• November 25: Geunyoung Kim

  • Title: High-dimensional contractible manifolds

  • Abstract:  In 1961, Mazur constructed a contractible, compact, smooth 4-manifold which is not homeomorphic to the standard 4-ball, using a 0-handle, a 1-handle and a 2-handle.  In this talk, for any integer n ≥ 2, we construct a contractible, compact, smooth (n+3)-manifold which is not homeomorphic to the standard (n+3)-ball, using a 0-handle, an n-handle and an (n+1)-handle. The key step is the construction of an interesting knotted n-sphere in Sn × S2 generalizing the Mazur pattern. As a corollary, for any integer n ≥ 2, there exists a smooth involution of Sn+3 whose fixed point set is a non-simply connected homology (n + 2)-sphere.

• November 18: Conor McCoid

  • Title: Mozart Ex Machina: A toy example of an artificial neural network

  • Abstract: Machine learning is a major new forefront in mathematics. Motivated to learn more on the topic and to prototype new ideas, I developed a small research project to construct simple neural networks out of publicly available data. The first stage of this project uses the music of Wolfgang Amadeus Mozart to build a random forest classifier. This talk explores the steps I took to code this classifier, some of the theory behind random forest models, and the hurdles that I needed to overcome. The talk is intended to be introductory, with no background knowledge assumed. If you would like to code along with me, you will need Python as well as the libraries scikit-learn and music21 and the files downloaded from http://www.piano-midi.de/mozart.htm.

• November 11: Dylan McGinley

  • Title: Introduction to Ricci Flows

  • Abstract: I plan to informally introduce some of the basics of the Ricci flow, including short time existence, singularity formation and its relation to some classical problems in geometry. This will be an informal talk, so feel free to ask any questions you might have.

• November 4: Ilgwon Seo

  • Title: Complete Proofs of Gödel’s Incompleteness Theorems

  • Abstract: In 1920s, One of the most influential mathematicians, David Hilbert proposed to formulate mathematics with axioms. In 1931, however, one of the results put a limitation on his program. This limitation is called Gödel’s Incompleteness theorem, which states our axiomatic system cannot prove its own consistency and is not even complete. In this talk, we will explore the key concepts and methods leading to this theorem—including the structure of recursive functions, encoding techniques, and logical formalizations—that collectively demonstrate the limits of formal systems in mathematics.

• October 24: Leona Tils

  • Title: Sign patterns and the nSMP

  • Abstract: Our project involves seeing which sign pattern matrices, matrices that have either +,- or 0 entries, have the Non-Symmetric Strong Multiplicity Property (nSMP). The goal of our project is to characterize nxn sign patterns which require the nSMP, those which allow the nSMP and those which do not allow the nSMP. We have managed to categorize all the matrices of order 2 and order 3 matrices so far.

• September 23: Hannah Nardone & Kyle Sung

  • First Speaker: Hannah Nardone

    • Title : Densities of Bounded Primes for Hypergeometric Series

    • Abstract: In this talk we will introduce hypergeometric series as solutions of a certain differential equation. These functions arise all throughout mathematics and physics.We will discuss when the solutions of these equations can be reduced modulo primes, a question which has applications in number theory. In certain cases the answer to this questions has a precise formula, and this project focuses on finding an analogous formula in a related case.

  • Second Spearker: Kyle Sung

    • Title : Learning How Machines Learn

    • Abstract: Machine learning is changing the world, so let’s learn about its theory and applications. We briefly review linear regression and discuss fundamental questions motivating deep learning: what are the “best” models and how do we find them? Then, we’ll discover how combining linear regressions and nonlinear “activation” functions can form a neural network (NN): a type of function inspired by the makeup of our brains.We conclude with some mathematical proofs and real-world results to learn more about NNs. This talk should be accessible to most second-year undergrads: first year multivariable differential calculus and linear algebra are handy prerequisites, but no knowledge of machine learning nor statistics is assumed. For those with more ML experience, some NN results may still be interesting.

• September 19:  Emma Coates

  • Title: Combined effects of heterogeneous transmission and disparate vaccination rates

  • Abstract: How can mathematical modeling be used as a powerful tool in public health? In this study, we present how deterministic models can be used to inform public health policies with a specific example of disparities observed in SARS-CoV-2 seroprevalence data and vaccination efforts to address the observed disparities. Ma et al. in 2021 modeled the impact of racial and ethnic disparities on COVID-19 epidemic disparities by fitting structured compartmental models to early outbreak seroprevalence data stratified by race and ethnicity from New York City. Ma et al. used these mathematical models to investigate whether disparate seroprevalence could be explained by differences in per-contact susceptibility or by differences in contact rates and patterns. Using the Ma et al. paper as a guide, we investigate how vaccine uptake combines with existing transmission disparities in modeled outcomes by developing a mathematical modeling framework for quantify potential effects of interventions aiming to reduce disparities through vaccination efforts.

• April 11: Emma Naguit

  • Title: Rainbow Connection in Graphs

  • Abstract: The concept of rainbow connection in graphs arose naturally as a solution to weaknesses in security systems discovered in the early 2000s. To introduce this topic, we will first discuss some basic graph theory ideas, then the notion of being rainbow-connected, as well as some examples, results in further research and other useful applications.

• April 4: Dylan McGinley

  • Title: Ricci flow on surfaces with a view toward uniformization

  • Abstract: Ricci flow has famously been applied in three dimensions to prove the Poincare conjecture and shed new light on 3D geometry. In this talk, I will explore to what extent these new tools may be applied to the classical problem of geometry in two dimensions. Many of the theorems initially proved in the turn of the 20th century can now be approached from the methods of Ricci flow. That we can derive such results from the Ricci flow indicates that it is in fact interesting in all dimensions and, indeed, the 4D case is now being pursued at the cutting edge of research.

• March 28: Kate Tretiakova

  • Title: Unraveling Mysteries: A Journey into Sangaku Geometry

  • Abstract: Embark on a captivating journey into Sangaku Geometry with our casual math seminar this week. Discover the rich history and cultural significance of these Japanese geometric puzzles, from their origins in the Edo period to their influence on modern mathematical thought. Delve into the elegant theorems and techniques underlying Sangaku problems and engage in interactive problem-solving to sharpen your geometric intuition and analytical skills. Let's uncover its secrets together!

• March 21: Hari Kunduri

  • Title: Geometric inequalities in general relativity

  • Abstract: I will give a brief introduction to Riemannian geometry. The model geometry is Euclidean space, which has zero curvature. Manifolds which approach Euclidean space in an appropriate sense are called 'asymptotically flat' (AF). AF manifolds are characterized by geometric invariants which have the interpretation in physics (general relativity) as energy and angular momentum of an isolated system. I will discuss ideas behind the proof of geometric inequalities satisfied by these invariants.

• March 14 (Pi day!): Cameron Franc

  • Title: Pi and special values of the Riemann zeta function

  • Abstract: In this talk we will discuss the relationships between the Riemann zeta function and pi.

• March 7: Dicle Mutlu

  • Title: Thinking locally

  • Abstract: In mathematics, a local property refers to information about a small piece within a structure, while a global property provides insights about the entire structure. In this talk, we will talk about the relationship between local and global information across various fields of mathematics, including instances where local details contribute significantly to our understanding of the global. 

• February 29: Chris Greyson-Gaito

  • Title: Why use mathematics in biology?

  • Abstract: As an empirically trained biologist, now moving into the mathematical ecology/economics world, I (Chris Greyson-Gaito) will explore the connections between mathematics and experimental biology. I will ask questions about why we use mathematics in biology and what are the similarities and differences. I will end with my take on the power of mathematics in biology.

• February 15: Anastasis Kratsios

  • Title: Digital Computers Break The Curse of Dimensionality: Adaptive Bounds via Discrete Geometry

  • Abstract: Many of the foundations of machine learning rely on the idealized premise that all input and output spaces are infinite, e.g.~R^d. This core assumption is systematically violated in practice due to digital computing limitations from finite machine precision, rounding, and limited RAM.  In short, digital computers operate on finite grids in R^d, not on all of R^d.  By exploiting these discrete structures, we show the curse of dimensionality in statistical learning is systematically broken when models are implemented on real computers.  Consequentially, we obtain new generalization bounds with dimension-free rates for kernel and deep ReLU MLP regressors, which are implemented on real-world machines.  

Our results are derived using a new non-asymptotic concentration of measure result between a probability measure over any finite metric space and its empirical version associated with N i.i.d. samples when measured in the 1-Wasserstein distance.  Unlike standard concentration of measure results, the concentration rates in our bounds do not hold uniformly for all sample sizes $N$; instead, our rates can adapt to any given $N$.  This yields significantly tighter bounds for realistic sample sizes while achieving the optimal worst-case rate of O(N^{-1/2})$ for massive.  Our results are built on new techniques combining metric embedding theory with optimal transport.  

Joint work with: Martina Neumann (TU Vienna) and Gudmund Pammer (ETH Zürich)

• February 8: Daniel Presta

  • Title: Stability, Uncertainty, and Sensitivity Analyses of the Goodwin Model and its Extensions

  • Abstract: We investigate the stability of various stock-flow consistent economic models and the potential causes for economic collapse therein. Using techniques from applied mathematics (local stability analysis, bifurcation theory), machine learning (random forests, boosting), and statistics (Sobol indices, Shapley Values), we analyze numerous economic models that each build upon the foundational Goodwin Model [Goodwin, 1967]. In addition, we incorporate a climate module for each economic system, and analyze public sector intervention through carbon taxes and abatement subsidies. In order to maintain a stable growth path and prevent a permanent economic contraction, we propose the implementations of an expansionary (green) monetary policy, increased public sector subsidies of abatement costs, and stricter carbon taxes.

• February 1: Jan Arulseelan

  • Title: It Depends On What the Meaning of the Word 'Is' Is: An Introduction To Equality And HoTT

  • Abstract: Equality, identity and equivalence may all seem extrememly simple to modern math students and schoolchildren. However, questions about identity have puzzled philosophers and inspired mathematics since at least the time of Aristotle. We will discuss some of the historical problems of equality including those involved in the development of modern set theory and some questions about extensionality. We will then discuss some basic ideas about homotopy type theory. Homotopy type theory is one of the most recent fields of study to be inspired by the concept of identity. We will describe higher identity types and work with infinity-categories intuitively. We will motivate our constructions with intensionality and Univalence.

• January 25: Armina Akhlagh Nejat

  • Title: Gacha, Genshin, & Gambling

  • Abstract: Many people have a love-hate relationship with the game "Genshin Impact". What is really interesting is the seemingly abusive relationshihp many have with the "gambling" or gacha system in this game. In this talk we'll take a closer look at the probability aspects of the gacha system and discuss the impact of these rates on player decision-making and spending habits.

• January 18: John Nicholson

  • Title: (In)formalized Mathematics: An introduction to the Lean theorem prover

  • Abstract: Computer proof assistants have recently gained traction in mainstream mathematical discourse. In this talk, we will delve into the concepts behind computer proof assistants, briefly survey the various proof assistants to choose from, and discuss some ongoing projects in the formalization of mathematics. We will conclude by proving some elementary results in Lean.

• April 13: Kürşat Sozer

  • Title: Exploring the interplay between higher categories and spaces: From category theory to homotopy hypothesis

  • Abstract: In this survey talk, we will explore the intricate relationship between higher categories and spaces. We will begin with an overview of the basics of category theory, including the definition of a category, functors, and natural transformations. We will then delve into the concept of higher cateogires by defining 2- and 3-categories, discussing some examples, and exploring the issues of strictness and weakness.

Next, we will introduce the descriptive definition of n-categories and the main principle of higher category theory. Equipped with these notions, we will make a connection between certain higher categories and certain topological spaces; namely, we will state Grothendieck's homotopy hypothesis. If time permits, we will also briefly touch on other connections between higher categories and spaces, specifically the stabilization hypothesis and the cobordism hypothesis.

• April 6: Mike Cummings

  • Title: Play With Your Food: Brussels sprouts and other games on graphs

  • Abstract: In 1967, over afternoon tea, John H. Conway and Michael S. Paterson derived the pencil-and-paper game of sprouts. Players alternate adding edges to an initial set of vertices until no additional allowable edge can be drawn. In this talk, we will play Sprouts, its variant Brussel Sprouts, and other games involving graphs, and discuss the underlying elementary graph theory. We will also discuss the Sprouts Conjecture and its relation to the game of Nim.

• March 30: Siegfried Van Hille

  • Title: Metamathematics: category theory

  • Abstract: In any field in pure mathematics one studies sets endowed with extra structure. One then starts developing the theory by defining interesting functions, namely ones that interact well with the structure. Examples are vector spaces and linear transformations, topological spaces and continuous maps, measure spaces and measurable function ... Actually anything you can possibly think about! In all these fields one also performs constructions with these structures such as taking products and limits of them. Category theory is a field in mathematics that abstractly studies structures purely in terms of the funtions between them. This talk aims to be a very gentle introduction to this field.

• March 23: Dicle Mutlu

  • Title: Ultrafilters

  • Abstract: We will talk about a notion from set theory called 'ultrafilter'. They have very nice applications in many areas. In this talk, we will discuss some of them, for example, Arrow's theorem from social choice theory.

• March 16: Elliot Kaplan

  • Title: Winning Ways for your Mathematical Plays: The Basics of Combinatorial Game Theory

  • Abstract: Combinatorial game theory emerged from the analysis of impartial games (games where both players can make exactly the same moves). Later, BerleKamp, Conway, and Guy introduced a more general framework for partizan games, including well-known games like chess and go, including mathematical games like Hackenbush. I will discuss the basics of this theory, including the classification of games and sums of games.

• March 9: Alexi Block Gorman

  • Title: "Wir müssenwissen, wir werden wissen": Impacts of the second world war on modern mathematics

  • Abstract: In this talk, we'll look at how the 2nd world war changed the landscape of academic mathematics, both directly and indirectly.

• March 2: Silas Vriend

  • Title: Some Folding Required: The Mathematics of Origami

  • Abstract: Origami, the Japanese art of paper folding, has a rich mathematical structure which has developed greatly in the last few decades. In this colloquium lecture, we will explore the connection between origami and the straightedge and compass (SE&C) constructions of Euclidean geometry. In particular, we will see why straight-crease single-fold origami is strictly more powerful that SE&C as a tool for constructing field extensions of the rationals. Origami paper will be provided, and there will be guided folding exercises throughout the talk!

• February 16: Michel Alexis

  • Title: Calculus teachers hate him because of this one weird trick, find out why!!

  • Abstract: I'll present the concept of dyadic decomposition, a simple but stupidly effective technique for quickly estimating various sums and integrals, all without computing a single anti-derivative. We'll discuss topics ranging from the p-test in Calculus to the Calderon-Zygmund decomposition for estimating averaging operators and the Hardy-Littlewood maximal function.

• February 9: Adele Padgett

  • Title: The math of mapmaking

  • Abstract: Have you ever wondered why the same piece of land can look different in different maps of the Earth? This is not just carelessness on the part of mapmakers! A sphere and a plane have different curvature, so Gauss’s Theorema Egregium tells us that they are not isometric. This means that there is no way to make a flat map of the Earth that preserves all distances. Instead, mapmakers often try to preserve other metric properties like area, local shape, or direction. In my talk, I will discuss the Theorema Egregium and some of the math involved in different map projections.