BECAUSE OF CONCERNS ABOUT THE SPREAD OF CORONAVIRUS, GIST WILL BE GOING ONLINE FOR THE REST OF THE SEMESTER, EFFECTIVE IMMEDIATELY. MORE INFORMATION WILL BE AVAILABLE SHORTLY.
Title: On flex points and bitangents
Abstract: We will prove the Plücker formulas to compute the number of flex points and bitangents of a smooth plane curve. Then we will explain a second method to compute the number of bitangents on a smooth plane quartic. This will help us to generalize the concept of bitangent to higher degree plane curves.
Title: Conjugacy Classes of Algebraic Groups
Abstract: The study of similar matrices is one of the cornerstones of linear algebra. From eigenvalues to invariant factors and rational canonical form, it's a subject that no graduate student can avoid. In this talk I will discuss how the theory of similar matrices generalizes to structure of conjugacy classes of an algebraic group, discussing topics ranging from the closed semisimple classes to the Kostant section into the regular classes.
Title: Equidistribution , Counting and Quadratic Forms
Abstract: An important concept in Ergodic theory is that of equidistribution. An elegant result of Eskin and McMullen relates the equidistribution of certain sets to the counting of lattice points in affine spaces. In the talk I will introduce the relationship between equidistribution and counting and show how it can be used in the case of integral quadratic forms.
Title: The Probabilistic Method
In this talk, I will describe a combinatorial technique called "the probabilistic method". It is a powerful, non-constructive approach to certify the existence of desired phenomenon. I'll give some examples of how this technique can be used to prove some cool results about graphs. No background in anything is required! All are encouraged to attend and ask questions.
Title: Eat, Iterate, Perron-Frobenius.
Abstract: People are always asking me: “Ethan, what should I know if I wanna be a dynamicist?” There are a lot of different ways I could answer this question, but one thing you’re bound to run into in topological dynamics is the Perron-Frobenius theorem. While purely a statement in linear algebra, the PF theorem has wide-ranging applications to a huge variety of dynamical systems. Clunky to state and bizarre to contemplate, the PF theorem can at first feel like an frustrating foil rather than a god-sent gizmo. In this talk I’ll first give the statement of Perron-Frobenius before digressing to a sequence of interesting applications demonstrating its utility, summarizing my own journey from first learning the theorem to gradually coming to love it. At the end of the talk we give a short proof of the theorem.
Title: Geodesics and Embedded Contact Homology
Abstract: I will discuss an application of embedded contact homology to a problem concerning geodesics in Riemannian surfaces.
Title: The inscribed rectangle problem
Abstract: In the talk, I will prove that given a Jordan curve in R^2, there are 4 points on the curve that together form the vertices of a rectangle. The proof involves some basic topology and a clever parameterization. I might talk briefly about some generalizations afterwards.
Title: Khovanov homology and link detection
Abstract: Khovanov homology is a combinatorially defined link invariant. One of the topological applications of Khovanov homology is to the problem of link detection. In this talk we will motivate studying Khovanov homology and link detection, sketch a proof that Khovanov homology detects the unknot, and if time permits review other known detection results.
Title: Fibered Knots and Contact Structures
Abstract: One of the fundamental ideas in modern contact topology is the Giroux correspondence between open book decompositions and contact structures. Via the Giroux correspondence, we can use contact structures to study fibered knots in 3-manifolds and vice versa. In my talk, I'll describe this correspondence and some recent work regarding fibered knots in S3 and the contact structures they support. The talk will be accessible to all levels of topology background. There will also be lots of pictures!
Title: So you wanna be a complex dynamicist?
Abstract: We introduce the basic concepts of complex dynamics, which is the study of iterated actions of rational maps on the Riemann sphere. We pay particular attention to the idea of the Julia set of a rational map, providing a wealth of pictures and a hands-on example. Time permitting, we introduce the Mandelbrot set and discuss open questions in this field. A familiarity with polynomials is assumed, as is an appreciation for cool images!
Title: Minimal model program for rational surfaces
Abstract: One major goal of algebraic geometry is to classify algebraic varieties up to isomorphism. A more reasonable goal is to instead classify algebraic varieties up to birational equivalence (isomorphism along a dense open set). In this talk, we will introduce the minimal model program, a procedure which takes as input an algebraic variety X and outputs a distinguished member of the birational equivalence class of X (a minimal model of X). We describe the classification of minimal models of smooth rational surfaces, and as a consequence, show that a minimal model need not be unique.
Title: On some collections of geodesics on a hyperbolic surface
Abstract: Geodesics, or locally distance-minimizing paths, are one of the most studied objects in two-dimensional hyperbolic geometry. They reveal, reflect, and in fact determine the geometric structure of an underlying surface. In this talk, I will give a brief overview and sketch proofs, when available, of select notable results about some collections of geodesics on a hyperbolic surface, going up in both cardinality and complexity. Some background in hyperbolic geometry is useful, but not required.
Title: An interesting solution to a graph theory homework problem.
Abstract: When I was an undergrad, I submitted an interesting solution to a homework problem in the graph theory course. I'll talk about it in the talk.
https://www.dpmms.cam.ac.uk/study/II/Graphs/2016-2017/graph20172.pdf
The problem I'm going to talk about is question 13. Come to the talk if you can't think of a way to solve it, especially for the last part "Does this remain true if G is uncountable?"!
Title: What is class field theory?
Abstract: I will be motivating class field theory using only basic field and Galois theory, basic commutative algebra, and elementary analysis. My approach is to generalize Dirichlet’s famous theorem on primes in arithmetic progressions to number fields. This traditional approach avoids the heavy adélic theory and cohomology that tend to obscure the underlying arithmetic. Ultimately, we will be able to state the 3 main theorems of global class field theory and see some consequences of these theorems.