#### Topology, Algebra, and Geometry Seminar

In Fall 23 the Topology, Algebra, and Geometry Seminar is held 1 - 2 PM on Fridays in RT 1516. To download notes from a given day's lecture, please click on the links below. For any questions, please contact the organizer Federico Galetto.

Date | Speaker | Title | Abstract / NOTES |
---|---|---|---|

Sep 22, 29, & Oct 13 | Greg Lupton | Higher Digital Homotopy Groups | An extensive literature on digital topology includes several treatments of the fundamental group in the digital setting. I will report on recent work that results from an AIM workshop and in which we develop a notion of a second (higher) homotopy group. We calculate this invariant to be $\Mathbb{Z}$ for a digital image that may reasonably be interpreted as a digital $2$-sphere. Although the overall development follows the same steps as the development of higher homotopy groups in topology, the calculation we give for our digital $2$-sphere involves some interesting combinatorial ingredients. I will mention connections with the (graph-theoretic) settings of A-homotopy theory and X-homotopy theory which we learned about at the AIM workshop. I will also speculate about several directions in which we intend continuing our work. (notes) |

#### 2022-23

Date | Speaker | Title | Abstract / NOTES |
---|---|---|---|

Apr 21 & 28 | Jonathan Scott | A Novel Estimate for Rational Lusternik-Schnirelmann Category | In this report on research in progress, we discuss a rational homotopy invariant for spaces that is closely related to rational cone length. This is joint work with Paul-Eugène Parent. |

Mar 24 | Mesut Sahin | Vanishing Ideals and Codes on Toric Varieties |
Motivated by applications to the theory of error-correcting codes, we give an algorithmic method for computing a generating set for the ideal generated by $\beta$-graded polynomials vanishing on a subset of a simplicial complete toric variety $X$ over a finite field $\mathbb{F}_q$, parameterized by rational functions, where $\beta$ is a $d\times r$ matrix whose columns generate a subsemigroup $\mathbb{N}\beta$ of $\mathbb{N}^d$. We also give a method for computing the vanishing ideal of the set of $\mathbb{F}_q$-rational points of $X$. We talk about some of its algebraic invariants related to basic parameters of the corresponding evaluation code. When $\beta=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical semigroup $\mathbb{N}\beta=\langle w_1,\dots,w_r \rangle$, $X$ is a weighted projective space and generators of its vanishing ideal is related to the generators of the defining (toric) ideals of some numerical semigroup rings corresponding to semigroups generated by subsets of $\{w_1,\dots,w_r\}$. Note: This talk will be on Zoom. Please contact the organizers for the link. |

Feb 17 & 24 | David Fernandez Breton | Combinatorics from a logical viewpoint |
Although logic and set theory are commonly associated with discussions about the foundations of mathematics, another no less important rôle that they fulfill is as a framework for rigurously studying the combinatorics of infinite structures. In this talk, I will illustrate the flavour of this kind of mathematics by exhibiting some examples, primarily stemming from the branch of combinatorics known as Ramsey theory, where one can appreciate the subtle variations in behaviour of the objects of study as one analyzes them from different logical viewpoints (countable objects, uncountable objects, what happens without the Axiom of Choice, etc.). Note: These talks will be on Zoom. Please contact the organizers for the link. |

Dec 2 | Gregory Lupton | Two Rank Conjectures in Topology (and Commutative Algebra) | Mark Walker has deduced a case of the toral rank conjecture as a consequence of deep results on certain rank conjectures in algebra. The toral rank conjecture (in simple form) asserts that the total dimension of the rational cohomology algebra of a space $X$ should be at least $2^n$ whenever a torus of rank $n$ acts freely on the space $X$. We investigate the cases of the Toral Rank Conjecture which may be deduced from Walker's reult applies within the setting of rational homotopy. It turns out that, for the spaces involved, we can retrieve the cases entirely within rational homotopy. In fact, for these cases, we can establish a slightly improved lower bound than that given by the TRC and also give a slightly improved lower bound on their cohomology than that given by another well-known conjecture in rational homotopy, namely the Hilali conjecture. (notes) |

Oct 18 | Henry Chimal-Dzul |
Special four-lecture course on finite rings |
Fundamental notions on rings Introduction to coding theory over finite rings Symmetric, frobenius, and quasi-frobenius Classification of finite rings |

Sep 9, 16, 29 | Eduardo Camps | Q-Borel ideals | Recurring topic for Fall 22 semester |

#### 2021-22

Date | Speaker | Title | Abstract / NOTES |
---|---|---|---|

May 6 | Gregory Lupton | Digital Topology: Speculations Towards Applications | notes |

May 6 | Nicholas Iammarino | Calculating jets in Macaulay2 | - |

Feb 18 & 25 | Jonathan Scott | Exact weights, path metrics, and algebraic Wasserstein distances | In Topological Data Analysis, the Wasserstein distance is a generalization of the Bottleneck Distance on the space of persistence diagrams. In this sequence of two talks, we generalize the Wasserstein distance, using a sort of path metric on functors whose indexing category is a measure space. |

Jan 28 & Feb 4 | Ivan Soprunov | Plücker-type inequalities for mixed areas and intersection numbers of tropical curves | Consider $n$ planar tropical curves and count their pairwise intersection numbers. It turns out that starting with $n=4$ these ${n\choose 2}$ numbers satisfy certain quadratic inequalities similar to the well-known Plücker relations for the Grassmannian. In the first part, I will state the result for $n=4$ and explain the relation between this problem and the problem of describing mixed area configuration spaces. In the second part, we will look at some ideas of the proof as well as a graph representation of spaces defined by Plücker-type inequalities. |

Nov 12 & 19 | Tianran Chen | Fireflies, power plants, and convex polytopes | The Kuramoto equation has been used extensively in the study of mysterious phenomena of spontaneous synchronization in biology (such as synchronized flashing of fireflies) and chemistry (such as chemical oscillators). The closely related power-flow equation is used to model the synchronization of power generators as well as the balancing of power generation and distribution, and it is studied intensively in electric engineering. In this talk, we explore how discrete geometry, Newton polytopes, and the theory of Bernshtein-Kushnirenko-Khovanskii bound brings new insights into the study of these two equations. This talk includes my joint work with Robert Davis and Evgeniia Korchevskaia. |

Oct 29 & Nov 5 | Eduardo Camps | Powers of Principal Q-Borel Ideals | Consider $R:=\mathbb{K}[x_1,\ldots,x_n]$ the polynomial ring over a field of characteristic zero and $Q$ a poset over $\{x_1,\ldots,x_n\}$. Given a monomial $m\in S$, we say that the ideal $I=( x_j*m/x_i\ |\ x_i|m,\ x_j\leq_Q x_i)$ is a principal $Q$-Borel ideal. These are a generalization of the concept of Borel ideals and the purpose of this talk is to get as much information as possible of the ideal using just the information to build it, this is the poset $Q$ and the monomial $m$. Specifically, we are going to look the symbolic powers of this ideals and we are going to prove that these coincide with the regular powers. |

Oct 15 & 22 | Delio Jaramillo-Velez | F-thresholds and test ideals of Thom-Sebastiani type polynomials | In this talk I am gonna make some comments about the roots of test ideals and F-thresholds. After this I am gonna focus on the case of the test ideal of a hypersurface. Finally, I'll explain part of a joint work with Luis Núñez-Betancourt and Manuel González Villa about the case of the F-thresholds of a Thom-Sebastiani polynomial. |

Sep 24 & Oct 1 | Yuriko Pitones | Resolution of a linear code | In this talk we will show two methods to associate a free resolution to linear codes. The first, via matroids and the associated Stanley-Reisner ideal, which gives us information about the generalized Hamming weights. The second, using a binomial ideal and Gröbner basis theory. |

#### 2020-21

Date | Speaker | Title | Abstract / NOTES |
---|---|---|---|

Apr 2, 9 & 23 | Ivan Soprunov | On the number of F_q-zeros of families of sparse polynomials | Fix a lattice polytope P and consider the family of polynomials whose monomials have exponents in P and whose coefficients are arbitrary elements of a finite field F_q. Our goal is to estimate the maximum number of F_q-zeros of the polynomials in such a family in terms of geometric/combinatorial invariants of P. When q is large, the maximum is given by polynomials with the largest number of factors. On the combinatorial side, this corresponds to “maximal Minkowski decompositions” inside P. I will first explain this connection and review the previously known case of bivariate polynomials (i.e. dim P=2) as well as the “classical” case of n-variate polynomials of given degree. I will then focus on the case of trivariate polynomials (i.e. dim P=3) where the combinatorics is much richer and new phenomena arise. The talk is based on a joint work with Kyle Meyer and Jenya Soprunova. (notes) |

Mar 5, 12 & 19 | Federico Galetto | Finite group actions on free resolutions | Minimal free resolutions are invariants of finitely generated graded modules over polynomial rings. Under mild conditions, group actions on modules extend to group actions on minimal free resolutions. After reviewing the basics, I will show how to determine characters of finite groups acting on free resolutions. (notes) |

Feb 19 | Graham Keiper | Regularity and h-Polynomial Degree of Toric Ideals of Graphs | I will discuss recent work of Favacchio, Van Tuyl and myself which demonstrates that we may construct toric ideals associated with graphs which have regularity r and h-polynomials of degree d for any 4 ≤ r ≤ d. The ideas involved rely on Groebner bases and combinatorics. I will also discuss how we can use this to recover a result of Hibi, Higashitani, Kimura, and O’Keefe. |

Jan. 29, Feb. 5 & 12 | Steven Gubkin | Sheaves on Sites | In preparation for understanding the definition of condensed sets (ala Scholze and Clausen), we will motivate and explore the properties of sheaves on a site. A sheaf of sets on a topological space is a contravariant functor from its poset of open sets to Sets which satisfies a gluing condition. The category of sheaves on a space is a kind of category called a topos which has extremely nice categorical properties. Generalizing, we then define a "site" which is a category with a notion of "covers" allowing sheaves to be defined on them. Sites generalize the poset of opens of a topological space, and sheaves on a site also form a topos. (notes) |

Oct. 30 & Nov. 20 | Jonathan Scott | Commutative Differential Graded Algebras with Lattices | Let X be a topological space; the image of its integral cohomology in its rational cohomology is called its lattice. We define an invariant, the “commutative differential graded algebra with lattice” of X, which encodes its rational homotopy type as well as its lattice structure. This is joint work with Don Stanley and Xiyuan Wang. (notes) |

Oct. 9 & 16 | Hiram Lopez | An introduction to evaluation codes | We will see how evaluation codes are related to commutative algebra. (notes) |

Sep. 11, 18, & 25 |
Federico Galetto |
Jets of graphs |
Inspired by a construction in algebraic geometry, I will define jets of graphs and discuss a few different properties in the context of combinatorial commutative algebra. (notes) |

#### Spring 2020

Date | Speaker | Title | Abstract / NOTES |
---|---|---|---|

Jan 31, Feb 7, and Feb 14 | Ivan Soprunov | Relations between Mixed Volumes | Recently Esterov and Gusev showed that the problem of classifying generic sparse polynomial systems which are solvable in radicals reduces to the problem of classifying collections of lattice polytopes of mixed volume up to 4. Given the value of mixed volume m and dimension n, there exist only finitely many collections (P_1,…,P_n) of n-dimensional lattice polytopes of mixed volume m, up to unimodular transformations. One reason for this is that the volume of the Minkowski sum P_1+⋯+P_n is bounded above by O(m^{2^n}), as follows by a direct application of the Aleksandrov-Fenchel inequality. I will show how one can employ more relations between mixed volumes to improve the bound to O(m^n), which is asymptotically sharp. We will also see the exact sharp bound in dimensions 2 and 3. This is joint work with Gennadiy Averkov and Christopher Borger. |

#### Fall 2019

Date | Speaker | Title | Abstract |
---|---|---|---|

Sept. 6th |
Greg Lupton (CSU) | A Fundamental Group in Digital Topology |
Notes from the seminar |

Sept. 13th | Greg Lupton (CSU) | A Fundamental Group in Digital Topology |
A \emph{digital image} is a finite subset of the integer lattice in $\mathbb{R}^n$ together with an adjacency relation. For instance, a 2-dimensional digital image is an abstraction of an actual digital image consisting of pixels. Then \emph{digital topology} refers to the many and ongoing efforts to use notions from (algebraic) topology to study and analyze digital images. As part of a broad program to bring notions of homotopy theory into the setting of digital topology, we have defined a fundamental group for an arbitrary digital image. Our construction of this digital fundamental group is intrinsic, in that it is defined in an essentially combinatorial way directly in terms of the digital image; it also differs in key aspects from previously defined versions of the fundamental group in digital topology. We show that our fundamental group of a digital image is isomorphic to the edge group of the clique complex of the digital image considered as a graph. The clique complex is a simplicial complex. As such, its edge group is isomorphic to the ordinary (topological) fundamental group of its geometric realization. These identifications mean that many familiar facts about the ordinary topological fundamental group may be translated into their counterparts for our digital fundamental group. We conclude that the digital fundamental group of any digital circle is $\Z$. We deduce the Seifert-Van Kampen Theorem in the setting of digital topology. We show that the (digital) fundamental group of every 2-dimensional digital image is a free group. These results have the potential to be useful in topology-related image processing operations such as thinning and feature extraction. |

Sept. 20th | Jonathan Scott (CSU) | Algebraic Factorization Systems for Algebras |
Model categories were created by Daniel Quillen to provide a general environment in which to do homotopy theory. The factorizations asked for by the axioms (for example, factoring a morphism into a weak cofibration followed by a fibration) are nowadays assumed to be functorial, or natural. Emily Riehl showed that, if the fibration half of the factorization had a certain algebraic structure (and the cofibrations a coalgebraic structure), then the solutions to the lifting problems demanded by the model structure can also be made functorial. Furthermore, she showed that any cofibrantly generated model category (such as the category of chain algebras) has such an algebraic system. We provide an explicit construction of such an algebraic system for chain algebras, using relatively classical techniques. This is joint work with P.-E. Parent (U. Ottawa) and K. Hess (EPFL). |

Sept. 27th |
Jonathan Scott (CSU) |
Algebraic Factorization Systems for Algebras | Cont'd |

Oct. 4th | Federico Galetto | Betti numbers in commutative algebra | Lecture Notes |

Oct. 18th |
Federico Galetto |
Linear Quotients | Lecture Notes |

Nov. 15th |
Federico Galetto |
Star Configurations | Lecture Notes |

#### Spring 2019

Date | Speaker | Title | Abstract |
---|---|---|---|

March 29 |
Viji Thomas (CSU) |
Kaplansky's Conjecture |
We will introduce Kaplansky's conjecture and talk about what all has been proved so far. |

#### Fall 2018

Date | Speaker | Title | Abstract |
---|---|---|---|

October 12 | Greg Lupton (CSU) | Digital Homotopy Theory |
I will describe work in progress, joint with John Oprea and Nick Scoville. An n-dimensional digital image is a finite subset of the integer lattice in R^n, together with an adjacency relation. For instance, a 2-dimensional digital image is an abstraction of an actual digital image consisting of pixels. Our work consists of developing notions and techniques from homotopy theory in the setting of digital images. In an extensive literature, a number of authors have introduced concepts from topology into the study of digital images. But some of these notions, as they appear in the literature, do not seem satisfactory from a homotopy point of view. Indeed, some of the constructs most useful in homotopy theory, such as cofibrations and path spaces, are absent from the literature. Working in the digital setting, we develop some basic ideas of homotopy theory, including cofibrations and path fibrations, in a way that seems more suited to homotopy theory. We illustrate how our approach may be used, for example, to study Lusternik-Schnirelmann category and topological complexity in a digital setting. One future goal is to develop a characterization of a "homotopy circle" (in the digital setting) using the notion of topological complexity. This is with a view towards recognizing circles, and perhaps other features, using these ideas. The talk(s) will include a survey of the basics on topological notions in the setting of digital images. |

October 19 | Greg Lupton (CSU) | Digital Homotopy Theory | Continued |

October 26 | Greg Lupton (CSU) | Digital Homotopy Theory | Continued |

November 9 | Viji Thomas (CSU) | A Group theoretical construction | We will introduce a group theoretical construction and show its relation to the Schur Multiplier (Second Homology group with coefficients in the integers), second stable homotopy group of the Eilenberg Maclane space and the Bogomolov multiplier. If time permits we will show its relation to a conjecture of Schur and also its relation to Noethers rationality problem. |