# AMS Student Chapter of IUPUI

## In this website we will share the info about the activities of the AMS student chapter of IUPUI.

## Our Graduate Student Chapter got featured in the May 2018 AMS Notice!

# Fall 2018

### Faculty Research Talks

# "How to Make the Slope Constant in a Dynamical System."

# by: Prof. Dr. Michal Misiurewicz

Abstract: ** **Surprisingly many models allow some kind of reduction to iterations of a continuous interval map. The complexity of such a system is measured by its topological entropy. To simplify the system even further, one can attempt to produce a map of constant (absolute value of the) slope, equal to the exponential of the entropy. In late 1970's Milnor and Thurston proved that such simplification is always possible when the map is piecewise monotone with finitely many pieces and the entropy is positive. I will describe the ideas of a proof, possible generalizations, and the problems that arise when we assume that the number of pieces of monotonicity is infinite.

# Wednesday, October 24th

# 10:30 am at SL 010

# Spring 2018

## Colloquium

### Faculty Research Talks

# "You Didn’t Think I Could Solve It, but I Can! A Survey of the Modern Theory of Integrable Systems"

# by: Prof. Dr. Alexander Its

Abstract: The term “Integrable Systems” usually refers to mathematical objects, most often differential equations, with special symmetry properties which allow to study them in a very detailed way and sometimes even to solve them in a closed form. The class of integrable systems includes several fundamental equations of nature, and the mathematical foundations of integrable systems go back to classical works of Liouville, Gauss, and Poincaré. In our days, the theory of integrable systems has become an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. In this talk, a brief history and state-of-the-art of the theory of integrable system together with the place the integrable systems occupy in the general area of mathematics will be presented.

# Monday, April 2nd

# 10:30 am at SL 055

# "Dynamics: from the torus to the Cantor set"

# by: Prof. Dr. Bruce Kitchens

Abstract: The torus can be thought of as an abelian group using coordinate-wise addition modulo 1 as the group operation. It has a topological and differentiable structure. The usual two-dimensional Lebesgue measure is invariant under the group operation and is the topological group’s Haar measure. There are continuous group automorphisms of the torus and they can be described by 2 × 2 matrices having integer entries and determinant ±1. The automorphisms preserve Lebesgue measure. We will examine the dynamics of one of these. It is the prototype for the Anosov diffeomorphisms of manifolds. There is a very nice geometric construction of a Markov partition for the map. It allows one to relate the dynamics of the map on the torus to a simply described homeomorphism of the Cantor set. The homeomorphism on the Cantor set is a topological Markov shift and can be described using the matrix that described the map on the torus. The advantage of this construction is that it allows one to easily analyze many important dynamical properties of the original map on the torus. The properties include periodic points, invariant measures and entropy.

# Monday, March 26th

# 10:30 am at SL 055

# "Transitionally Commutative Structures on Rank 2 Real Vector Bundles"

# by: Dr. Bernardo Villarreal

Abstract: A transitionally commutative structure (tc structure) is a lift of the classifying map of the bundle to the classifying space for commutativity denoted BcomG. In this talk I will focus on G=O(2), by constructing explicit tc structures on the trivial bundle over the 2 sphere, and showing that (surprisingly) the tautological bundle over the Grassmannian of 2-planes in R^n (n>3) does not admit any tc structure.

This is part of joint work with O. Antolín-Camarena and S. Gritschacher, and part of work in progress with D. Ramras.

# Monday, February 19th, 26th

# 10:30 am at SL 055

# Fall 2017

### Faculty Research Talks

# "The Proof of the Shapiro-Shapiro Conjecture "

# by: Professor Evgeny Mukhin

Abstract: I will recall how a study in the area of mathematical physics unexpectedly led to a proof of a long standing conjecture in real algebraic geometry. This talk is based on several papers written together with Prof. Tarasov (IUPUI) and Prof. Varchenko (UNC, Chapel HIll) a few years ago.

# Tuesday, November 7th

# 06:00 pm at SL 010

### Mini Courses

* 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).

**Lecture Notes:** here

**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.

# Spring 2017

# "Solutions of the cubic Fermat equation in algebraic number fields"

# Part II

# by : Professor Patrick Morton

Abstract : I will discuss my work in trying to prove that the cubic Fermat equation x^3 + y^3 = z^3 has nontrivial solutions in every quadratic field K=Q(sqrt(-d)) in which -d < 0 and d = 2 (mod 3). This was conjectured by Aigner in 1955, but is still an open problem. I will show how to find solutions in a large number of these quadratic fields (at least 50% of them) using modular functions and Galois theory. In particular, I will show how these techniques lead to the solution

(-4+29sqrt(-17))^3 + (-4-29sqrt(-17))^3 = 70^3

in the field K = Q(sqrt(-17)).

**Below you can view the notes that the talks are based upon.**

# Tuesday, April 25th

# 12:00 pm at LD018

## Special Workshop

## Here you can view\download the slides of the workshop :

## Day 1:

## Day 2:

# Special Announcement

# HTML Workshop

## By: Michel Tavares

*The AMS graduate student chapter of IUPUI announces an special workshop on HTML programming. This workshop consists of three sessions on 17th, 19th and 21st of April 2017. The goal of the workshop would be to help each participant to have a personal website written by him\herself at the end of the workshop. The workshop will be held at UL1130 (Library Computer Instruction Room) at 4:30 p.m. on each of the dates mentioned above. The outline of the topics discussed in the workshop is given below*

* *

# Day 1 (Monday April 17)

## · Web Development

### o Introduction

## · HTML

### o Introduction

### o Attributes

### o Elements

### o Tags

### o Paragraphs

### o Headings

### o Lists

### o Links

### o Styles

### o Text Formatting

### o Images

### o Tables

### o Forms

# Day 2 (Wednesday April 19)

## · Cascading Style Sheets

### o Introduction

### o Colors

### o Backgrounds

### o Tables

### o Lists

### o Position

### o Transitions

### o Animations

## · JavaScript - Part I

### o Introduction

### o Syntax

### o Variables

### o Operators

### o Strings/Arrays

# Day 3(Friday April 21)

## · JavaScript - Part II

### o Loops

### o Functions

### o Events

### o Objects

## · Miscellaneous Topics

### o Responsive Web Design

### o Frameworks (JQuery, etc.)

### o Browser Testing

*Attending the workshop(including the refreshments served at the end of each session) is totally free for IUPUI students but due to the limited space(~30 participants), our plan is to choose the participants based on the first RSVPs that we receive. So if you would like to attend the workshop please an email to rgharakh@iupui.edu and include your name/department in the body of your email, also include the date(s) that you might not be able to attend the workshop.*

*Note that this invitation will be sent to most departments in the school of science, so make sure to RSVP as soon as you decide to attend the workshop.*

*Best regrads,*

*Roozbeh Gharakhloo*

*President *

*AMS graduate student chapter of IUPUI*

### Faculty research talks :

# "Spray Geometry"

# by: Professor Zhongmin Shen

Abstract: A spray on a manifold can be viewed as a collection of systems of second order ODEs. The solutions are called the geodesics of the spray. A spray can also be viewed as a collections of parametric curves (called geodesics) in the manifold with the following properties: 1) for every tangent vector v at a point p, there is a unique geodesic c(t) with c(0)=p, c’(0)=v; 2) for any geodesic c(t) and any positive number k > 0, c(kt) is still a geodesic. No distance measure is associated with the spray. Every Rieannian metric determines a spray. In this talk, I will introduce, curvatures for sprays and discuss their geometric meaning. Basic knowledge on manifolds, vector fields, differential forms on manifold are required.

# Thursday, April 20th

# 12:00 pm at LD018

* 2-Contact Embeddings in Dimension Three *, by : Professor Olguta Buse

Abstract: In joint work with D. Gay, we introduce the concepts of capacity and shape for a three dimensional contact manifold relative to a transversal knot. We will explain the connection with the existing literature and provide our main computation for the shape in the case of lens spaces L(p,q). The main tool used here are rational surgeries which will be explained through their toric interpretations based on the continuous fraction expansions of p/q. We will discuss possible parallels with the study of ellipsoid embeddings in four dimensions.

* 3- Quantum integrable models, what mathematics are they about?, *by : Professor Vitaly Tarasov

Abstract : After a brief gentle introduction into quantum mechanics and integrability, I will try to explain what kind of problems in linear algebra, representation theory (it is "in between" abstract algebra and linear algerba), combinatorics, complex analysis, etc. arise under the umbrella of quantum integrable models. I hope to keep the content within general mathematical background, but it would be useful to know in advance the basic idea of a tensor product.

* 4- 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.

# Fall 2016

### Faculty research talks :

Abstract : It has long been known to mathematicians and physicists that while one full turn of an object in 3-space causes tangling, two full turns can be untangled. Physical demonstrations of this fact include Dirac's belt trick and the Indonesian candle dance. Mathematically, this is a topological feature of the rotation group SO(3). In this talk, we will explore a geometrically defined untangling procedure, leading to interesting conclusions about the minimum complexity of untanglings. Along the way, we'll see quaternions, degrees of continuous mappings, and animations of our geometric untangling. This is based on joint work with David Pengelley.

### Mini Courses

* 1. On the Theory of Partial Differential Equations in Sobolev Spaces* , by : Andrei Prokhorov, IUPUI.

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.

* 2. Grassmannians, Positroid Strata, and Total Positivity* , by : Chris Fraser. IUPUI.

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 :

### Qualifying Exam workshops

- Complex Analysis, by : Andrei Prokhorov
- Real Analysis, by : Michael Pilla
- Topology, by : Ahmad Barhoumi