The Directed Reading Program (DRP) at UIC is a program in which undergraduate students are paired with graduate student mentors and pursue independent reading projects. The presentation session for the Summer 2024 iteration of the DRP will be held on Monday, September 9 from 1:30PM to 3PM. The first presentation will take place in SEO 612 in the first half-hour, and the remaining three presentations will take place in SEO 636 in the ensuing hour. The schedule for the presentation session is as follows:

**Wait, There is More than One Geometry?**

Juan Jose Rosendo, mentored by Amelia Pompilio

Abstract: When taking a high school level geometry class, we are exposed to the reality of shapes. It is all we know, until one decides to learn more about a "geometry." Therefore, we will explore the differences between various geometries, including Euclidean, Hilbert, Spherical, Hyperbolic, and Projective geometry. A brief introduction will be given for each as well as a fun fact of why each one differs from the other.

**Fantastic Fractals**

Janien Hammonds, mentored by Lisa Cenek

Abstract: Fantastic Fractals explores the paradoxical nature of fractals by building on the concept of the Cantor set. I use the Koch curve and Koch island to exemplify this characteristic.

**Pólya's Recurrence Theorem & Electric Networks**

Marcelo Lozano, mentored by Karoline Dubin

Abstract: "A drunk man will return home, but a drunk bird may get lost forever." Will a point wandering randomly on an n-dimensional grid return to its initial position? George Pólya's Recurrence Theorem tells us that simple random walks are recurrent on one and two dimensional lattices and are transient in higher dimensions. This talk will apply the language of electric networks to develop intuition for this famous result (and outline one of its proofs).

**Why Lebesgue? Understanding the Limitations of Riemann Integration**

Sebastian Tous, mentored by Nick Christo

Abstract: While Riemann integration provides a straightforward and geometrically intuitive approach to finding the area under a curve, it falls short in cases involving discontinuous functions and is insufficient for nice general convergence theorems to hold. In this presentation, I will discuss the motivation behind Lebesgue integration, its relation to the Riemann integral, and some of its limitations.

Undergraduate students are not expected to propose their own project, but they are welcome to if they so desire. A list of texts to consider, compiled by the DRP at The University of Texas at Austin, can be found here.

Undergraduate students interested in a first experience with mathematics outside the classroom, as well as undergraduate students who may have undergone challenges throughout their journey with mathematics as a result of their identity or life circumstances, are highly encouraged to apply.

The timeline for Fall 2024 iteration of the DRP is as follows:

The current list of mentors for the DRP is as follows:

**Darius Alizadeh**

**Julian Benali**

**Nick Christo**

**Karoline Dubin**

**Michael Gintz**

**Vignesh Jagathese**

**Shin Wook Kim**

**Katie Kruzan**

**Sacha L'Heveder**

**Sixuan Lou**

**Clay Mizgerd**

**Abhijeet Mulgund**

**Theo Sandstrom**

**Jaegeon Shin**

**Chirag Singhal**

**Duan Tu**

**Jagerynn Verano**

**Ping Wan**

Information regarding older presentation sessions can be found below:

This program is hosted by the UIC student organization Symbols of Inclusion and co-organized by Nick Christo and Michael Gintz. Questions regarding the program can be directed to Michael Gintz at mgintz2@uic.edu.

Juan Jose Rosendo, mentored by Amelia Pompilio

Abstract: When taking a high school level geometry class, we are exposed to the reality of shapes. It is all we know, until one decides to learn more about a "geometry." Therefore, we will explore the differences between various geometries, including Euclidean, Hilbert, Spherical, Hyperbolic, and Projective geometry. A brief introduction will be given for each as well as a fun fact of why each one differs from the other.

Janien Hammonds, mentored by Lisa Cenek

Abstract: Fantastic Fractals explores the paradoxical nature of fractals by building on the concept of the Cantor set. I use the Koch curve and Koch island to exemplify this characteristic.

Marcelo Lozano, mentored by Karoline Dubin

Abstract: "A drunk man will return home, but a drunk bird may get lost forever." Will a point wandering randomly on an n-dimensional grid return to its initial position? George Pólya's Recurrence Theorem tells us that simple random walks are recurrent on one and two dimensional lattices and are transient in higher dimensions. This talk will apply the language of electric networks to develop intuition for this famous result (and outline one of its proofs).

Sebastian Tous, mentored by Nick Christo

Abstract: While Riemann integration provides a straightforward and geometrically intuitive approach to finding the area under a curve, it falls short in cases involving discontinuous functions and is insufficient for nice general convergence theorems to hold. In this presentation, I will discuss the motivation behind Lebesgue integration, its relation to the Riemann integral, and some of its limitations.

Undergraduate students are not expected to propose their own project, but they are welcome to if they so desire. A list of texts to consider, compiled by the DRP at The University of Texas at Austin, can be found here.

Undergraduate students interested in a first experience with mathematics outside the classroom, as well as undergraduate students who may have undergone challenges throughout their journey with mathematics as a result of their identity or life circumstances, are highly encouraged to apply.

The timeline for Fall 2024 iteration of the DRP is as follows:

- September 9: Program begins, students are paired with mentors
- September 16: Project proposals due
- October 25: Project updates due
- November 22: Titles and abstracts due
- December 6: Presentations

The current list of mentors for the DRP is as follows:

Darius Alizadeh is a 5th year PhD student who studies groups through their actions on geometric objects, aka geometric group theory. Any topic where you can draw pictures and think in shapes is a good fit for him, including manifolds, algebraic topology, and more. He would love to explore some interesting topic outside the standard curriculum. Math is a big world to explore!

Julian Benali is a PhD student studying commutative algebra, though he also has interests in logic, topology, and category theory. Julian is looking to work with an experienced undergraduate on a reading project in pure math. The direction is up to the student, but potential topics could include tight closure theory, the chain conjecture, recent developments in intuitionist math, or applications of topos theory.

Nick Christo is a 5th-year PhD candidate studying probability, combinatorics, and statistical physics. I have in mind reading through a few chapters of a textbook (e.g., graph theory, probability texts), reading through papers is also an option on any MCS related topic. Reading independently can be a challenging experience that one has to almost learn how to do and would be happy to help make that an easier experience.

Karoline Dubin is a 5th-year PhD student interested in probability theory, statistical physics, and computer science. She is interested in topics of math with physical motivation and algorithmic applications, and is excited to explore such topics with undergrads.

Michael Gintz is a third-year PhD student studying commutative algebra and algebraic geometry. He'd be happy to introduce students to any number of topics related in whole or in part to either of these fields, and would be most interested to go through a textbook at the student's level to introduce them to a topic.

Vignesh Jagathese is a 4th year PhD student studying Commutative Algebra and Algebraic Geometry. Vignesh likes to use algebraic formalism to encode and understand singularities, or how "bad" a given point or region in a space is, and would thoroughly enjoy working with a UIC undergraduate on such topics. Algebra is widely applied in many fields of mathematics, so there are a lot of different topics to study! Vignesh, for instance, also likes thinking about other applications of commutative algebra such as number theory, arithmetic geometry, or just algebra for its own sake. Ultimately, though, he is happy to mentor undergraduate students and guide them towards papers covering topics that they have interest in.

Shin Kim is a 4th-year PhD student studying Complex Geometry. He is interested in reading with undergraduate students on any topic that they are interested in.

Katie Kruzan is a 3rd year PhD student studying Mathematical Computer Science with particular interest in Algorithms and Graph Theory. She is interested in reading papers//texts with undergraduate students on topics related to their interest, and also help them learn the skills around getting the most bang-for-the-buck around reading academic texts.

Sacha L'Heveder is a 2nd-year PhD student studying group theory and dynamics. He has familiarity with Geometric Group Theory (fun with shapes), Measured Group Theory (fun with measures), Functional Analysis (fun with functions), and Dynamics (fun, but do it a lot of times). He looks forward to working with an undergraduate on a reading around a topic of their interest.

Sixuan Lou is a 5th-year PhD student studying Algebraic Geometry. He likes to think about algebro-geometric problems and is also familiar with other topics in Math/CS. He would like to read with undergraduate students on topics of their interests (preferably related to geometry, topology, algebra, number theory, etc). You could be reading a paper, lecture notes or a textbook on the topic of interests. He is looking forward to talk to you!

Clay Mizgerd is a 3rd-year PhD student working in probability and combinatorics. In addition to these fields, he especially enjoys number theory, and has a broad background in analysis and algebra. He is happy to read on any field of pure math, and looking forward to learning some new math together.

Abhijeet Mulgund is a 4th year PhD student studying machine learning, information theory, and complexity theory. He studies applications of deep learning to error-correcting codes. He is interested in reading on topics related to complexity theory, machine learning, or probability. These fields form a bridge between applied and pure math, which can lead to some very interesting problems.

Theo Sandstrom is a 2nd-year PhD student primarily studying commutative algebra and algebraic geometry, but with broad interest across mathematics (particularly in topics with geometric flavor). He is happy to work with undergraduates on any topic in this vast realm. Possible topics include: an introduction to X, where X in {algebraic geometry, commutative algebra, homological algebra}; elliptic curves; proof assistants (e.g., Lean, Coq, Agda); algebraic topology; knot theory; foliation theory; p-adic analysis.

Jaegeon is a fifth-year PhD student studying complex geometry, advised by Professor Julius Ross. Complex geometry is a mix of algebra and complex analysis, using techniques from both disciplines to solve geometric problems. He is very passionate about teaching and mentoring students who want to learn more about math beyond the textbooks. Learning math is never easy, but it can be very fun and rewarding with a mentor!

Chirag Singhal is a third year Mathematics PhD student at UIC. His research interests lie in Number Theory and in particular, Arithmetic Geometry. He also has some side interests, which include Theoretical Computer Science and Combinatorics.

Duan Tu is a 5th year PhD student studying Mathematical Computer Science. Her current research focuses on machine learning theory, but her interests also involve probability and discrete math in general. Through the Directed Reading Program, she wants to help undergraduate students break the seemingly steep barrier to advanced math and math research. She is excited to learn something new together with the student!

Jagerynn is a fifth year phd student in geometry. She is passionate about antiracism and immigration reform.

Ping Wan is a 5th-year PhD student studying Geometric Group Theory. They spend most of their research time drawing triangles, pentagons, or random curves. They are interested in reading papers or surveys with undergraduate students on topics about hyperbolic spaces and/or infinite groups.

Information regarding older presentation sessions can be found below:

This program is hosted by the UIC student organization Symbols of Inclusion and co-organized by Nick Christo and Michael Gintz. Questions regarding the program can be directed to Michael Gintz at mgintz2@uic.edu.