Events

Upcoming Events

Gerald A. Goldin — What is Deep Mathematical Understanding?

Nov 16, 2026, 5:30pm in Hill 705

Mathematics is often taught as a well-defined system of symbolic representation, with increasingly complex constructs. As learners we study established patterns, as well as rules or procedures for performing calculations and finding solutions to problems. We learn definitions and systems of axioms. We prove many theorems based on them. We cover a lot of ground, centuries of mathematical development in a relatively short time.

As we do this, we hopefully acquire some understanding of the abstract concepts behind the definitions, the ideas behind the proofs, and ways of thinking that are precise and logical. For some, understanding seems to occur rapidly and spontaneously; for others, more slowly or only after explanation and discussion.

This talk will be an informal exploration of what it means to understand a mathematical idea or concept, what it means to do so deeply, and how such understanding can be acquired. I have come to believe that all of us, from novices to experts, can gain mathematical power from such an inquiry. We will look at one or two examples together, in topics accessible to all.

Michael Kiessling — What is a well-posed problem, and why is well-posedness important?

Nov 23, 2026, 5:30pm in Hill 705

When faced with the problem of solving a system of linear algebraic equations, students early on learn that there can be zero, one, or more solutions, possibly infinitely many, how to decide which is the case, and how to compute them (if possible). Much emphasis is given to problems for which there exists a unique solution. This theme is continued when one learns about linear ordinary differential equations, then linear partial differential equations, and perhaps linear integral equations. While these differential / integral equations can be seen as infinite-dimensional extension of linear algebraic problems, working in infinitely many dimensions tends to create the problem of well-posedness. In this talk I explain what this means and why it is an important concept, using plenty of examples.

Past Events

FALL 2026

SPS × RUMA Jeopardy Night

Nov 12, 2026, 6:30pm in Serin 330 W

SPS and RUMA will be hosting a joint jeopardy night this Thursday at 6:30 pm in Serin 330 W! Join us for some fun math and physics questions while eating free pizza.

RUMA Seminar: Jason Liu — How Hard is an Estimation Problem?

Nov 2, 2026, 6:00pm in Hill 705

You are given data drawn from some unknown distribution, and are tasked with estimating a parameter of a distribution—for example, the mean or variance. As a clever member of RUMA, you propose a consistent estimator for this parameter: given enough data, it will be very close to the true parameter with high probability. Then you wonder, how good is my estimator? Is there an estimator that is better than mine?

These questions lead us to the minimax risk, which tells us the fundamental difficulty of an estimation problem. More precisely, it quantifies the worst-case estimation error that any estimator must have. In this presentation, I will introduce an information-theoretic technique to bound the minimax risk, and explain how to prove that an estimator is minimax optimal.

RUMA Seminar: Michael Fiore — Infinite Games and Vaught's Conjecture

Oct 29, 2026, 5:00pm in Hill 705

How do you tell two structures apart up to isomorphism? We explore this question using a game between a Duplicator and his adversary, the Spoiler. Connections between winning strategies for this game and infinitary logic let us explore an open problem in model theory about counting structures.

Kurt Bryan — Making Do With Less: The Mathematics of Compressed Sensing

Oct 14, 2026, 6:00pm in Hill 705

Suppose a bag contains 100 marbles, each with mass 10 grams, except for one defective off-mass marble. Given an accurate electronic balance that can accommodate anywhere from one to 100 marbles at a time, how would you find the defective marble with the fewest number of weighings? You may well have thought about this kind of problem before and know the answer. But what if there are two bad marbles, each of unknown mass? Or three or more? In these cases an efficient scheme isn’t so easy to figure out. Is there a strategy that’s both efficient and generalizable?

The answer is “yes,” at least if the number of defective marbles is sufficiently small.

Surprisingly, the procedure involves a strong dose of randomness. It’s a nice example of a new and very active research area called “compressed sensing” (CS), that spans mathematics, signal processing, statistics, and computer science, and has many surprising applications. In this talk I’ll explain the central ideas, which require nothing more than straightforward linear algebra and a bit of probability. I’ll then show variety of applications.

RUMA Seminar: Alon Danai — Numbers! Geometry!

Sep 21, 2026, 6:00pm in Hill 525

The Geometry of Numbers is a field of math dedicated to studying number theory using geometric tools. It was pioneered by Hermann Minkowski in 1896 when he discovered a theorem about when convex regions must contain a lattice point. We will go over this result and use it to prove famous results in number theory such as Fermat’s theorem on sums of two squares and Lagrange’s theorem which states that any integer is the sum of four squares.

Samanthak Thiagarajan — How the Fourier Transform Studies the World Through Waves...

Sep 17, 2026, 5:00pm in Hill 705

The Fourier Transform, at least for me, was always a bit mysterious. People usually say the FT is a tool for decomposing complicated waves into simpler ones, but what does that actually mean and how do you do that? In this talk, hopefully we will make this a bit clearer and we’ll see some fun applications to problems in number theory, PDEs, and understand how the FT is super useful for music, signal processing, and more.

Alex Kontorovich — Interactions of Formalization, AI, and Mathematics

Sep 10, 2026, 5:00pm in Hill 705

We’ll discuss Lean verification, reliability, and related issues as they pertain to evaluating mathematics produced with/by AI.

Math Major Q&A

Sep 4, 2026, 6:00pm in Hill 705

We will have four panelists with various experience in math: an undergraduate student, a graduate student, a professor, and a person in industry. Come to ask all your questions about being a math major!

SPRING 2026

Estimathon

May 1, 2026, 6:00pm in Hill 705

This is a team-based contest that combines trivia, game theory, and mathematical thinking. Teams have 30 minutes to work on a set of 13 estimation problems, the winning team being the one with the best set of estimates. Come join us for a fun time and food!

Liza Ter-Saakov — Matroids and the Lattice of Flats

May 1, 2026, 12:30pm in Hill 425

Matroids are a very interesting object in combinatorics/algebra that have been rising in popularity as something to study in the past decade, including in work whose authors have received Field Medals. They can be defined in many equivalent ways (cryptomorphisms), and this talk introduces a couple of them. We will focus on the flat definition of a matroid, and explore the lattice that one obtains from the flats. We’ll introduce basic lattice theory then investigate what we can say about matroids from their lattices and what we can say about lattices from their matroids.

Some questions one could ask: Can we characterize the lattices that come from flats of matroids? (yes!) Can we predict how changing a matroid changes the lattice of flats? (depends on the change!) Does augmenting the lattice by [insert way here] give us another matroid, and if so, are there relations between the new and old matroid? (sometimes!) Do lattices of matroid flats have any special properties? (yes, some of which are topology related!) There are a plethora of exciting questions to ask and this talk seeks to pose many, and answer most.

Ish Shah — Arithmetic progressions in infinite sets

Apr 23, 2026, 5:00pm in Hill 705

Suppose you pick an infinite set of natural numbers. Must it always contain a pattern of the form $a$ , $a + d$ , $a + 2 d$ ? The answer ends up being no—though if the set is “large enough” in some sense, then such patterns are unavoidable. But what precisely does “large enough” mean, and what happens if we look for longer patterns like 4-term or 5-term progressions? In this talk, we will explore these questions and see how they lead to remarkable results lying at the intersection of combinatorics, number theory, and Fourier analysis.

Zheng-Chao Han — The Ubiquity of Bessel Functions

Apr 16, 2026, 5:00pm in Hill 705

Often viewed merely as daunting power series or mysterious integral representations, Bessel functions are actually natural, geometric basis vectors rooted in the structure of solution space of the Laplace or Helmholtz equations. They appear in many different contexts of mathematics, sometimes in unexpected ways, and the myriad varieties of identities or recursive relations make them even more mysterious.

This talk will attempt to use their roles as building blocks of solutions of the Laplace or Helmholtz equations, often in different coordinate representations, to help uncover some of these mysteries. We will use examples from wave mechanics & superposition, solutions of these equations in different dimensions, their roles in spectral geometry in the Euclidean as well as in the hyperbolic space to exhibit their ubiquity and their relations among different representations.

Natasa Sesum — Flowing geometry, the PDE of shapes and spaces

Apr 10, 2026, 3:00pm in Hill 525

In the talk we will introduce geometric flows, such as the curve shortening flow, the mean curvature flow and the Ricci flow. We will talk about some properties of these flows.

Skip Garibaldi (IDA/CCR, IPAM) — Surprisingly large payouts in New Jersey Pick-It

Apr 6, 2026, 5:00pm in Hill 705

For the first decade of their existence, modern legal lottery games in the US were a commercial failure. That all changed when New Jersey introduced Pick-It, the first successful legal game. The popular story is that Pick-It was successful because it imitated the popular illegal numbers game, but there is more to the story than that. When we examined the payouts from the game, we discovered something mysterious. The payouts started surprisingly high and then declined over the first ten years of the game, while the rules of the game remained unchanged. We’ll use a combination of probability, data science, and historical spelunking to explain this strange phenomenon and highlight creative design features of the game that helped its commercial success.

The math in this talk will use some probability of the sort covered on the SOA P exam (Math 477).

This talk represents joint work with Anthony Gamst.

Malcolm Gabbard — Shoelaces in the 4th Dimensions: An Introduction to Low-Dimensional Topology

Mar 12, 2026, 5:00pm in Hill 705

Topology is a large field of mathematics which centers around creating spaces from sets and then considering deformations of these spaces. In this talk we will briefly introduce some foundational definitions and results you would find in an introduction to topology class. With this foundation we will quickly build up to modern questions in low-dimensional topology. In particular, we will learn what knots in the 4th dimension are, how we can visualize them, and why we should care.

Ian Jauslin — Schrödinger's cat and the measurement problem in quantum mechanics

Mar 3, 2026, 5:00pm in Hill 705

Schrödinger’s cat is one of the most well known illustration of quantum mechanics, but when Erwin Schrödinger introduced the thought experiment in 1935, he did so to point out a significant flaw of quantum mechanics: the wave function alone cannot be a full representation of reality. The problem at the root of the issue is that measurement in quantum mechanics cannot be the product of the natural evolution of any system. This is called the “measurement problem”. In this talk, I will explain where it comes from, what it’s implications are, and describe several ideas to get around it.

RUMA × SPS LaTeX Workshop

Feb 25, 2026, 7:30pm in Serin 330W

RUMA and the Society of Physics Students (SPS) are excited to host a $L A T E X$ workshop! $L A T E X$ is a typesetting system which is widely used for digital mathematical writing, and is a very useful tool for anyone who works with mathematics (including adjacent fields, such as physics and computer science). Join us to learn about some basic features and quality of life improvements in $L A T E X$ !

Wolfram Research × RUMA Designing & Visualizing Mathematical Structure

Feb 11, 2026, 5:00pm in Hill 705 and on Zoom

Do you want to learn how to use Mathematica’s powerful geometry and visualization tools? This workshop will cover expression trees, geometric design in Mathematica, and how analytic regions can be converted into validated meshes. More information about the event and speaker can be found on this document.

To follow along, Mathematica can be installed for free using your Rutgers credentials here.

The event will be held over Zoom at the time above, and there will also be an in-person screening in Hill 705, as well as an option to join virtually. The Zoom link will be shared in a follow-up announcement. We hope to provide Wolfram merchandise while supplies last. See you there!

FALL 2025

Estimathon

Dec 5, 2025, 6:00pm in Hill 705

REU Panel

Dec 4, 2025, 6:00pm in Hill 705

Are you interested in doing research this summer, perhaps through a Research Experiences for Undergraduates (REU) program? Join us for the RUMA REU Panel if you’d like to know more about the application process, what it’s like doing summer research, or anything else about REUs! Come hear from people who have been on both sides of math REUs—undergraduate participants, as well as professors who have served as mentors or program overseers—or ask your own questions!

RUMA Seminar: Michael Fiore — Mapping the Universe: Models, Stability, and Independence

Nov 25, 2025, 6:30pm in Hill 705

Classical model theory studies the relationship between statements written in first-order logic and set-theoretic structures which interpret them. Beginning primarily with the work of Shelah, much of work in the field became dedicated to the herculean task of classifying all first-order theories according to their behavior, categorized in meaningful ways by certain “dividing lines.” We will introduce two characterizations of “good behavior” in first-order theories: stability and NIP, and present a two-cardinal theorem of Lachlan and Shelah that demonstrates what can be achieved within stable theories.

Tim Bates — An Introduction to Spatial Graphs

Nov 7, 2025, 4:00pm in Hill 525

A planar graph is a connected graph which can be drawn in the plane without any edges crossing (we call this a planar embedding of the graph). We know that any planar embedding will satisfy Euler’s formula ( $v - e = 2 - f$ ), so no matter how the graph is embedded, I can tell you exactly how many complementary regions of the graph there will be. So we are able to express topological information about the embedding from combinatorial information of the graph. Any graph can be embedded into 3-dimensional space (we call such an embedding a spatial graph), and while we have more freedom when embedding into 3-dimensional space, a lot of topological information about the embedding depends on the structure of the abstract graph. For example, I’ll prove that the complete graph on 6 vertices is intrinsically linked (a result due to Conway and Gordon).

Exploring Actuarial Science & Health Insurance with Cigna Healthcare

Nov 4, 2025, 6:00pm in Livingston Student Center 202AB

The Rutgers Actuarial Club is hosting an information session about actuarial science and health insurance with Cigna Healthcare. This event is a great opportunity for undergraduate students interested in applying math, statistics, and economics to real-world problems and exploring career paths in actuarial science, analytics, and data-driven decision-making. Pizza will be provided at the event! If you are interested in attending, please fill out this form.

RUMA Seminar: Sean LeClair — Amenability and Thompson's Group F

Oct 29, 2025, 6:00pm in Hill 705

In the 1920’s, Alfred Tarski and Stefan Banach proved that given a three dimensional sphere, you could partition the sphere into two proper subsets that are identical to the original sphere. This result motivated the study of amenable groups, or groups that admit a left invariant measure. One of the biggest open problems in the study of Amenable Groups is whether or not Thompson’s $F$ group is amenable. We give a brief overview of the history of paradoxical sets, as well as introduce amenable groups and Thompson’s $F$ group. At the end we will discuss recent results concerning the amenability of $F$ .

RUMA × RUSEC × RU Linguistics Cryptography and Ciphers

Oct 28, 2025, 6:00pm in School of Communications & Information Room 222

Cryptography is used in a multitude of different ways on the internet, with techniques (many mathematical) helping protect valuable information by obscuring information. We will be giving an overview of cryptography, including, common algorithms and ciphers, as well as hashing functions and how they work! Afterwards, we will have a small activity with ciphers 🧩. Pizza will be provided 🍕

RUMA Seminar: Alon Danai — Impartial Vector Games

Oct 22, 2025, 6:00pm in Hill 705

Impartial combinatorial games are two player, turn-based games where the allowed moves depend only on the position of the game, not the player. We will go over how to solve these games, how to add games, and how to classify games using the Sprague-Grundy theorem. Then we’ll talk about vector games, which are a specific class of impartial games that are played on a grid. We will go over a few examples and recent results about vector games.

RUMA Seminar: Kyan Valencik — About the Mutation Graph of Projective Three-space

Oct 16, 2025, 6:00pm in Hill 525

Under mirror symmetry, there is believed to be a correspondence between Fano manifolds and Laurent polynomials. When this happens, we say that a Laurent polynomial $f$ is a mirror partner for a Fano manifold $X$ . There is a change of variables known as a “mutation” which yields a family of mirror partners for $X$ from the polynomial $f$ . From this, we can construct a graph of all possible mutations and ask questions about the graph’s structure. In this talk, we examine the mutation graph of $f = x + y + z + 1 / (x y z)$ corresponding to complex projective three-space.

RUMA Seminar: Trisha Kothavale — Quantum Groups over Fields of Positive Characteristic

Oct 9, 2025, 6:00pm in Hill 525

Groups are mathematical objects that describe the symmetries of classical systems. For quantum systems, groups alone may not capture all the symmetries. To address this, one studies quantum groups, which are Hopf algebras whose representation theory encodes quantum symmetries. We will first define Hopf algebras and quantum groups. Then, we will present a new example of a quantum group in prime characteristic and the steps involved in constructing it.

RUMA Seminar: Ish Shah — Nearly holomorphic modular forms of half integer weight

Oct 1, 2025, 6:00pm in Hill 525

What is a modular form, and why should we care about them? We will answer these questions, and then discuss various generalizations of the definition. After introducing these generalizations and giving some intuition as to why they are worth considering, a new result about them from a recent REU project will be presented. The proof of this result, and how computers were able to aid with it, will be touched upon.

Matthew Young — The conjecture of Birch and Swinnerton-Dyer

Sep 25, 2025, 6:00pm in Hill 525

An elliptic curve is a cubic equation which can be written in the form $y^{2} = x^{3} + a x + b$ with $a$ and $b$ integers. A basic question is to describe the set of rational solutions to this equation, and a particularly interesting sub-question is to ask when there are infinitely many solutions. There is a conjecture of Birch and Swinnerton-Dyer from the 1960’s which would give a satisfying answer to this question; its solution is worth $1,000,000 from the Clay Math Institute. I will give a survey of their conjecture and discuss what is known.

Facundo Mémoli — The Gromov-Hausdorff distance between spheres

Sep 19, 2025, 6:00pm in Hill 705

A very natural question in geometry (and in many applications) is: How can we measure how different two shapes are?

One way to do this is by using distances between metric spaces, such as the Gromov–Hausdorff distance, which is often invoked in areas like data classification. However, even though distances of this type are widely used, their exact values are known only in a few very special cases.

In this talk, I will focus on the case of spheres (with their usual round, geodesic distance). I will describe how one can obtain lower bounds for the Gromov-Hausdorff distance between spheres, and in certain cases prove that these bounds are sharp by constructing optimal correspondences. But many challenges remain: for most pairs of spheres the exact value of the distance is still unknown, and even the right techniques for tackling these cases are not yet clear. This makes the problem a rich source of open questions at the intersection of geometry, topology, and analysis.

Interestingly, these results connect with a classical topological result called the Borsuk–Ulam theorem—but in a new setting where we have to deal with discontinuous functions.

Math Major Q&A Session

Sep 5, 2025, 6:00pm in Hill 705

Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have a professor, a current math graduate student, a graduated math major in industry, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come.

SPRING 2025

Robert Dougherty-Bliss — Expensive Computations

May 2, 2025, 2:00pm in Hill 423

Speaker: Robert Dougherty-Bliss (Dartmouth College)

Abstract: The Fibonacci numbers (0, 1, 1, 2, 3, 5, …) are defined by

\begin{matrix} F (0) & = 0 \\ F (1) & = 1 \\ F (n) & = F (n - 1) + F (n - 2), n \geq 2 \end{matrix}

In theory this tells you everything about them. In practice, many questions are computationally infeasible without being smarter. For example, what are the ten most significant digits of $F (2^{1000})$ ? I’ll show you some unexpected places that computations get expensive (sequences, polynomials, summation), and what to do about it.

Estimathon

Apr 25, 2025 in Hill 705

Fioralba Cakoni — Seeing the Unseen with Mathematics -- A Journey into Inverse Problems

Apr 23, 2025, 5:00pm in Hill 705

Speaker: Fioralba Cakoni

Abstract: Plato’s Allegory of the Cave describes chained “prisoners” attempting to understand reality by observing shadows projected onto the walls.

Kac famously posed the question: “Can you hear the shape of a drum?” In this talk, we discuss examples of inverse problems, where mathematics is used to determine underlying causes or parameters that produce a desired or observed effect. Solving an inverse problem is like knowing the answer and trying to find the question. I challenge you to try this exercise prior to coming to the lecture.

Mario Szegedy — Repeated averages on graphs

Apr 17, 2025, 6:00pm in Hill 705

Speaker: Mario Szegedy

Abstract: Imagine an undirected graph $G$ with $n$ nodes, where one node holds 1 kg of gold, while the rest start with nothing. We explore a simple redistribution process: every second, a random edge is chosen, and the gold at its two endpoints is evenly split between them. Over time, the gold naturally spreads, and it is easy to see that after a long enough time, every node will hold roughly $1 / n$ kg of gold.

But how long does this take? While the final outcome is clear, determining the speed of convergence turns out to be a much harder question. In this work, we provide a complete answer for all graphs G (up to a constant factor).

Try solving the case of a cycle graph! Even though the problem seems simple, don’t be discouraged—the proof of the answer (which I do not disclose here:) is rather elaborate to our best knowledge.

Joint with Ramis Movassagh and Guanyang Wang.

Henryk Iwaniec — PRIMES!, PRIMES!, PRIMES!

Apr 8, 2025, 2:00pm in Hill 423

Speaker: Henryk Iwaniec

Abstract: I will describe old and new results about the distribution of prime numbers, some classical methods (sieve methods) and modern ideas (random matrix models for zeros of the Riemann zeta function).

Talk notes (PDF)

Paul Ellis — Advanced Hat Puzzles

Apr 4, 2025, 11:00am in Hill 423

Speaker: Paul Ellis

Abstract: We explore a variety of Hat Puzzles, of a variety of difficulties, which use a variety of techniques to solve. Here’s a warm-up: 3 people each have a hat with a real number. No one can see their own hat, but they can see the others. They each silently write down an interval and give it to the proctor. They may not talk during the trial, but they can agree on a strategy ahead of time. How can they guarantee that at least one of them guesses an interval containing the number on their own hat? Here’s a trickier one: There are infinitely many boxes, each containing a random real number. There are 10 logicians. Each logician may open as many boxes as they like, then guess the number in one of the boxes they did not look in. Then they have to close the boxes for the next logician’s turn. No talking between turns. Show that it is possible for at least 9 of them to make correct guesses.

RUMA × RUCP Five Minute Talks

Apr 1, 2025, 6:00pm in Hill 705

What if you were giving a RUMA talk, but with no math involved at all? It must be April 1… For this event, you can volunteer to give a short (5 minute) talk on any topic of your choosing–except math or competitive programming! Even if you don’t want to present, you can still learn about the diverse array of interests among the members of our clubs.

Michael Kiessling — Paradoxes in Mathematics

Mar 27, 2025, 6:00pm in Hill 705

Speaker: Michael Kiessling

Abstract: Mathematics is full of famous paradoxes; think of Zeno’s paradox of Achilles and the Tortoise, or the Banach—Tarski paradox, to name only two of the many famous ones. But what makes a paradox a paradox? What is so intriguing about them? In my talk I review some of the well-known paradoxes. I also present some that are not well-known, but none-the-less intriguing, too, that I myself ran into during my professional career as a mathematical physicist. Incidentally, I also try to answer the two questions I raised above.

RUMA × SPS LaTeX Workshop

Mar 26, 2025, 7:30pm in Serin 330

RUMA and the Society of Physics Students (SPS) are excited to host a $L A T E X$ workshop on Wednesday, March 26, 7:30-8:30 PM in Serin 330! $L A T E X$ is a typesetting system which is widely used for digital mathematical writing, and is a very useful tool for anyone who works with mathematics (including adjacent fields, such as physics and computer science).

Pi Day

Mar 13, 2025, 6:28pm in Hill 323

Stop by the math lounge (Hill 323) to enjoy some pie and participate in a competition to recite the most digits of pi from memory! There will be a prize (water bottle).

RUMA Seminar: Trisha Kothavale — Categorification in Representation Theory

Mar 12, 2025, 6:00pm in Hill 705

Speaker: Trisha Kothavale

Abstract: The underlying ideas of categorical representation theory involve lifting representations to higher levels of structure by viewing them in terms of categories and functors. This is especially important in studying representations of quantum groups and Lie algebras. We will introduce the representation theory of associative algebra using module theory and define what it means to categorify a representation. We will then give some examples of how these categorifications shed more light on the structure of algebraic objects.

Filip Dul — A Geometer's Perspective on Black Hole Thermodynamics

Mar 11, 2025, 6:00pm in Hill 705

Speaker: Filip Dul

Abstract: Black hole thermodynamics is an increasingly relevant topic at the conjunction of classical and quantum field theory. Mathematicians—mostly geometric analysts and homological algebraists—are an increasingly relevant group in its study. In this talk, I will give an accessible overview of black hole physics and thermodynamics while introducing the necessary mathematical tools. Time permitting, I will explain how some modern topological tools may help us unravel the mystery a little bit further.

Kristen Hendricks — The Jones Polynomial and Categorification

Mar 6, 2025, 6:00pm in Hill 705

Speaker: Kristen Hendricks

Abstract: The Jones polynomial is a powerful invariant of knots and links discovered by V. Jones in the late 1980s. We will review a simple construction of this polynomial and talk about what it can tell us about the geometry of the knot. We then discuss a modern refinement (or “categorification”) of the Jones polynomial, a graded vector space called Khovanov homology, and talk about some of the four-dimensional information it detects. Along the way, we say a few things about the philosophy of categorification.

RUMA Seminar: Valentina Pappano — Mathematics of Origami

Mar 5, 2025, 6:00pm in Hill 705

Speaker: Valentina Pappano

Abstract: When is it possible to fold an origami crease around a vertex? Can an entire crease pattern fold if each vertex can fold? This talk will introduce the basics of the mathematics of origami and some of its important theorems and lemmas. Then, we will see new results in origami involving graph theory and combinatorics. No previous knowledge required.

Lev Borisov — What is a Scheme?

Feb 26, 2025, 5:00pm in Hill 705

Speaker: Lev Borisov

Abstract: In the 1960s Grothendieck and his collaborators developed the theory of schemes, which revolutionized algebraic geometry and remains its preferred technical foundation to this day. In this introductory talk, I will aim to explain the basic definition of a scheme and provide some examples that illustrate the theory.

RUMA Seminar: Nadya Belova — Abelian Categories

Feb 19, 2025, 6:00pm in Hill 705

Speaker: Nadya Belova

Abstract: Many theorems in homological algebra apply to a specific class of categories called abelian categories. We will build up an understanding of the basic objects in category theory (limits, functors, products, kernels, etc.) in order to understand abelian categories. We will then formulate and prove some basic results in homological algebra.

RUMA × RUCP Board Game Social

Feb 12, 2025, 3:30pm in BSC-115

Come join us in BSC-115 on Wednesday, February 12 at 3:30-6:30 PM for a relaxed experience to socialize and play board games with your fellow club members. There is no inherent competitive programming or mathematics in the event, but if you’d like to discuss these please visit!

FALL 2024

Estimathon

Dec 12, 2024 in Hill 705

RUMA Seminar: Ava Ostrem — Mapping Class Groups

Dec 10, 2024, 5:30pm in Hill 705

Speaker: Ava Ostrem

Abstract: A common object in undergraduate group theory classes is the dihedral group, which is the group of symmetries of a polygon. The concept of a symmetry group can be extended to other mathematical objects besides polygons. In this talk, we explore mapping class groups, which are the groups of symmetries of surfaces. We will study the construction of mapping class groups and sketch the proof that mapping class groups of compact, orientable surfaces are generated by Dehn twists.

Rutgers Math Trivia Night

Dec 3, 2024 in Hill 705

Come join us for a Kahoot game with prizes for the top three winners! The Kahoot will mostly contain various questions about the Rutgers math department and various math events in the community. No actual knowledge of math itself is needed or expected; this is a fun (and maybe informative) event!

Abhishek Mallick — The 4th Dimension, where Geometry Meets Topology

Nov 25, 2024, 2:00pm in Hill 005

Speaker: Abhishek Mallick

Abstract: The 4th dimension is wild, for example: while there is only one Euclidean space in every other dimension, there are infinitely many ‘different’ 4-dimensional Euclidean spaces!! Moreover, one of the most fascinating open questions in modern mathematics, unsolved for over 40 years, asks whether there exist two different 4-dimensional spheres. Could you be the one to solve this mystery someday? Start here.

Sameer Erramilli — Toeing the Line Bundle: Topology and Quantization in Electromagnetic Systems

Nov 20, 2024, 7:00pm in virtually

Speaker: Sameer Erramilli

Abstract: Paul Dirac asked in 1931 what magnetic monopoles would look like in a quantum world, and discovered something intriguing: their existence alone would force the electric charge to be quantized. Completely unrelatedly, physicists in the 1970s tried applying strong magnetic fields to transistors much like the ones in your computer, and made the amazing discovery that the conductance induced in them was exactly quantized. Both of these results were groundbreaking on their own. But when we compare them, we find some striking similarities that suggest a deep link between quantization and topology! In this talk, we’ll use these physical systems as a starting point to introduce the concept of a line bundle and its associated Chern class. Along the way, we’ll catch a glimpse of the great interplay between quantum systems and algebraic topology. I’ll assume comfort with multivariable calculus, as well as some conceptual familiarity with the basics of electromagnetism and quantum mechanics.

RUMA Seminar: Ish Shah — The Only Scale You'll Need: The Lebesgue Integral

Nov 19, 2024 in Hill 705

Speaker: Ish Shah

Abstract: You have a meter-long pipe with a really funny property: at a rational distance from the endpoints (in meters), the linear density is 1 kg/m. Everywhere else, it is half of that. What is the mass of this pipe? Is this even defined? In this talk, we answer this question by introducing Lebesgue measure on the real line and the integral it gives rise to. We will then explore important theorems and further applications of Lebesgue integration, demonstrating how powerful of a tool it can be. If time permits, we will briefly discuss some details about the $L^{1}$ space of functions. A background in undergraduate-level real analysis is helpful, but not strictly necessary for understanding much of the material.

RUMA Seminar: Sean LeClair — Constructible Sets

Nov 12, 2024 in Hill 705

Speaker: Sean LeClair

Abstract: Can the size of the reals be determined solely by the axioms of ZFC? Sadly, the answer is no: the Continuum Hypothesis can neither be proved nor disproved in standard mathematics. We will go over the proof of the relative consistency of ZFC and the Continuum Hypothesis, as well as the effects of the Continuum Hypothesis on infinite trees.

Stephen Miller — Sphere Packing

Nov 7, 2024 in Hill 425

Speaker: Stephen Miller

Abstract: How many spheres can you fit in a box and what does that have to do with the Fourier transform? In 2016 these questions were fully answered in dimensions 8 and 24 using single-variable calculus. We’ll give an overview, including the tantalizing connection with number theory.

RUMA Seminar: Nilay Tripathi — Introduction to Homology Groups & Applications

Nov 6, 2024 in Hill 705

Speaker: Nilay Tripathi

Abstract: Algebraic topology is a field of mathematics that studies topological spaces through algebraic invariants. In this talk, we discuss homology groups, one of the most widely studied and important topics in modern topology. We will give an introduction to homology groups, some elementary computations, and some applications to see what we can learn from them.

Ian Jauslin — Entanglement, Quantum Non-locality, and Bell's inequalities

Nov 1, 2024 in Hill 705

Speaker: Ian Jauslin

Abstract: The world is not local! In the sense that one can design an experiment in which taking an action at one location can have an instantaneous effect on a system that is arbitrarily far away. This is rather surprising… In fact, it is a huge paradigm shift: for the better part of the twentieth century, most physicists operated under the assumption that the world is local. But there is now little room for doubt: experiments have been carried out that show that locality is, in fact violated. This came up as a theoretical possibility in a paper by Einstein, Podolsky, and Rosen, who noted that the current understanding of quantum mechanics violates locality. They concluded quantum mechanics needs to be completed in order to preserve locality. However, as Bell later discovered, this cannot be done: quantum mechanics cannot be made to be local. In this talk, we will discuss the reasons why.

RUMA Seminar: Liza Ter-Saakov — The p-adics and Other Local Fields

Oct 30, 2024 in Hill 705

Speaker: Liza Ter-Saakov

Abstract: Everyone knows about the real numbers, and many know that they are actually a completion of the rationals (meaning that every Cauchy sequence converges). Comparatively, the $p$ -adic numbers are hardly studied, even though they are also a completion of the rationals, and a very interesting one at that. This talk will introduce the $p$ -adics, where they come from, and some important theorems about them, as well as completions of other fields that arise in similar manners. These completions turn out to be what we call local fields!

Christopher Woodward — Triangles and More

Oct 24, 2024 in Hill 705

Speaker: Christopher Woodward

Abstract: I will talk about a few of my favorite formulas for triangles such as (i) formulas describing when triangles exist (ii) formulas describing their area (iii) generalizations to higher dimensions such as tetrahedra (iv) generalizations to higher dimensions such formulas describing the possible eigenvalues of a sum of matrices. The last is work with Allen Knutson and Terry Tao from two decades ago.

RUMA Seminar: Maxwell Goldberg — The Power of Gröbner Bases

Oct 23, 2024 in Hill 705

Speaker: Maxwell Goldberg

Abstract: Polynomials show up in many different disciplines and contexts, so being able to solve them is an important task. For example, in high school we learn the quadratic formula, and in linear algebra, we see how to solve linear systems of equations. How do we solve systems of multivariable polynomials? This question seems ugly, but it can be solved rather eloquently with Gröbner bases. In this talk, we will be exploring the world of classical algebraic geometry to develop some theory behind Gröbner bases and their profound implications. I won’t be assuming a background in algebra!

Play-Doh Social

Oct 17, 2024, 6:00pm in Hill 323

RUMA is having a Play-Doh social to destress, have fun, and explore topology!! Come to the math lounge (Hill 323) to socialize and relieve some stress from your Thursday evening, all while homotoping some Play-Doh! We will have some toys for the Play-Doh, too.

RUMA Seminar: Jay Patwardhan — The Lickorish-Wallace Theorem

Oct 16, 2024 in Hill 705

Speaker: Jay Patwardhan

Abstract: How do we classify objects in topology? This talk will take a leisurely walk through the world of manifolds and the classification of low dimensional objects. Ultimately, we will explain the role of knots in classifying 3-manifolds through the Lickorish-Wallace theorem.

RUMA Seminar: Trisha Kothavale — Tensor Categories

Oct 9, 2024 in Hill 705

Speaker: Trisha Kothavale

Abstract: Tensor categories arise naturally in many areas of representation theory and Lie theory, and they are often thought of as the categorical analog of a ring. There are various types of tensor categories, and studying them allows us to categorify a lot of the tools that are used in representation theory. We will define a tensor category and see how it gives rise to a categorical interpretation of the concept of the trace of a matrix.

RUMA Seminar: Nadya Belova — The Fundamental Group

Oct 2, 2024 in Hill 705

Speaker: Nadya Belova

Abstract: Using intuition and tools from both algebra and topology, the aptly named field of algebraic topology converts topological problems into algebraic ones. It’s a very pretty subject typically only covered at the very end of an undergraduate degree in mathematics (if at all!). This is a shame, as it has many exciting theorems and questions to explore. We will motivate an initial exploration of the subject by playing with its simplest object: the fundamental group.

Sheldon Goldstein — What is Quantum Theory?

Sep 18, 2024 in Hill 705

Speaker: Sheldon Goldstein

Abstract: Physicists believe that at a fundamental level our world is governed by quantum theory. But despite the fact that quantum theory has been around for almost a century, it remains extremely controversial, and there is little agreement among physicists about its interpretation. There is a good reason for this: it is not at all clear what the physical theory that is quantum theory actually says about the world.

Math Major Q&A Session

Sep 6, 2024 in Hill 705

Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have a professor, a current math graduate student, a graduated math major in tech, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come.

SPRING 2024

Charles Kenney — Introduction to the Probabilistic Method

Apr 25, 2024, 3:00pm in Hill 005

From the statistics of card and dice games, many people know that counting (combinatorics) is an important component of probability. But did you know the relationship goes both ways? Many combinatorial problems have only ever been solved by the probabilistic method, a way of using randomness to understand discrete structures. In this lecture I’ll introduce the core ideas of the probabilistic method and illustrate their use for some classical problems of combinatorics.

Shadi Tahvildar-Zadeh — Castles in the Air: Physical Theories and Their Mathematical Foundations

Apr 18, 2024, 5:30pm in Hill 525

Physical theories are good examples of Thoreau’s “castles built in the air”: fantastical structures dreamed up by visionary physicists, often with scant regard for logic or rigor. The task of laying the foundations for these castles invariably falls to the mathematicians, who proceed by first digging a very deep hole in the ground. In this talk I will give a couple of examples of research in mathematical physics, all done in collaboration with Rutgers undergraduates, that illustrate how math can not only provide that foundation, but also when taken seriously, can point to the path for further progress in theoretical physics.

Dewan Chowdhury — Exploring Geometric Flows: An Introduction to Curve-Shortening Flow

Apr 11, 2024, 6:10pm in Hill 705

Geometric flows represent a fascinating area of study in mathematics, blending geometry and analysis to understand the evolution of geometric objects over time. One of the most intuitive flows is the curve-shortening flow (CSF), in which a curve evolves by moving in the direction of its normal vector, with speed proportional to its curvature. In this talk, we will examine many of the concepts central to the theory of geometric flows through an exploration of the CSF. Only calculus is assumed.

Feng Luo — The story of circle packing

Apr 4, 2024, 5:00pm in Hill 005

A circle packing in the plane is a collection of round disks with disjoint interior. I will tell you some of the fascinating theorems concerning circle packing and their relationship to modern 2-dimensional conformal geometry (e.g., the Riemann mapping theorem and the uniformization theorem). These results relate combinatorics (planar graphs) with geometry. The contributions of Koebe, Thurston, Andreev, and He-Schramm will be discussed.

Alex Kontorovich — Applications of Thin Orbits

Mar 28, 2024, 6:00pm in Hill 705

We will discuss some recent advances in our understanding of various problems at the intersection of dynamics, geometry, and number theory.

Pi Day

Mar 7, 2024, 6:28pm in Hill 703

Pi day is coming up over spring break, so we’re celebrating a week early! There will be pie and tea!

Zheng-Chao Han — A glimpse into some mathematics behind soap films and soap bubbles

Mar 5, 2024, 4:00pm in Hill 705

Soap films and bubbles appear abundantly in daily life. We will explore some mathematics used to describe these commonly encountered yet fascinating geometric objects. What are the underlying laws in nature that govern their formation? How are they studied mathematically? What kind of mathematical tools can be brought to bear on these problems? Some of the questions have relatively easy answers, but these are challenging and not so easy questions. In this talk I will try to provide a glimpse into a small collection of such problems and discuss some of the mathematical background for tackling such problems.

Sam Spiro — Solving Math Problems with Anime

Feb 28, 2024, 6:00pm in Hill 705

The minimal superpermutation problem asks how many symbols a string must have in order to contains all $n!$ permutations of ${1, \dots, n}$ as substrings. The Haruhi problem asks for the most efficient way to watch a certain anime. Remarkably, these two problems turn out to be equivalent. Even more remarkably, the anime community has made substantially more progress than the mathematicians! In this talk, we discuss an improved lower bound to the minimal superpermutation problem discovered on 4chan, as well as the anime theoretic motivation for this problem. As time permits, we also discuss a problem involving Rock, Paper, Scissors which was also inspired by anime. The talk will be entirely self contained and assume no prior knowledge of anime.

Kristen Hendricks — A first look at knots and symmetries

Feb 23, 2024, 4:00pm in Hill 705

A mathematical knot is a simple object — take a piece of string, tie it up however you like, and glue the ends together so you can’t untie it. But these deceptively easy objects to describe and fiddle with are key to understanding deep geometric questions, many not nearly so accessible. We’ll introduce knots and consider some possible measures of how complicated a knot is, before turning our attention to one of my favorite topics, possible symmetries of knots. In the end, we’ll see how different types of symmetry have wildly different relationships with how “complicated” the knots involved are.

Kimberly Weston — Using probability and differential equations to solve social and economic problems

Feb 6, 2024, 6:30pm in Hill 705

What models make good models for asset prices? Financial equilibrium theory provides one such answer — good models are those that are formed by equating supply and demand. Studying financial equilibria requires the use of several areas of math, and this talk will introduce you to some of the related areas and terminology. Financial equilibria can be characterized by solutions of systems of backward stochastic differential equations (BSDEs). We will discuss what BSDEs are, the structure of relevant BSDE systems, and how BSDE solutions can be useful in solving social and economic problems through financial equilibria.

Simon Thomas — Large Cardinals and The Axiom of Determinancy

Jan 31, 2024, 5:30pm in Hill 705

Let $𝒩$ be the space of infinite sequences of natural numbers $(a_{0}, a_{1}, \dots, a_{n}, \dots)$ ; and for each subset $X \subseteq 𝒩$ , let $G (X)$ be the two player game in which:

Players I and II alternately choose natural numbers $b_{n}$ , $c_{n}$ respectively;
Player I wins if and only if $(b_{0}, c_{0}, b_{1}, c_{1}, \dots, b_{n}, c_{n}, \dots) \in X$ .

A subset $X \subseteq 𝒩$ is said to be determined if one of the two players has a winning strategy in the game $G (X)$ . In this talk, we will consider the question of which subsets $X \subseteq 𝒩$ are determined. As we will see, this question is intimately related to existence of large cardinals; i.e. cardinals that are so large that they can prove the consistency of their own nonexistence.

Tom Benhamou — Filters and Ultrafilters - a mathematical approach to smallness and largeness

Jan 24, 2024, 4:00pm in Hill 525

In this talk, we will present the notion of filters and ideals along with several key examples. These notions provide an abstract framework to different notions of “small” and “large” across mathematics. Then we will present the notion of ultrafilters, some of the most fundamental results in the area, and classical applications, such as the ultrapower constructions. If time permits, we will also show some applications to infinite combinatorics.

FALL 2023

Paul Ellis — Hyperbinary Numbers and the Calkin-Wilf Tree

Dec 11, 2023, 3:00pm in Hill 323

Hyperbinary is the same as binary except you are allowed to use the digit ' $2$ '. So for example, $21$ and $101$ both represent the number five. It turns out that if you make a sequence where $f (n)$ is the number of ways to represent the number $n$ in “hyperbinary” form, then the sequence $f (n) / f (n + 1)$ will list out every positive rational number with no repeats! Come see why!

Estimathon

Dec 1, 2023, 5:00pm in Hill 705

Time: 5:00pm to 7:00pm

Brought to you by RUMA and AWM

This is a team based contest that combines trivia, game theory, and mathematical thinking. Teams have 30 minutes to work on a set of 13 estimation problems, the winning team being the one with the best set of estimates. Come join us for a fun time and food!

Dinner will be provided, and (limited) T-shirts will be available!

Benjamin Gunby — Generating Functions

Nov 17, 2023, 6:00pm in Hill 705

Encoding a sequence as a power series seems like a strange thing to do on the surface. However, it turns out to be widely applicable in counting, approximating, and comparing sequences. We will talk about how to count things like the Fibonacci sequence, zigzag permutations, Catalan paths, and more using this powerful tool.

Dima Sinapova — Infinity and the Powerset Function

Oct 27, 2023, 3:00pm in Hill 705

Set theory is the study of infinite objects. After Cantor discovered that the cardinality of the reals is strictly bigger than the cardinality of the natural numbers, a natural question emerged: is there anything in between? The continuum hypothesis (CH) states the answer is no, and it became the famous Hilbert’s first problem. CH was shown to be independent of the usual axioms of mathematics (ZFC) by works of Gödel (in 1940) and Cohen (in 1964). That means that neither CH, nor its negation, can be proven from the usual axioms. Since then, modern set theory investigates ZFC constraints (i.e. “what is necessary”) versus consistency results (i.e. “what is possible”). I will survey some classical results and then discuss recent developments in the study of infinite sets.

Doron Zeilberger — Gambling on Gambling

Oct 9, 2023, 7:00pm in Hill 705

The mathematics of gambling is not always easy, but I will show that it is a good gamble to study the mathematics of gambling.

Robert Dougherty-Bliss — Proofs by Example

Oct 2, 2023, 7:00pm in Hill 705

You have been told that mathematical statements require “rigor” and “proof.” This is an overstated belief. Broad classes of combinatorial, algebraic, and even analytic statements can be established purely based on empirical evidence. These techniques are not as well-known as they should be. We will share in the joy of proof by example with a series of problems about sums, integrals, and recurrences.

Michael Kiessling — Optimal Social Distancing, and such: a Mathematical Perspective

Sep 11, 2023, 6:00pm in Hill 705

At the height of the pandemic it was advised to practice social distancing. The rule was to leave a certain distance of closest approach between you and the next person. But how many people then fit into a room of given square footage, and shape? Mathematicians have studied such optimization problems and many of their variations for a very long time, for various reasons and under various names. They are surprisingly rich, interesting, and they are difficult to solve, many of them of the type “NP hard”. One of these problems gave rise to Stephen Smale’s 7th problem worth the attention of mathematicians in the 21st century.

In my talk I will introduce you to some of the mathematical questions that are being asked, some of the surprising answers that have been found, and to the challenges one faces.

Math Major Q&A Session

Sep 8, 2023, 7:00pm in Hill 705

Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have a professor, a current math graduate student, a graduated math major in tech, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come.

SPRING 2023

William Chen — Groups, numbers, and moduli stacks of elliptic curves

Apr 21, 2023, 4:30pm in Hill 525

Let $G$ be a finite group. Given a generating pair $(x, y)$ of $G$ , consider the three transformations sending $(x, y)$ to $(y, x)$ , $(x, y^{- 1})$ , and $(x, x y)$ . Two generating pairs of $G$ are said to be Nielsen equivalent if one can be obtained from the other by applying a sequence of such transformations (or their inverses). A natural problem, dating back to the 50’s, asks for a description of the Nielsen equivalence classes of $G$ . While simple to state, the problem is surprisingly deep, and touches on topics ranging from number theory to topology and algebraic geometry. An interesting case is when $G = S L (2, p)$ , in which case Nielsen equivalence classes are related to Markoff numbers, and the moduli of $S L (2, p)$ -covers of elliptic curves. In this talk we will try to sketch some of these relationships.

Short Undergraduate Special (SUS) Lectures

Apr 7, 2023, 7:30pm in Hill 705

Why is the Earth round? How do we stop climate change? What even is an RU Screw? In honor of April Fools’ Day, we will have attendees and board members alike answer the deep questions plaguing the world in the most serious way possible — through mathematics!

Dr. Corrine Yap — Intersections of Combinatorics and Statistical Physics

Mar 22, 2023, 5:30pm in Hill 705

This talk will introduce how statistical physics (also known as statistical mechanics) can provide a useful perspective on problems in combinatorics and computer science, and vice versa. In recent years, mathematicians have tackled questions about phase transitions in particle systems by analyzing graph structures, and produced counting and sampling algorithms by using physics-inspired polymer models. We’ll start with an introduction to some combinatorially relevant models such as the Ising, Potts, and hardcore models and explore a variety of questions and tools that lie in the intersection of combinatorics and statistical physics.

Paul Ellis — The Four Numbers Game

Mar 9, 2023, 7:30pm in Hill 705

Write any four numbers at the corners of a square. Next, write the difference of each pair at the midpoint of the corresponding side. Connect these midpoints to make a smaller square inscribed in the original one. Repeat the process. What happens? Always? Can you prove it?

Michael Beals — Generalized Functions and Generalized Derivatives

Feb 3, 2023, 5:30pm in Hill 705

How do we make sense of the derivative of a discontinuous function? How do we make sense of the sum of a non-converging series of functions? How do we take half of a derivative of a function? And why would we want to do any of these things? The answer to that last question is, perhaps, because the physicists and mathematicians of the nineteenth century needed the results. And the resulting theory of generalized functions changed analysis in the twentieth century. We will consider how it all works.

FALL 2022

Priyadip Mondal — Tetrahedral decomposition of hyperbolic knot and link complements

Dec 7, 2022, 7:30pm in Hill 705

The study of geometry and topology largely rests on the study of geometric structures of manifolds. In this talk, we will concentrate on hyperbolic structures of 3-manifolds. A hyperbolic 3-manifold is a geometric object which locally looks like the hyperbolic 3-space (i.e., the upper-half space), and consequently, they inherit a hyperbolic metric. One property that significantly distinguishes the (hyperbolic) geometry in the hyperbolic 3-space from the standard Euclidean geometry is that, unlike the standard Euclidean geometry, the shortest distance between points is not always given by straight line segments between them. The hyperbolic 3-manifolds that we will primarily see in our talk are the hyperbolic knot and link complements. The hyperbolic structure of a hyperbolic knot or link complement is often best understood in terms of its decomposition into ideal tetrahedra (if it has one). This talk will be devoted to exploring hyperbolic knot and link complements through such tetrahedral decompositions. If time permits, we will also talk about some 3-manifold softwares that people use to understand such tetrahedral decompositions. (Since the talk will be geometric in nature, it will be accompanied by a lot of pictures, and no prior knowledge of hyperbolic geometry at any level will be assumed.)

Lars Ruthotto — Neural Network Approaches for High-Dimensional Optimal Control

Dec 5, 2022, 5:00pm

This talk presents recent advances in neural network approaches for approximating the value function of high-dimensional control problems. A core challenge of the training process is that the value function estimate and the relevant parts of the state space (those likely to be visited by optimal policies) need to be discovered. We show how insights from optimal control theory and – in the stochastic case - the fundamental relation between semi-linear parabolic partial differential equations and forward-backward stochastic differential equations can be leveraged to achieve these goals. To focus the sampling on relevant states during neural network training, we use the Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. Our approaches can handle both stochastic and deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem.

We will present several numerical experiments for deterministic and stochastic problems with state dimensions of about 100 and compare our methods to nonlinear optimization approaches and existing approaches.

Robert Dougherty-Bliss — When does a sum have an answer?

Nov 21, 2022, 7:00pm in Hill 705

Many sums have an answer, like $1 + 2 + \dots + n$ . How do we find these answers? Do all sums have answers? For example, does $1! + 2! + \dots + n!$ have an answer? What is an “answer” anyway?

These questions are analogous to what you learn to ask in Calculus II: which integrals have answers, and how do you find them? Fortunately, the state of affairs for summation is much nicer. Unfortunately, it is less known to most people! I will try to rectify that by showing us some summation techniques and answering a few of the above questions.

Eilidh McKemmie — The Galois Group of a Random Polynomial

Nov 8, 2022, 6:00pm in Hill 525

A Galois group is an algebraic structure which we can assign to a polynomial and gives us information about the polynomial. For example, Galois theory explains why we have a quadratic equation to find roots of quadratic polynomials, but no similar equation to find roots of polynomials of degree 5 or more. We will consider some different ways of picking a random polynomial and look at what the Galois group of a random polynomial is likely to be.

Yi-Zhi Huang — Representation Theory of Geometric Objects and Quantum Theory

Oct 28, 2022, 6:00pm in Hill 525

Quantum mechanics and quantum field theory play a fundamental role in physics. Mathematically, they can be viewed as the representation theory of geometric objects, that is, the theory about how to study geometric objects using vector spaces and linear transformations. I will give an introduction to this mathematical approach to quantum theories.

Paul Ellis — Modular Origami and the Platonic Solids

Oct 27, 2022, 6:00pm in Hill 703

We will be working mostly with the Sonobe units. By folding a bunch of these units and fitting them together, we can construct some interesting shapes, including all the platonic solids. If time permits, we will also talk about why there are only five of these. I will bring plenty of other interesting origami models to share, in case we have more time.

Ian Jauslin — Statistical Mechanics

Oct 12, 2022, 7:00pm in Hill 705

Many of us were taught that the world around us is made of invisible, microscopic particles such as molecules, electrons or atoms, and that the physical laws that bind them explain the observations we make on a human scale: for instance, temperature is explained as molecules moving, rotating and vibrating; electrical currents are accounted for as electrons zipping through a metal. But how much do we really understand about the relationship between the invisible microscopic world and the observable macroscopic one? Can we formulate natural mathematical models for the microscopic world, for which we can prove macroscopic physical laws?

Dennis Kriventsov — From optimization to free boundaries

Sep 27, 2022, 6:00pm in Hill 525

Calculus of variations is an area of analysis concerned with minimizing energies, generally over large collections of objects like functions or sets. I will describe how with some ideas from Calculus I (and maybe some hidden machinery) you can successfully find minimizers to such energies and then try to understand what they look like. Depending on the energy, this leads directly to partial differential equations and to free boundary problems (which are like partial differential equations satisfied by set boundaries). These subjects contain the most successful analysis developments of the past century—but also some basic questions which we just do not know how to answer.

Michael Beals, Stephen Hu, Adam Jamil, Brittany Gelb — Math Major Q&A Session

Sep 16, 2022, 7:00pm in Hill 525

Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have Professor Beals, a current Rutgers graduate student, a math major now in tech, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come. Don’t miss out on the first event of the semester!

SPRING 2022

Natasa Sesum — Ancient Solutions in Geometric Flows

Apr 27, 2022, 5:30pm

We will discuss the Ricci flow, importance of ancient solutions for the flow and classification results of ancient solutions in certain cases.

Recording password: WswS?.F1

Recording

Colin Fan — Holomorphic Nullstellensatz

Apr 20, 2022, 7:00pm in Hill 525

Hilbert proved his Nullstellensatz (German for “theorem of zeros”) in 1893, establishing a duality (1-1 correspondence) between geometry and algebra in the setting of polynomials. In particular, the Nullstellensatz allows us to use algebra (easier) to study geometry (harder).

Being more ambitious, we can ask whether or not we can establish a duality between geometry and algebra in the setting of holomorphic functions. This question is intrinsically interesting, and is further motivated by the need to study complex manifolds. Thankfully, Rückert, in 1931 established his analytic Nullstellensatz, being the first one to introduce algebraic tools to complex analytic geometry.

In this talk, I hope to keep the prerequisites at a minimum with little to no proofs and highlight the beauty of both the Hilbert and the Rückert Nullstellensatz, as well as the difficulties one may face in the study of holomorphic functions of several variables.

James Propp, UMass-Lowell — Tiling problems, old and new

Mar 30, 2022, 3:30pm

A classic puzzle is to show that an $8 \times 8$ square cannot be tiled by $1 \times 2$ and $2 \times 1$ rectangles if two opposite $1 \times 1$ corner-squares are removed. A less famous puzzle is to count how many tilings there are if you DON’T remove those corners. This is an example of a dimer problem, first considered by physicists, at the intersection of graph theory and enumerative combinatorics. I’ll review some highlights in the theory of dimers, displaying elegant formulas for the number of tilings and striking pictures of the kinds of long-range order that can arise spontaneously.

Then I’ll discuss beautiful work of Conway, Lagarias, and Thurston applying combinatorial group theory to the study of tiling problems, yielding criteria for distinguishing between doable and non-doable tiling problems. I’ll apply the method to obtain a surprising result about a class of tiling problems that are ALMOST never doable.

Finally, bringing the two strands together, I’ll discuss my recent empirical work counting tilings related to trimer models. None of the methods that I know of for obtaining exact formulas for dimers apply here, yet the data show a wealth of patterns suggesting that there are deep theorems to be proved. Especially mysterious are some patterns involving the 2-adic numbers.

This talk should be understandable to undergraduates. No background in graph theory, enumerative combinatorics, combinatorial group theory, or $p$ -adic analysis will be required, and you won’t need to know what a dimer or a trimer is when you arrive (though you will when you leave).

Sangjun Ko — A topological proof of Euclid's Theorem

Mar 23, 2022, 7:30pm in Hill 705

Euclid’s Theorem, which states that there are infinitely many primes, is a classic theorem whose proof is studied by every student who is beginning to learn about proofs. As an undergraduate, Abel Prize winner Hillel Furstenberg came up with a different proof which takes ideas from point-set topology. We will give a quick introduction to basic point-set topology and present Furstenberg’s proof, and, if time permits, discuss some properties of the so-called “evenly spaced integer topology” as well as other similar constructions.

Konstantin Mischaikow — We have Data and Computers, why do we need Math?

Mar 2, 2022, 7:30pm in Hill 705

With today technology we can collect massive high dimensional sets of data from experiments and generate massive high dimensional sets of data numerically, but at the end of the day it is still a finite set of points. Assume that we are trying to understand a continuous process that generated the data. For example a process that can be modeled by a differential equation. In this case we probably want to be able to extract a continuous function, which raises the question how can one go from finite data to continuum objects with some sense of certainty.

To be a bit more concrete this talk will focus on how to go from finite data to the identification of a periodic orbit using algebraic topology.

Prerequisites for this talk: Calculus and Linear Algebra

Brian Pinsky — Weakening the Axiom of Choice

Feb 23, 2022, 7:00pm in Hill 705

You and your countably many friends are being held captive by an evil set theorist. Tomorrow, they will give each of you a hat that is one of countably many colors (multiple people may have the same color). Then you will all simultaneously try to guess your hat’s color. You can see everyone else’s hat, but not your own. You lose if infinitely many of your friends guess wrong.

Assuming the axiom of choice, your friends have a winning strategy. However, assuming the axiom of choice lets the evil set theorist cheat and banach-tarski the finite set of friends who guess wrong into infinitely many friends, and you still lose.

Is there an axiom that is strong enough for your friends to win, but too weak for the set theorist to cheat? In this talk, we’ll explore strictly weaker versions of the axiom of choice, and hopefully find an answer.

Ian Coley — What is homotopy theory?

Feb 3, 2022, 7:30pm in Hill 705

Multivariable calculus naturally leads to the study of manifolds, spaces that look like our first friend $ℝ^{n}$ . But what if we went the other way, where we stopped thinking about rigid spaces and starting thinking more floppily? I’ll introduce you to homotopy theory: its origins, some cool little tools, and how it ties in with both algebra and geometry.

FALL 2021

Yunus E. Zeytuncu (University of Michigan - Dearborn) — Spectral Geometry of CR manifolds

Dec 6, 2021, 7:00pm

In this talk, we will look at the relation between the spectrum of differential operators and the geometry of underlying manifolds. In particular, we will talk if one can hear the shape of CR manifolds. No prior knowledge of differential equations and geometry is necessary; we plan to construct everything from scratch.

Recording password: *$c1Ysx=

Recording

Olena Kozhushkina (Ursinus College) — On Everywhere Surjective Functions

Dec 2, 2021, 7:00pm

Analysis is full of examples verifying some kind of pathological property — continuous nowhere differentiable functions; linear discontinuous functions; differentiable nowhere monotone functions, and others. In this talk, we will focus on another interesting example — everywhere surjective functions.

A function $f : ℝ \to ℝ$ is called everywhere surjective, if for every non-empty interval $(a, b)$ , the image $f ((a, b))$ is the entire $ℝ$ . Such function is doomed to be discontinuous at every real number, and in particular, provides a “strong” counterexample to the converse of the Intermediate Value Theorem. Namely, $f$ satisfies the Intermediate Value Property — for any two numbers $x_{1} < x_{2}$ in $ℝ$ , if $y$ is a real number between $f (x_{1})$ and $f (x_{2})$ , then there exists a real number $c$ in $(x_{1}, x_{2})$ such that $f (c) = y$ — but $f$ is not even close to being continuous.

The existence of everywhere surjective functions was first noticed by H. Lebesgue. In fact, there are plenty of such functions. The set $S$ of everywhere surjective functions is $2^{c}$ -lineable; that is, $S \cup {0}$ contains a vector subspace of the largest possible dimension, $2^{c}$ .

Recording

Nicholas Scoville (Ursinus College) — Discrete Morse theory as an introduction to topology

Nov 16, 2021, 7:00pm

Like algebra, geometry, and number theory, topology is one of the main branches of mathematics today. Yet unlike algebra, geometry, and number theory, most people are not aware of what topology is. Because of its abstract nature, explaining the main concepts found in topology can be challenging. In this talk, we will discuss some of the main ideas in topology via an introduction to discrete Morse theory. Discrete Morse theory is a tool used in topology, but viewing this tool from the right point of view can shed great light on the important ideas in topology. We will look at two original research problems in discrete Morse theory that I have worked on with undergraduate students with many pictures, animations, and no equations.

Recording password: BP5cwg&%

Recording

Daniel Krashen — Algebraic structures and the algebraic structures on which they live

Oct 20, 2021, 7:00pm in Hill 525

Mathematical objects often have two lives. On the one hand, they are something like creatures living in complex ecosystems, but on the other hand, they can also be like the planets on which these creatures live. Sometimes these “planetary” algebraic structures can be given additional geometric or topological features, giving a range of new tools from other mathematical disciplines.

In this talk I will try to push this metaphor, and use it as a lens through which to view some open problems in the theory of quadratic forms and division algebras. In particular, we will see how these problems connect to ideas in geometry, such as the (in)famous “hearing the shape of the drum” problem.

Purdue Mathematics Society intercollegiate math social

Oct 16, 2021, 7:00pm

The event will include:

Games/Icebreakers
Research Talks
Problem Solving Sets (where people can present solutions in groups)
Presentation of Solutions

Louis Gaudet — The circle method in analytic number theory

Sep 27, 2021, 6:00pm

Analytic number theory is a part of mathematics that studies properties of the integers and the prime numbers via tools from analysis (like calculus). The circle method, first developed by Hardy and Ramanujan in 1918, is an amazing tool in this field for studying “Diophantine problems” (i.e. solving equations with whole numbers). We’ll take a look at the basic ideas of the method in the context of some famous examples, like the Goldbach problem about sums of two primes.

Recording

Harlin Lee — Topics from Spectral Graph Theory

Sep 21, 2021, 7:00pm

Spectral graph theory is the merging of linear algebra and graph theory to provide a novel understanding of graph structure in relationship to various matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. In this lecture, we will develop the mathematical machinery of SGT and explore some of its most spectacular results.

Math Major Q&A Session

Sep 8, 2021, 7:00pm

Speakers: Michael Beals, John Kerrigan, Brooke Ogrodnik, Aaron Scheiner

Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have Professor Beals, Professor Kerrigan, a former graduate student now in industry, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come. Don’t miss out on the first event of the semester!

SPRING 2021

Fioralba Cakoni — Given the answer how do you find the question?

Apr 20, 2021, 6:00pm

Inverse Problems are problems where causes for a desired or an observed effect are to be determined. They arise in many real life applications. Mathematical methods for solving inverse problems are rich and challenging. In this presentation we will discuss some toy examples and some more realistic ones to get a feeling for mathematics behind inverse problems.

Recording

Matthew Issac — A Tour of Universal Properties

Apr 20, 2021, 6:00pm

In this lecture, we will examine “universal properties,” a concept which lurks in the background of many math classes. Moreover, these special properties can be used as definitions, once we have shown what we have defined indeed exists. We will begin with familiar examples (the empty set) and move onto more exotic constructions (the tensor product). Moreover, we will give some formal notion to what exactly it means to satisfy a “universal property,” and what are the consequences of this.

Brandon Gomes — An Introduction to Logic and Formal Reasoning

Mar 30, 2021, 7:00pm

Logic is the foundation on which all of mathematics, the sciences, and philosophy are built. In this talk we explore what logic is and discuss how to study logic as its own branch of mathematics. We will cover historical and modern theories of logic as well as the intersection of logic with computer science and computability theory.

Eric Ling, Department of Mathematics — Spacetime Singularities in Black Holes

Mar 12, 2021, 6:00pm

In 2020 Sir Roger Penrose (a mathematician) won the Nobel Prize for his theoretical work on spacetime singularities in black holes. In this talk we give an introduction to the differential geometry techniques used to prove his theorem.

Recording password: !p16tM4b

Recording

Mariusz Mirek, Department of Mathematics — Fourier Analysis in Additive Combinatorics

Mar 2, 2021, 7:00pm

We will discuss the proof of famous Roth’s theorem asserting that every subset of integers with non-vanishing density contains infinitely many arithmetic progressions of length at least three.

Doron Zeilberger, Department of Mathematics — Asking as Few Questions as Possible

Feb 24, 2021, 6:30pm

Any fool can figure out the answer if he or she (or it) is allowed to ask many questions, even if the answer to each question is Yes or No. But figuring out how to ALWAYS get the right answer in as few questions as possible, is not so easy.

Recording password: 5nm%BWsG

Recording

Nicholas McConnell, RU '21 — On the Matrix Exponential

Feb 11, 2021, 7:00pm

The exponential function is an important function in mathematics. Among other reasons, $y = e^{a t}$ solves the differential equation $\frac{d y}{d t} = a y$ , $y (0) = 1$ . However, using both differential equations and linear algebra, one can take an $n \times n$ matrix $A$ , and establish a matrix $y = \exp (t A)$ that depends on a parameter $t$ . Then the differential equation $\frac{d x}{d t} = A x$ , $x (0) = x_{0}$ (where $x \in ℝ^{n}$ ) has solution $x = \exp (t A) x_{0}$ .

Linear operators (which are not typically thought of using matrices) can also be exponentiated, and whenever $T$ is a linear operator, $\exp (t T) x_{0}$ is the solution to $\frac{d x}{d t} = T x$ , $x (0) = x_{0}$ . The main example of this is, if $D = \frac{d}{d x}$ then $\exp (t D) f (x) = f (x + t)$ . We will see the definition, basic properties and use of exponentiation of linear operators and matrices.

A Lie group is a smooth manifold which has a group structure where the multiplication and inverse are smooth maps (e.g., $ℝ^{n}$ under addition; $T^{n}$ (where $T = {z \in ℂ : | z | = 1}$ ) under multiplication; $GL (n, ℝ)$ under multiplication; $GL (n, ℂ)$ ; $SL (n, ℝ)$ ; $SL (n, ℂ)$ ; $O (n)$ ; $SO (n)$ ; $U (n)$ ; $SU (n)$ ). If $G$ is a Lie group, the tangent space $𝔤$ at the identity is a Lie algebra, as left-invariant vector fields are closed under the Lie bracket and correspond bijectively to vectors at the identity. There is an exponential map $\exp : 𝔤 \to G$ such that for each $X \in 𝔤$ , the map $t \mapsto \exp (t X)$ is the unique smooth homomorphism $f : ℝ \to G$ such that $f^{'} (0) = X$ . For $GL (n, ℝ)$ , $GL (n, ℂ)$ , and their subgroups, this is merely the usual matrix exponential.

Recording password: .PmmOdV9

Recording

Roberta Shapiro, Graduate Student — Mugs and Donuts: An Introduction to Surface Topology

Feb 3, 2021, 8:00pm

What is a surface? What are all* the types of surfaces that exist in the wild? How do we tell them apart? We will answer these questions and more** as we take a first dive into the surface topology.

*not all

**BYOQ: bring your own questions

Recording password: XXL!h.32

Recording

Board Game Night with RUCP

Jan 29, 2021, 8:00pm

Join us for Among Us, Skribbl.io, and Codenames. The event will be hosted over discord.

FALL 2020

RUMA Q&A Session

Dec 12, 2020, 4:00pm

Speakers:

Robert Beals, Chair of the Undergraduate Honors
John Kerrigan, Part Time Lecturer
Brooke Logan, Department of Mathematics, graduate student
Adam Jamil, RU ’21, Mathematics and Computer Science

This will be a talk all about the math major and the different paths it can lead you on. With perspectives from a wide variety of mathematical perspectives, come prepared with questions!

Nick Backes, Graduate Student — p-adic Numbers

Nov 30, 2020, 7:00pm

The rational numbers are fantastic; however, they have gaps. You can make sequences of rational numbers which should converge, but they do not converge to a rational number. Thus, we introduce the real numbers as a completion of the rational numbers. However, convergence of a Cauchy sequence depends on a chosen norm (or absolute value, if you prefer). As it turns out, there is more than norm on the rationals, and if the rationals are completed with respect to a different norm, then we obtain the $p$ -adic numbers. The $p$ -adics are somehow a different way to fill the gaps between rational numbers. The $p$ -adics have unexpected and beautiful geometric, analytic, and number-theoretic properties.

Recording password: ya0x^Q8i

Recording

Konstantin Matveev, Department of Mathematics — A Surprising Journey of Longest Increasing Subsequences

Oct 8, 2020, 7:00pm

I will talk about an eventful story of longest increasing subsequences. It will touch on a combinatorial geometry problem that has led to a 68 years long marriage, on computer experiments run by one of the key participants of the Manhattan project, on representation theory of symmetric groups, on random matrices, and on statistical physics.

Recording password: k5d?&p8%

Recording

Simon Thomas, Department of Mathematics — The Axiom of Choice

Sep 30, 2020, 6:30pm

In this talk, I will discuss the Axiom of Choice and its striking consequences.

Recording password: MK4Md^*H

Recording

Kristen Hendricks, Department of Mathematics — Classical and Modern Invariants of Knots

Sep 23, 2020, 7:00pm

We’ll give a brief introduction to what knot theory is and why you might be interested in it, and talk about some classical invariants of knots and what they detect. We’ll then introduce a modern invariant called Heegaard-Floer knot homology from the early 2000s, and talk about its properties and its relationship to classical invariants.

Recording password: t!aeE9p!

Recording