Understanding the Boundary of the Cubic Main Hyperbolic Component, by: Ricky Simanjuntak, Indiana University Bloomington.
Abstract: For degree-two polynomials, the main hyperbolic component corresponds to the main cardioid in the Mandelbrot set. In the case of cubic polynomials, this set has two complex dimensions and is homeomorphic to an open 4-ball. Conjecturally, hyperbolic components are the stable parameters among rational maps. Understanding the boundary of this set is central to studying the bifurcation locus and hyperbolic components in general.
I will explain how I use the Blaschke product model to describe the boundary of the main hyperbolic component. The insights gained reveal a decomposition of the boundary set into:
“The tame part”: Homeomorphic to an open solid torus.
“The wild part”: Non-orientable fractals, fibered over a Möbius band, conjectured to be not locally connected.
Furthermore, I will identify several regions in the wild part and show evidence of parabolic implosion in parameter space. If you are seeking a thesis problem in Complex Dynamics that is analytical in nature, this talk will be highly beneficial.
Poster: can be accessed from here.
A transfer principle for abelian varieties, by: Nathan Lowry, Indiana University Bloomington.
Abstract: The Lefschetz Principle states, roughly, that any statement about algebraic varieties over an algebraically closed field of characteristic 0 can be verified by assuming the field to be the field of complex numbers. In this talk, we use basic model theory to formalize and prove an extension of the Lefschetz Principle to other large classes of fields (e.g. real closed fields, p-adically closed fields) by restricting our attention to abelian varieties and their group of rational points.
Poster: can be accessed from here.
Some Aspects of Symblic Dynamics, by: Diyath Pannipitiya, IUPUI.
Slides for this talk can be accessed from here.
Reductions in Matrix Painleve Equations, by: Andrei Grigorev, IUPUI.
Abstract: Painlev´e equations appear in different areas, in particular, they appear in the context of isomonodromic deformations, related mathematical physics, or in the context of birational geometry. Here we will try to explain a construction relating certain matrix generalizations of Painlev´e equations with matrix Painlev´e equations of different sizes. We will follow the example of matrix Painlev´e II. The talk is based on work joint with M. Bershtein and A. Shchechkin.
A Taste of Lagrangian Intersection Theory, by: Karim Boustany, University of Notre Dame.
Abstract: The study of symplectic manifolds is inherently global, on account of the classical fact that they possess no local invariants. The study of their distinguished submanifolds on the other hand turns out to be a much more fruitful venture. A particular class of submanifolds, the so-called Lagrangian submanifolds, is of central importance. Indeed, many questions about symplectic geometry can be recast as questions about Lagrangian submanifolds. One of these questions concerns the intersections of Lagrangian submanifolds, and this has given birth to many powerful ideas and techniques in the field. In this talk, we will discuss some facets of the Lagrangian intersection question, and touch on some recent progress towards Lagrangian packing problems in four dimensions. No prior knowledge other than familiarity with the basic machinery of smooth manifolds is assumed.
On Riemann-Hilbert Problems, by: Kenta Miyahara, IUPUI.
Abstract: In this talk, I will first discuss general aspects of Riemann-Hilbert Problems and then the inverse monodromy problem of Painleve II as an example of RHPs.
Reference: Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painleve transcendents, Mathematical Surveys and Monographs, vol. 128, American Mathematical Society, Providence, RI, 2006, The Riemann-Hilbert approach. MR 2264522.
What’s anisotropic Sobolev space and how to use it!, by: Ekaterina Shchetka, University of Michigan, Ann Arbor.
Abstract: In this talk, we will give a very friendly introduction to anisotropic Sobolev spaces. We will give all definitions as well as intuition behind. We’re going to focus on concrete examples and discuss why anisotropic Sobolov spaces might be useful for spectral theory and dynamics. No prerequisites are assumed.
An Invitation to Topological K-Theory, by: Virgil Chan, IUPUI.
Abstract : Topological K-theory is the study of Abelian groups generated by vector bundles. It immediately produced a lot of striking results in mathematics after being introduced in 1959, such as the Atiyah-Singer Index Formula (Analysis), the proof of Frobenius Conjecture (Algebra), and the computation of the maximal number of linearly independent vector fields on spheres (Topology). In these lectures, I will give the definitions in Topological K-theory, discuss the fundamental results, and provide some computational results.
Introduction to Differential Geometry Through General Relativity, by: Patricia Marcal, IUPUI.
Abstract : The goal is to study Einstein’s field equation and the Schwarzschild solution from a mathematician point of view. The purpose is to use our goal as an excuse to understand basics of differential geometry. No background in physics is necessary (as I do not have it) and little knowledge in math is expected (mainly linear algebra and multivariable calculus).
Padé Approximants, by: Ahmad Barhoumi, IUPUI.
Abstract :In these talks, I will define Padé approximants and discuss some of their basic properties and applications. In the final talk, I will exhibit how one can arrive at an asymptotic formula for the error of approximation in a specific setting.
References:
Baker, George A., and Peter Graves-Morris. Padé approximants. Vol. 59. Cambridge University Press, 1996.
Baker, George A.,Nikishin, E. M. , and Sorokin, V. N. . Rational approximations and orthogonality. American Mathematical Society, 1991. and Peter Graves-Morris. Padé approximants. Vol. 59. Cambridge University Press, 1996.
Introduction to Category Theory Through Algebra and Topology, by: Virgil Chan, IUPUI.
Abstract: Modern mathematics are often formulated in terms of categorical theoretic languages. An advantage of this formulation is that some complicated concepts can be understood in terms of commutative diagrams; and theorems can be stated in a short and elegant way. I will introduce the basics of category theory, with motivations and examples from algebra (as covered in MATH 553) and topology (as covered in MATH 572).
References:
Samuel Eilenberg and Saunders MacLane. "General Theory of Natural Equivalances". In: Transactions of the American Mathematical Society 58.2 (1945), pages 231--294. URL: http://www.jstor.org/stable/1990284
Paul G. Goerss and John F. Jardine. Simplicial Homotopy Theory. Modern Birkhäuser Classics. Birkhäuser, 2010.
Lars Hesselholt. Lecture Notes for Arithmetic Algebraic Geometry II. URL: https://www.math.nagoya-u.ac.jp/~larsh/teaching/S2015_AG/lecture.pdf
Sauders Mac Lane. Categories for the Working Mathematician. 2nd edition. Graduate Texts in Mathematics 5. Springer, 1998.
On the Theory of Partial Differential Equations in Sobolev Spaces, by: Andrei Prokhorov, IUPUI. (4 lectures)
Abstract: The goal of these talks is to give a short introduction in the theory of differential operators acting in Sobolev spaces. The goal is to prove the theorems of Fredholm for elliptic differential operator of second order with Dirichlet boundary condition. On the first part of the course we plan to give the review of Sobolev spaces and prove the Rellich theorem about compactness of embedding of Sobolev spaces for bounded domains. Then we will show the Fredholm theorem for compact operators in Hilbert space and in the last part we will apply it to the elliptic differential operator of second order with Dirichlet boundary condition.
References:
a) M. S. Birman, M. Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space, 1987, D. Reidel Publishing Company, Dordecht, Holland.
b) O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, 1985, Springer-Verlag New-York Inc.
Grassmannians, Positroid Strata, and Total Positivity, by: Chris Fraser. IUPUI. (4 lectures)
Abstract: The Grassmannian Gr(k,n) is the projective algebraic variety that parameterizes k-dimensional subspace of a fixed n-dimensional vector space. It has a stratification given by special subvarieties known as positroid varieties. The combinatorics encoding this stratification is rich and elegant. I will give an examples-based introduction to the positroid stratification, touching on connections with total positivity, cluster structures, and physics (scattering amplitudes and the amplituhedron).
Some useful notes:
Rigidity Results in Symmetric Spaces, by: Seongjun Choi (Purdue University)
Abstract: Symmetric spaces are types of Riemannian manifolds with special symmetry feature that allows Lie-theoretic characterization. In this talk, I will briefly survey rigidity results on both real rank 1 case and higher rank cases, starting with Mostow rigidity theorem and Margulis' superrgidity, and then move toward on volume entropy rigidity.