\u00a0<\/strong>Rees Algebras and Rational Plane Curves<\/h2>\n\n\n\nabstract: Consider the polynomial ring\u00a0R=k[x,y]\u00a0in two variables over a field\u00a0k\u00a0and I\u00a0an ideal presented by a class of 3\u00a0x 2 matrices (Hilbert-Burch) with column degrees\u00a0d_1 \\leq d_2. In this talk we will say a few words on how one can translate information about the defining equations of the Rees Algebra of I\u00a0and Rational Plane Curves of degree\u00a0d_1+d_2=d. In particular, how one can translate between the configuration\u00a0of the singularities of a Rational Plane Curve\u00a0C\u00a0and the bi-degrees of the defining equations of the Rees Algebra\u00a0R(I).<\/p>\n\n\n\n
\n\n\n\n04\/09\/2024 – Jason Mitrovich: The Plateau Problem and Brouwer\u2019s Fixed Point Theorem<\/h2>\n\n\n\n
abstract: This seminar will introduce the Plateau problem, which is physically motivated by soap films and is named after the physicist Joseph Plateau. First, we will state the general Plateau problem before narrowing down to two simpler cases. Then we will sketch out how standard analysis techniques, as well as fixed points, can be used to solve the simpler cases. Afterwards, we will return to the general setting; however, to make progress in this setting, geometric measure theory (GMT) was developed to solve this problem and ultimately was used to resolve the Plateau problem. To close, we will discuss some results on the regularity of such surfaces as well as the higher mapping problem.<\/p>\n\n\n\n
\n\n\n\n04\/16\/2024 – Khoi Nguyen: Urbano Theorem and Minimal Surfaces in S^3<\/h2>\n\n\n\n
abstract: Studying minimal surfaces in different Riemannian manifolds is a recurring theme in the field of geometric analysis. In this talk, I will state a result characterizing closed, orientable minimal surfaces of the 3-sphere. I will start by reviewing the basic theory of minimal surfaces and explaining the relevant terms in the understanding of Urbano theorem. After that, I will state and give a proof of the theorem, which highlights the role of the Clifford torus in studying such minimal surfaces.<\/p>\n\n\n\n
\n\n\n\n04\/23\/2024 – Postdoc Panel <\/h2>\n\n\n\n
abstract: \u00a0This session will focus on how to apply for research postdocs in math. First, we’ll quickly go over the requirements for postdoc applications along with the typical timeline of what happens when, and then we’ll open up a Q&A with current ND students who are about to start research postdocs next year. Students from all years are welcome but it will be particularly relevant to students about to enter the final year of the Ph.D.<\/p>\n\n\n\n
\n\n\n\n04\/30\/2024 – Lorenzo Riva: The Why, What, and How of higher categories<\/h2>\n\n\n\n
abstract: One fundamental technique in mathematics is the process that we all know as “abstraction” but that in some cases goes by the name “categorification”: we take some theorem\/calculation\/construction X and we realize it as the shadow of a more complicated thing Y in a way that makes the concept X more natural, or its construction more elucidating, or its applications more useful in other areas of math. In this talk we will use the homotopy theory of topological spaces and categorify it to obtain a flavor of categorical objects called higher categories, an extension of ordinary categories. This will be an expository talk and no previous knowledge of homotopy theory (besides some basic topology and algebra) will be assumed.<\/p>\n\n\n\n
\n\n\n\n05\/03\/2024 – Jiayi Shen: Liftableness of torsion elements in GL_n(Z)<\/h2>\n\n\n\n
abstract: Consider the representation $\\text{Out}(F_n)\\rightarrow\\text{GL}_n(\\mathbb{Z})$. We would like to answer the question whether all torsion elements in $\\text{GL}_n(\\mathbb{Z})$ can be lifted to torsion elements in $\\text{Out}(F_n)$. We use a geometric approach towards torsion elements in $\\text{Out}(F_n)$ and use algebraic number theory to study torsion elements in $\\text{GL}_n(\\mathbb{Z})$. For a matrix with prime order $p$ where $p\\leq 19$, it is always liftable. For $n\\geq 22$, there always exists an order $23$ matrix that can not be lifted to $\\text{Out}(F_n)$.<\/p>\n","protected":false},"excerpt":{"rendered":"
01\/30\/2024 – Audriana Houtz: Connectivity of the Cut-System Complex abstract: We will start with an introduction to cut-systems on a handlebody. Then prove that the cut-system complex where cut-systems are related by simple moves is connected and if time allows we will prove that it is simply connected. 02\/06\/2024 – Connor Malin: Transforms in calculus […]<\/p>\n","protected":false},"author":4504,"featured_media":0,"parent":55,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-652","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/652","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/users\/4504"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/comments?post=652"}],"version-history":[{"count":6,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/652\/revisions"}],"predecessor-version":[{"id":677,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/652\/revisions\/677"}],"up":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/55"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/media?parent=652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}