4-manifolds, the relevant condition becomes knot and link concordance.    In some sense, the nicest class of links are the ones called boundary links; like a knot, they bound disjointly embedded orientable surfaces in Euclidean space, called a multi-Seifert surface.  The strategy to understand link concordance, starting with Levine in the 60s, was to first understand link concordance for boundary links and then to try to relate other links to boundary links.  However, this point of view was foiled in the 90’s when Tim Cochran and Kent Orr showed that there were links (with all known obstructions vanishing i.e. Milnor's invariants) that were not concordant to any boundary link.   In this work, Chris Davis, Jung Hwan Park, and I consider weaker equivalence relations on links filtering the notion of concordance, called n-solvable equivalence.  We will show that most links are 0- and 0.5-solvably equivalent but that for larger n, that there are links not n-solvably equivalent to any boundary link (thus cannot be concordant to a boundary link).   This talk will be accessible to a general audience and there will be a lot of pictures!  This is joint work with C. Davis and J.H. Park.

 

Wednesday, October 19, 2022

Topology Seminar

Franco Vargas Pallete, Yale

Peripheral birationality for 3-dimensional convex co-compact $PSL_2\mathbb{C}$ varieties

Abstract: It is a consequence of a well-known result of Ahlfors and Bers that the $PSL_2\mathbb{C}$ character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding $PSL_2\mathbb{C}$ character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.

 

Wednesday, October 12, 2022

Topology Seminar

Emily Stark, Wesleyan

Graphically discrete groups and rigidity

Abstract: Graphically discrete groups impose a discreteness criterion on the automorphism groups of graphs the group acts on. These groups are well suited to studying rigidity problems in geometric group theory. Classic examples of graphically discrete groups include abelian groups and fundamental groups of closed hyperbolic manifolds; nonabelian free groups are non-examples. We will present new families of graphically discrete groups and demonstrate this property is not a quasi-isometry invariant. We will discuss rigidity phenomena for free products of graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

 

Friday, October 7, 2022

Algebra Seminar

Jakub Krasensky, Charles University, Prague

Sums of integral squares in number fields

Abstract:  For any ring R, its Pythagoras number is the smallest number P(R) such that any sum of squares in R can be written as a sum of at most P(R) squares. Lagrange's celebrated four-square theorem can be stated as P(Z)=4. We will look for analogous results in other number fields; the most interesting ones turn out to be totally real fields, i.e. such that the image of all their complex embeddings is in fact a subset of the real numbers.

While the Pythagoras number of a number field is easy to determine, the Pythagoras number of its ring of integers is usually unknown. We will discuss the available results and the basic ideas behind them. In particular, to obtain upper bounds for Pythagoras numbers, we introduce the so-called g-invariants, which are similar to the Pythagoras number, but squares of numbers are replaced by squares of linear forms. The study of g-invariants, sometimes called the quadratic Waring's problem, is far from solved even in the case of rational integers; however, we will see some nontrivial results.

 

Tuesday, October 4, 2022

Logic Seminar

Alex Van Abel, Wesleyan

Asymptotics of the Spencer-Shelah Random Graph Sequence

Abstract:  This talk is based on work in progress. In combinatorics, the Spencer-Shelah random graph sequence is a variation on the independent-edge random graph model. We fix an irrational number $a \in (0,1)$, and we probabilistically generate the n-th Spencer-Shelah graph (with parameter $a$) by taking $n$ vertices, and for every pair of distinct vertices, deciding whether they are connected with a biased coin flip, with success probability $n^{-a}$. On the other hand, in model theory, an $R$-mac is a class of finite structures, where the cardinalities of definable subsets are particularly well-behaved. In this talk, we will introduce the notion of "probabalistic $R$-mac" and present an incomplete proof that the Spencer Shelah random graph sequence is an example of one.

 

 

 Friday, September 30, 2022

Algebra Seminar

John Cullinan, Bard College

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions.

Abstract:  Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$.  Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where $K$ is an $\ell$-power extension of $k$.  We have previously shown that, under mild rationality hypotheses, the case of interest is when $\ell=2$ and $K$ is the unique quadratic extension of $k$.
In this talk we determine the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E'$ over $\mathbf{Q}$ and asking for what proportion of primes $p$ do we have $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and $E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.  We also determine how the discriminants of the endomorphism rings are distributed at these primes.  This is joint work with Nathan Kaplan of UC Irvine.

 

Wednesday, September 28, 2022

Topology Seminar

Yu-Chan Chang, Wes

The RAAG Recognition Problem for Bestvina–Brady Groups.

Abstract: Given a finite simplicial graph, the associate right-angled Artin group (RAAG) is generated by the vertex set of the graph, and two generators commute if they are connected by an edge. The RAAG Recognition Problem asks whether a given group is (isomorphic to) a RAAG. Some algebraic structures of the given group are solutions to this problem. For example, if a group is not finitely presented, then it cannot be a RAAG. Some quasi-isometry invariants of groups also provide solutions, such as Dehn functions. However, this recognition problem is difficult in general, even when the given group has a very simple presentation.

 

Bestvina–Brady groups (BBGs) are subgroups of RAAGs, and they provide important (counter)examples in group theory and topology. In this talk, we will discuss the RAAG Recognition Problem for BBGs. In particular, we will see a complete solution to this problem for the BBGs associated to 2-dimensional flag complexes. This is joint work with Lorenzo Ruffoni.

 

Friday, September 23, 2022

Algebra Seminar

Jeff Yelton, Wesleyan

Clusters, semistable models, and inertia actions for hyperelliptic curves

Abstract:  Given a hyperelliptic curve C : y^2 = f(x) over a global field K, I will relate the “clustering” of its branch points with respect to a prime P of K to the construction of a semistable model of the curve at that prime.  This in turn sheds light on actions of inertia subgroups of the absolute Galois group Gal(K) which are associated to C.  All of this has a very nice description for primes P of residue characteristic not 2, and newer results suggest a somewhat analogous situation when the residue characteristic is 2.

 

Wednesday, September 21, 2022

Topology Seminar

Zvi Rosen, Florida Atlantic University

Neural Codes and Oriented Matroids

Abstract: Fix n convex open subsets of Euclidean space, representing stimuli that trigger specific neurons in the brain. Then, record all possible subsets of [n] whose intersection is nonempty, and not covered by other sets. This combinatorial object is called a "convex neural code." In this talk, we relate the emerging theory of convex neural codes to the established theory of oriented matroids, which generalize systems of signed linear relations. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.

 

Friday, September 16, 2022

Algebra Seminar

Wai Kiu Chan, Wesleyan

On the exceptional sets of integral quadratic forms

Abstract:  A collection S of equivalence classes of positive definite integral quadratic forms in n variables is called an n-exceptional set if there exists a positive definite integral quadratic form which represents all equivalent classes of positive definite integral quadratic forms in n variables except those in S.  In this talk, I will describe a recent joint work with Byeong-Kweon Oh which shows that for any given positive integers m and n, there is always an n-exceptional set of size m and there are only finitely many of them.

 

 

Tuesday, September 13, 2022

Logic Seminar

Scott Mutchnik, UC Berkeley

NSOP_2 Theories

Abstract:  Model theory has been described as a "geography of tame mathematics," creating a map of the universe of first-order theories according to various dividing lines, such as tree properties or order properties. While some regions of this map, such as the stable theories or simple theories, are well-understood to varying degrees, as we progress outward it even becomes open whether some regions are empty or not. Extending the NSOP_n hierarchy of Shelah [1995] defining an ascending chain of strong order properties for n > 2, Džamonja and Shelah [2004] introduce two further tree properties, NSOP_1 and NSOP_2, and ask whether the implications between NSOP_1 and NSOP_2 and between NSOP_2 and NSOP_3 are strict. We have answered the first of these questions, showing that the class NSOP_1 coincides with NSOP_2. We discuss this result and some aspects of its proof, which incorporates ideas from various other regions of the model-theoretic map such as the NSOP_1, NSOP_3 and NTP_2 theories.