Trivial notion

Fall 2023

Schedule

Abstracts

• • Sep 122023
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  Borromean Rings, Chainmaille, and Genuine Equivariant Homotopy Theory

  Natalie Stewart

  The topological space of chain maille weaves classifies geometric embeddings of circles into 3-space. This space of weaves, together with the evident action of the isometry group of R^3, is one of the primary morivating examples in the study of genuine equivariant homotopy theory. After developing some of the basics of this field, we prove a genuine equivariant refinement of the classical fact that the borromean rings can't be made out of geometric circles. If enough time remains, we'll introduce a geometricaly interesting operad, called toroidal weaves, over which the weaves form a (genuine equivariant) right-module, conditioned on a geometric conjecture. We may discuss a surprisingly simple presentation of the operad and right-module structure, conditioned on the same conjecture.
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• • Sep 192023
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  Representations of the Finite Group $SL_2(F_q)$

  Wyatt

  The complex representations of the finite group $SL_2(F_q)$ were first classified by Jordan and Schur in 1907. Over the next several decades, modern ideas from representation theory and algebraic geometry further elucidated the structure of these representations, culminating in a construction in a 1974 paper of Drinfeld. In this talk we will try to construct at least half of the representations of $SL_2(F_q)$. Along the way we'll encounter important ideas and techniques such as Chevalley decomposition, local class field theory, the Bruhat decomposition, Mackey's induction-restriction formula, Whittaker models, and the Selberg trace formula.
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• • Sep 262023
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  Can you (flat-)fold it?

  Kush

  Given a square paper whose side is one unit long, can you fold the paper so that, upon projecting the folded state from a certain angle, the perimeter of the projection is larger than 4? If so, how? (Hint: Yes, yes you can). The same mathematical ideas that we will use to find a solution to the above question are the same ideas used to design complex origami models! In this talk, we will discuss the mathematics of crease patterns and paper-folding.
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• • Oct 3rd2023
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  Equipartitions of (Z/NZ)^x and cyclotomic polynomials

  Daishi

  In this talk we look at what happens when the set $S=\{0\le m\le n\mid (m,n)=1\}$ is subdivided into equal subintervals. We will prove that the parity of the size of these subsets is captured by the values of cyclotomic polynomials at roots of unity. We use this fact as a guiding principle to find nice properties about the geometry of the aforementioned set.
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• • Oct 102023
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  ALgebraic set theory

  Grant

  I will do some set theory with sets of fuzzy elements. This will give short proofs of the Jordan curve theorem and the Grothendieck generic flatness lemma.
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• • Oct 172023
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  The Hegelian Taco

  Taeuk

  We will give a brief introduction to the Hegelian dialectic and its influence on philosophical and political thought, with a focus on the central notion of sublation, or Aufhebung. Then we will discuss Lawvere’s formalization of Aufhebung using category theory.
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• • Oct 242023
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  Exceptional Isomorphisms

  Sanath

  There are a bunch of neat coincidences in low dimensions. I'll talk about that, and try to also say something about the historical context for some of them.
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• • Oct 302023
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  The Twisor Line

  Dylan

  Over $\C$, the $n$th singular cohomology group of a variety has a canonical mixed Hodge structure. If we restrict to smooth projective varieties, we get a pure Hodge structure. It was observed by Simpson that the category of pure Hodge structures can be formulated geometrically as a category of vector bundles on a curve, and so can the association of a Hodge structure to variety. I'll talk about how this is done using the twistor line.
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• • Nov 72023
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  Period three implies chaos

  Sina

  Suppose that f is a self-map on a closed real interval and that f has a point with orbit 3. Then, it turns out that f must have periodic points of every order! Moreover, there is an uncountable subset of the interval where the map behaves “chaotically”. I will talk more about what chaos means in this setting and prove the above facts i.e. the famous “Period three implies chaos” theorem.
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