Talk 1:
Title: Characteristic p methods
Speaker: Trevor Arrigoni
Venue: Snow 564
Date and time: 07th September 2023, 12pm
Talk 2:
Title: An Introduction to Differential Operators
Speaker: Christopher Wong
Venue: Snow 564
Date and time: 14th September 2023, 12pm
Talk 3:
Title: The Dedekind-Weber Paper
Speaker: Cheng-Pang Shih
Venue: Snow 564
Date and time: 21st September 2023, 12pm
Abstract: In two of Riemann's papers, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, and Theorie der Abel'schen Functionen, he gave geometric/physical intuitive proofs of what would later be known as the Riemann-Roch theorem, and also Abel's theorem. Theorie der algebraischen Functionen einer Veränderlichen, also known as the Dedekind-Weber paper, aimed to prove these theorems rigorously, using Dedekind's previous development on algebraic number theory as a basic tool. In the process, they clarified several properties in Riemann's papers, and paved way for modern algebraic geometry. This talk aims to give a survey on this paper using John Stillwell's annotated translation to introduce several original concepts developed by Dedekind and Weber and to give historical context on how the usual topics in an algebra course came to be important concepts in algebra and in particular algebraic geometry.
Talk 4:
Title: Matlis Duality
Speaker: Ryan Hunter
Venue: Snow 456
Date and time: 05th October 2023, 12pm
Abstract: In 1958, Eben Matlis proved an anti-equivalence between the categories of Noetherian and Artinian modules over a complete local ring. This result comprised the first half of his thesis, which was written while working under Irving Kaplansky. This result can be used to prove Grothendieck's local duality theorem which establishes a relationship between certain Ext and local cohomology modules.
In this talk I plan to define some of the basic terms needed to understand the statement of Matlis' theorem, then give a proof of two of the three parts of the theorem. I will send the seminar organizers some notes for the talk which explain the concepts of injective modules, essential extensions, and completion of rings. The notes will also contain proofs of each of the lemmas I use, as well as the proof of Matlis' theorem. Attendees unfamiliar with any of the above terms are encouraged to read through the notes before the talk begins.
Talk notes with detailed proofs
Talk 5:
Title: Macaulay2 Workshop
Presenters: Abraham Pascoe, Ritika Nair
Venue: Snow 456
Date and time: 12th October 2023, 12pm
Abstract: This is a beginner's guide to Macaulay2, and it is structured in a follow-along way. So, we strongly recommend you bring your laptop to the session to be able to get some practice!
Talk 6:
Title: Castelnuovo-Mumford Regularity of Subspace Arrangements
Speaker: Sreehari Suresh Babu
Venue: Snow 456
Date and time: 19th October 2023, 12pm
Abstract: We discuss the paper “A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements” by Harm Derksen and Jessica Sidman. We aim to sketch the proof of the following cool result: if X is an arrangement of d linear spaces in P^n, then I(X) is d-regular.
Talk 7 :
Title: Gröbner bases and the Buchberger Algorithm
Speaker: Jon Theodore Tremblay
Venue: Snow 564
Date and time: 26th October 2023, 12pm
Abstract: A large portion of computational and referential aspects in the theory of polynomial rings arise from the use of Gröbner bases. In this talk I aim to define and give sufficient conditions for the existence and uniqueness of Gröbner bases. Additionally I hope to explore Buchberger’s algorithm for computing Gröbner bases, which can be viewed as a multivariable, nonlinear analogue of Euclid’s algorithm. With this in mind we will (see) its applications in solving polynomial systems of equations and it’s used in both theoretical and real world problems.
Talk 8:
Title: Flag varieties
Speaker: Aaron Ortiz
Venue: Snow 456
Date and time: 16th November, 12pm
Abstract: Flag Varieties are of great importance in areas such as algebra, algebraic combinatorics, and geometry. We will introduce complete flag varieties and focus on their algebraic structure and interpretations. We will then introduce partial flag varieties, their rich algebraic structure, and see what happens to our underlying algebra if we play around and change our vector spaces that make up our flags. At the end, we will discuss modern research in the area, open problems, and where to get started if anyone is interested in learning more about the subject.
Talk 9:
Title: Homology and Cohomology in Algebraic Topology
Speaker: Andrew Lin
Venue: Snow 456
Date and time: 30th November, 12pm
Abstract: The ideas of homological algebra find their roots in algebraic topology, where homology and cohomology arose from a desire to study topological objects (like manifolds or simplexes) by studying their “holes.” This will be formalized by the concept of an “n-chain,” which lends itself to the study of homology of chain complexes. However, homology leaves certain things to be desired, leading us to study cohomology. If time permits, we will look at the basic ideas of Poincaré duality, a theorem which relates homology and cohomology groups of certain manifolds.
Talk 10:
Title: Introduction to Linkage
Speaker: Monalisa Dutta
Venue: Snow 456
Date and time: 07th December, 12pm
Abstract: We will introduce the classical theory of linkage of ideals and modules and peek into some basic properties of them.
Talk 11:
Title: Introduction to Local Cohomology
Speaker: Ryan Hunter
Venue: Snow 564
Date and time: 09 February, 11am
Talk 12:
Title: Local Cohomology and Global Cohomology
Speaker: Sreehari Suresh Babu
Venue: Snow 564
Date and time: 16 February, 11am
Talk 13:
Title: The Secrets of a Spectral Sequence
Speaker: Abraham Pascoe
Venue: Snow 456
Date and time: 23 February, 11am
Abstract : A spectral sequence is a family of chain complexes whose differentials induce chain complexes on the homology groups of the family. In this talk, we will explore the world of spectral sequences, understand some basic facts about them, and look at a few examples of spectral sequences that arise from studying local cohomology modules.
Talk 14:
Title: An Introduction to Invariant Theory
Speaker: Christopher Wong
Venue: Snow 564
Date and time: 22 March, 11am
Talk 15:
Title: Toric Varieties and Zariski-Nagata Type Theorems
Speaker: Jordan Barrett
Passcode: 321129
Date and time: 05 April, 11am
Abstract : The Zariski-Nagata theorem is a classical result which expresses the nth symbolic power of a radical ideal I in a polynomial ring over a perfect field in terms of the nth regular powers of the maximal ideals in mSpec(I). In this talk I will state a well-known Zariski-Nagata type theorem for projective varieties, give a brief crash course on toric varieties, and discuss my work on characterizing which toric varieties satisfy a Zariski-Nagata result.