WVU Algebra Seminar Talks

Past Seminar/Colloquium Talks

▪ Kazuma Shimomoto, Nihon University, Japan

• Monday, February 18, 2019, ARM 315, 5:00pm-6:00pm
• Title: What is perfectoid commutative ring theory?
• Abstract: In 2011, Peter Scholze brought brilliant innovation to the world of arithmetic geometry over p-adic fields. While the basic theory of perfectoid spaces was developed from the scratch in his Ph.D. thesis, he succeeded in proving the ``Deligne's weight-monodromy conjecture" in many non-trivial cases. However, the main tools used in perfectoids are certain big commutative rings. In this talk, I will start with some historical remarks on the birth of perfectoid spaces and present some basic andimportant examples of perfectoid rings to get familiar with their ideas. If time permits, I will also talk about some recent results in perfectoid ring theory.
• Click here for the flyer of the talk.

▪ Michelle Homp, University of Nebraska - Lincoln

• Friday, November 30, 2018, ARM 315, 3:30pm-4:30pm
• Title: Master of Arts for Teachers: A Mathematics Degree Designed with Teachers in Mind
• Abstract: It is widely recognized that effective mathematics teaching requires highly specialized content knowledge. Whether preparing future teachers or providing professional development for veteran teachers, the CBMS publication, The Mathematical Education of Teachers II, recommends that content in advance mathematics courses ``should be tailored to the work of teaching, examining connections between middle grades and high school mathematics as well as those between high school and college."
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▪ David Jorgensen, University of Texas at Arlington

• Thursday, November 29, 2018, ARM 315, 4pm-5pm
• Title: Beyond matrix factorizations
• Abstract: Matrix factorizations of polynomial functions were introduced and studied by David Eisenbud in 1980. In the 1990's, Maxim Kontsevich proposed that matrix factorizations provide a good mathematical model of certain objects in string theory. We give a basic overview of matrix factorizations, discuss their major applications, and present some recent further directions.
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▪ Jugal Verma - IIT Bombay, India

• Thursday, September 20, 2018, ARM 315, 4pm-5pm
• Title: Milnor numbers of hypersurface singularities,mixed multiplicities of ideals and volumes of polytopes
• Abstract: If $H$ is an analytic surface defined by $f=0$ in $\mathbb C^{n+1}$ with an isolated singularity at the origin,then the colength of the Jacobian ideal $J(f)$ is called its Milnor number. B. Teissier refined this notion by a sequence of $\mu^*(H)$ Milnor numbers of intersections of $H$ with general linear spaces of dimension $i$ for $i=0,1,\dots, (n+1).$ J. J. Risler and Teissier showed that this sequence coincides with the mixed multiplicities of the maximal ideal and $J(f).$ They proposed conjectures about log-convexity of the $\mu^*(H)$ which were solved by B. Teissier, D. Rees-R. Y. Sharp and D. Katz. These give rise to Minkowski inequality and equality for the Hilbert-Samuel multiplicities of ideals. Mixed multiplicities are also connected with volumes of polytopes and hence to counting solutions to polynomial equations.
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▪ Yongwei, Yao - Georgia State University

• Friday, August 17, 2018, ARM 315, 4pm-5pm
• Title: Lech's inequality, the Stuckrad-Vogel conjecture, and uniform behavior of Koszul homology
• Abstract: Let $(R, \mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $\displaystyle{\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}}$ is bounded below by ${1}/{d!e(\overline{R})}$, where $\overline{R}=R/\operatorname{Ann}(M)$. Moreover, when $\widehat{M}$ is equidimensional, this set is bounded above by a finite constant depending only on $M$. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of St\"{u}ckrad-Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules. This is joint work with Patricia Klein, Linquan Ma, Pham Hung Quy, and Ilya Smirnov.
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▪ Tokuji Araya - Okayama University of Science, Japan

• Thursday, August 16, 2018, ARM 315, 4pm-5pm
• Title: An introduction to Path Algebras
• Abstract: Let $k$ be a field and let $Q$ be a quiver (i.e., $Q$ is a directed graph where loops and multiple arrows between two vertices are allowed, namely a multidigraph). The path algebra $kQ$ of $Q$ is a $k$-algebra whose basis is the set of paths in $Q$. Since any finite dimensional $k$-algebra is Morita equivalent to a factor algebra of some path algebra, it is very significant to investigate path algebras. In this talk, we will recall the definition and some properties of path algebras. This talk will be accessible to graduate students.
• Click here for the flyer of the talk.

▪ Robin Baidya - Georgia State University

• Thursday, April 14, 2018, ARM 315, 4pm-5pm
• Title: Six ways to compare modules
• Abstract: How many copies of one object exist in another? How many copies of one object are required to build another? If we can build an object using a particular method, to what extent is that method unique? These questions pervade mathematics, and they must be interpreted differently in different contexts. We will approach these questions as they pertain to modules. To answer the first question, we will study how modules embed and factor. For the second question, we will consider the various ways through which a module can be generated and cogenerated. We will investigate the question of uniqueness as it applies to the splitting of modules and to the problem of cancellation. By the end of the talk, we will have covered two ways to interpret each of our three questions, yielding six ways to compare modules. This talk will be accessible to graduate students.
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▪ Naoki Taniguchi - Waseda University, Japan

• Tuesday, March 6, 2018, ARM 315, 4pm-5pm
• Title: Almost Gorenstein rings
• Abstract: My talk is based on joint works with Shiro Goto and Ryo Takahashi. For the last fifty years, Commutative Ring Theory has been mainly concentrated in the study of Cohen-Macaulay rings and modules. While tracking the development, we often encounter non-Gorenstein Cohen-Macaulay rings in the field of not only Commutative Algebra, but also Algebraic Geometry, Representation Theory, Invariant Theory, and Combinatorics. On all such occasions, we have a natural query of why there are so many Cohen-Macaulay rings which are not Gorenstein. Gorenstein rings are defined by locally finite self-injective dimension, enjoying beautiful symmetry. However there is a substantial estrangement between two conditions of finiteness and infiniteness of self-injective dimension, and researches for the fifty years also show that Gorenstein rings turn over some part of their roles to canonical modules. It seems, nevertheless, still reasonable to ask for a new class of non-Gorenstein Cohen-Macaulay rings that could be called \textit{almost Gorenstein} and are good next to Gorenstein rings.
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▪ Henrik Holm - University of Copenhagen, Denmark

• Friday, March 9, 2018, ARM 315, 4pm-5pm
• Title: Prime Ideals in commutative and non-commutative rings
• Abstract: Prime ideals are important in commutative algebra (e.g. localization and Krull dimension), in algebraic geometry (e.g. affine schemes), and in number theory (e.g factorization in Dedekind domains). This talk -- which is about prime ideals, their generalizations, and their uses -- has three parts: In the first part, I will talk about certain aspects of prime ideals in commutative rings. In the second part, I will explain elements of Kanda's recently developed theory of (so-called) atoms. The notion of atoms is a useful and interesting generalization of prime ideals to non-commutative rings (and to abelian categories). In the third part, I will explain work in progress, joint with R. H. Bak, on how to actually compute/determine the atoms for certain types of non-commutative rings.
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▪ Thomas Polstra - University of Utah

• Tuesday, November 7, 2018, ARM 315, 4pm-5pm
• Title: When are two rings similar?
• Abstract: Inspired by results of Samuel, Hironaka, Cutkosky and Srinivasan, and others which give criteria for rings which are quotients of a common power series ring to be isomorphic, it is natural to investigate when two rings, which are the quotient of a common power series ring, are ``similar". In this talk we will discuss recent collaboration between the speaker and Ilya Smirnov which investigates this problem in rings of prime characteristic.
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▪ Jai Laxmi - IIT Bombay, India

• Tuesday, November 14, 2017, ARM 313, 4pm-5pm
• Wednesday, November 15, 2017, ARM 315, 4pm-5pm
• Thursday, November 16, 2017, ARM 313, 4pm - 5pm
• Title: Tate resolutions and deviations of graded algebras
• Abstract: Click here for the flyer of the talk.

▪ Sylvia Wiegand - University of Nebraska - Lincoln

• Wednesday, October 25, 2017, ARM 315, 4pm-5pm
• Title: Prime ideals in rings of power series
• Abstract: We discuss partially ordered sets that arise as the set of prime ideals of low-dimensional commutative Noetherian integral domains consisting of polynomials and power series.
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▪ Roger Wiegand - University of Nebraska - Lincoln

• Tuesday, October 24, 2017, ARM 315, 4pm-5pm
• Title: Rigid ideals in complete intersection domains
• Abstract: In 1994 Craig Huneke and I made the following conjecture, for a finitely generated module $M$ over a one-dimensional local domain $R$ (for example, $R$ might be the local ring of a singular point on an algebraic curve, or a localized ring of algebraic integers): If both $M$ and the tensor product of $M$ with its dual $M^{\ast} = \Hom(M,R)$ are torsion-free, then $M$ must be free. If the ring $R$ is Gorenstein (sort of a symmetry condition), the conjecture is equivalent to the following assertion: If $M$ is rigid (that is, every short exact sequence $0\to M \to X \to M \to 0$ splits), then $M$ is free. Over the past five years many authors have settled special cases of these conjectures, but they are still open. In this talk I will describe recent progress on the conjecture by Craig Huneke, Srikanth Iyengar and me, for the case when $M$ is an ideal of the ring.
• Click here for the flyer of the talk.

▪ Hiroki Matsui - Nagoya University

• Wednesday, April 12, 2017, ARM 313, 4pm-5pm
• Title: Relations among $n$-syzygy modules, $n$-torsionfree modules, and modules satisfying $(S_n)$
• Abstract: In this talk, we will consider three classes of modules over a commutative Noetherian ring; $n$-syzygy modules, $n$-torsionfree modules, and modules satisfying Serre's condition $(S_n)$. There are some relations among these classes of modules. For example, $n$-torsionfree modules are $n$-syzygy modules, and if $R$ satisfies ($S_n)$, then $n$-syzygy modules satisfy $(S_n)$. Therefore a natural question arises: when are two of these classes equal? This question has already been considered by several authors, such as Auslander-Bridger, Evans-Griffith, Araya-Iima, and Goto-Takahashi. The aim of this talk is to connect these results by using certain local Gorenstein conditions. This talk is based on a joint work with Ryo Takahashi and Yoshinao Tsuchiya.
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▪ Hiroki Matsui - Nagoya University

• Wednesday, April 5, 2017, ARM 313, 4pm-5pm
• Title: Thick subcategories of modules and characterizations of local rings
• Abstract: Let $(R, \textbf{m}, k)$ be a commutative Noetherian local ring. We say that a non-empty full subcategory of $\textbf{mod} R$ (the category of all finitely generated $R$-modules) is a \emph{thick subcategory} if it is closed under taking direct summands, and if it satisfies 2-out-of-3 property with respect to short exact sequences. For example, the full subcategory \textbf{FPD} of $\textbf{mod} R$ consisting of modules that have finite projective dimension is a thick subcategory of $\textbf{mod} R$. It is a classical result that $R$ is regular if and only if \textbf{FPD} contains the residue field of $R$. This result has been generalized to various kinds of classes of local rings. %(e.g., Gorenstein local ring and Gorenstein dimension.) Motivated by this, we will recall the necessary definitions and discuss when a given thick subcategory of $\textbf{mod} R$ contains the the residue field of $R$. The talk is based on a joint work with Hayato Murata.
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▪ Ananthnarayan Hariharan - University of Connecticut

• Tuesday, March 28, 2017, ARM 315, 4pm-5pm
• Title: Idealizations and connected sums
• Abstract: We will begin with an introduction to Gorenstein rings using partial derivatives. Two special constructions are idealizations and connected sums. The goal of this talk is to understand the connection between them. This talk will have many illustrative examples, and should be accessible to graduate students.
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▪ Jerzy Weyman - University of Connecticut

• Thursday, March 16, 2017, ARM 315, 4pm-5pm
• Title: From quiver representations to cluster algebras
• Abstract: In this talk I will give a survey of quiver representations, i.e., representations of oriented graphs, and the connections of this theory to different branches of mathematics. In particular I will discuss the applications to representations of general linear groups and the connections to cluster algebras that emerges in last decade. No previous knowledge of quivers or cluster algebras is required.
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▪ Tokuji Araya - Okayama University of Science, Japan

• Wednesday, February 8, 2017, ARM 315, 4pm-5pm
• Title: An introduction to the representation theory of commutative rings
• Abstract: The Auslander-Reiten quiver is one of the crucial tools to investigate the category of finitely generated modules over noetherian rings. It is an oriented graph whose vertices are indicated by indecomposable modules, and whose arrows are indicated by irreducible homomorphisms. In this talk, we will recall the notion of Auslander-Reiten quiver and calculate some examples. This talk will be accessible to graduate students.
• Click here for the flyer of the talk.

▪ Sean Sather-Wagstaff - Okayama University of Science, Japan

• Wednesday, February 8, 2017, ARM 315, 4pm-5pm
• Title: Finiteness conditions in algebra
• Abstract: Mathematical objects that satisfy finiteness conditions are frequently easier to work with than arbitrary objects. For instance, finite dimensional vector spaces are nicer in many ways than infinite dimensional ones. In this talk, I will discuss a variety of classical finiteness conditions in algebra and other areas: what they are and why they are useful. I will conclude with some new results answering a modified version of a question of Huneke about finiteness of associated primes in local cohomology. This talk will be accessible to graduate students.
• Click here for the flyer of the talk.