GIST will be online this semester.
Title: An Introduction to p-adic Groups and the Local Langlands Correspondence
Abstract: In this talk we will take a brief look at algebraic groups defined over p-adic fields. By defining a natural grading, inherited from the p-adic valuation, we can construct a class of supercuspidal representations that are useful in providing explicit examples of the Local Langlands Correspondence.
Title: Some Link Detection Results
Abstract: Here at BC, our favourite things are ``links" and ``Heegaard Floer homology". I will begin this talk by telling you what links are, and not really telling you what Heegaard Floer homology is. I will then discuss some new results stating that Heegaard Floer homology recognizes various links. This is joint work in progress with Gage.
Title: From intervals to surfaces to 3-manifolds: A dynamical evolution
Abstract: Dynamics is, broadly speaking, the attempt to understand objects by acting on them—or using them to act on something else. This unassuming idea quickly leads to rich theories connecting once-disparate areas of math, and indeed undergirds many recent advances. In this talk we will sketch one such example of this evolution, beginning with two classic interval transformations and using them to illuminate connections between number theory and geometry in 2 and 3 dimensions. Time permitting, we will highlight a few open questions in these areas, and motivate an upcoming talk in the Geometry/Topology/Dynamics research seminar by André de Carvalho.
Title: The symmetric groups and representation stability
Abstract: The symmetric groups are some of the first examples of finite groups and their representation theory is well studied. Often, a family of representations can be found with a family member for each symmetric group. In this talk we will recall the basics of representations of the symmetric group and discuss a property these families of representations may have called "representation stability". We will motivate this property by discussing some connections between representation stability and classical homological stability.
Title: Sheaves, Flag Varieties and Link Invariants
Abstract: I'll discuss a correspondence bridging two seemingly disparate areas of math: stratifications of the Grassmannian, and (Legendrian) link invariants given by collections of sheaves. I'll present everything in very broad strokes and will use lots of pictures (like 20 of them!) so you can mostly just sit back and enjoy the slides without thinking too hard.
Title: Doubly Slice Links
Abstract: A knot is called doubly slice if it is the cross section of an unknotted sphere in S^4. In this talk we talk about various generalizations of this property to links, how to distinguish them, and some constructions.
Title: Isospectral but Non-Isometric Hyperbolic Surfaces via Sunada's Construction
Abstract: Some information about the geometry of a Riemannian manifold can be read off from the spectrum of the Laplacian. One suggestive clue is a theorem of Huber's: the eigenvalues of the Laplacian operator on a closed hyperbolic surface and the lengths of all primitive closed geodesics determine each other. One can then ask if the spectrum can completely specify the geometry? Unfortunately, the answer is no; Sunada (1985) gave a general construction of isospectral but non-isometric hyperbolic surfaces. In this talk, I will define all these terms and go over Sunada's construction using one explicit example throughout.
Title: An Introduction to Heegner Points and the Gross-Zagier Formula
Abstract: A large part of number theory deals with analyzing rational points on elliptic curves by combining algebraic and analytic techniques. In this talk, we will do just that by using Heegner points to produce certain special rational points. We will then analyze these rational points by sketching a statement of the Gross-Zagier formula, which connects central derivatives of certain L-functions with the arithmetic theory of heights.
Title: (Ribbon) Cobordisms between lens spaces
Abstract: Lisca classified when two 3-dimensional lens spaces are rational homology cobordant. In this talk, I will discuss what this means, and provide an overview of his proof. I will then discuss how putting a certain condition on the handle structure of the cobordisms involved affects this classification.
Title: Odd Spoof Perfect Factorizations
Abstract: A perfect number is one that is equal to the sum of its (proper) divisors; 6 = 1 + 2 + 3. The even ones are easy to understand, but the odd ones don't seem to exist and no one can prove it. BUT there are some numbers that sort of "fake" being odd perfect, and we call these spoofs. Here's a Quanta article about the paper I contributed to where we thought a little harder about odd spoofs: link
Title: Inscribing rhombi in a closed loop in $\mathbb{R}^2$
Abstract: Recently I've proved that every Jordan curve (which is a fancy way to say a closed loop in \mathbb{R}^2$) inscribes uncountably many rhombi. I'll talk about it in the talk. The proof is extremely elementary. The proof is all undergrad level point-set topology and epsilon deltas, except for one tiny technical detail where I needed to use Alexander duality and Mayer–Vietoris sequence, which I won't spend more than 2 minutes to go through. So, even if you forget about Alexander duality and Mayer–Vietoris sequence, you'll still be able to understand at least 58 minutes of the talk. Last week we had some spooky perfect numbers that are not actually perfect numbers. This week we'll have some spooky paths that are not actually paths. The main content of the proof is to construct the theory of these spooky paths. Here's my paper: https://arxiv.org/abs/2010.05101
Title: Applications of the classical non-holomorphic Eisenstein series
Abstract: Eisenstein series are important examples of classical automorphic forms. Langlands was led to his functoriality conjectures by studying Eisenstein series and their constant terms. In this talk, we are going to use some of the known properties of the non-holomorphic Eisenstein series to find the volume of the fundamental domain (quotient of the upper half plane by SL(2,Z)).
Title: Machine Learning and Recommender Systems
Abstract: Ever wonder how services such as YouTube and Netflix recommend their next video or movie for you? Here's the time to find out! I'll start by explaining a traditional method called collaborative filtering in recommender systems, and if time permits I'll briefly discuss how deep learning has been enhancing and improving recommender systems in the past few years, following parts and bits from the paper Deep Neural Networks for YouTube Recommendations by Covington et al. (2016).
Title: Mission non-orientable: Constructing derived scheme in 50 minutes
Abstract: This is a non-technical talk on derived algebraic geometry, the idea of which might trace back to as early as the 1960s, while the modern foundation is best known to be laid down by Lurie. We will keep our style informal and non-rigorous, but instead focus on the curious history leading to this area, as well as its recent impact on the geometric Langlands program, p-adic geometry, and homotopy theory. The intended audience include both number theory oriented and topology oriented people. Although the title suggests the definition of derived schemes might occur towards the very end, this might be a little bit too much to ask for.