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"Balancing Efficiency, Physics, and Robustness: What Goes into Designing a Good Algorithm?"

Abstract: Many physical systems can be described by partial differential equations (PDEs). Yet, analytic solutions are oftentimes unavailable, and lab experiments have financial and time restrictions. This motivates the need for numerical solutions. When numerically solving PDEs, we must consider the storage requirements, computational complexity, physical structures, and much more. In this talk, I will overview the ingredients we consider when constructing an algorithm to numerically solve PDEs, followed by my current research endeavors.

"Counting Number Fields and Predicting Asymptotics"

Abstract: A guiding question in number theory, specifically in the subfield called arithmetic statistics, is: How many number fields are there? Number fields are vector spaces over the rational numbers that include both the rational numbers and the roots of a fixed polynomial f(x) under the operations of addition and multiplication. Like other fields, every nonzero element of a number field has a multiplicative inverse. If we allow ourselves to vary over polynomials of a fixed degree n, we can refine the question to the way number theorists like to study it: How many number fields of a fixed dimension n are there? And, if we filter the family of polynomials not only by degree but on the types of symmetries the roots have dictated by what's known as the Galois group, we arrive at the questions surrounding Malle's Conjecture: precisely, how does the count of number fields of degree n whose normal closure has Galois group G grow as their discriminants tend to infinity? In this talk, we will discuss the history of this question including its connection to the inverse Galois problem and take a closer look at the story in the case that n = 2,3,4, i.e. the counts of quadratic, cubic, and quartic fields.

"The Moon Tilt Illusion and Perspective Geometry"

Abstract: In this talk, we'll explore the wonders of how geometry and perspective art can help us look at the world in new ways (literally). We'll use this geometry to dig into a controversy about classical art (was Ivins correct that Dürer messed up perspective in his iconic etching of St. Jerome?). And then we'll use our tools to analyze an illusion that most of us haven't even thought to contemplate. The Moon Tilt illusion confuses the viewer about the direction of illumination of a waxing or waning moon. We give several examples of this phenomenon and explain how the illusion arises from standard (but surprising) aspects of perspective projections. Familiar perspective drawings and photographs of objects such as clocks and cubes help us further analyze the unfamiliar explanations of pictures of illuminated portions of spheres.

“An Introduction to Flows on Hyperbolic Manifolds”

Abstract: This talk will be an approachable introduction to hyperbolic space. We will cover both geodesic and unipotent flows on the hyperbolic plane and its quotients, with the hope of contextualizing ergodic dynamics.

“Examples of Finite Genus Surfaces Embedded in R3 with Anosov Geodesic Flow”

Abstract: The types of dynamical behavior that can occur in Hamiltonian (i.e. measure preserving systems) range from simple and well-behaved (integrable system) to strongly chaotic (Anosov) with varying levels of “chaoticness” in between.

We show how many of these behaviors can arise from geodesic flows on surfaces. However, some of these surfaces are abstract surfaces that do not exist in three-dimensional Euclidean space. We explore examples of surfaces that exist in three-dimensional space and show that such systems can be strongly chaotic (Anosov). The geometry of our three-dimensional space does not limit chaoticness! We finish by giving an explicit estimate for the genus (# of holes) of these examples.

“Quotients of Special Classes of Positroids”

Abstract: A matroid is a structure that abstracts linear independence. There is a special kind of matroid called positroids, and there has been great interest in characterizing quotients of positroids. In this talk, we will talk about the definitions of matroids, matroid quotients, positroids, and two combinatorial objects that are in bijection with positroids. After that, we give a complete characterization for positroid quotients of adjacent ranks. When the larger matroid is a uniform matroid, we give a characterization for positroids with any rank that are quotients of it. At the end, we will provide several interesting follow-up open problems.

Spring Break - No Colloquium

"The Game of Cops and Robbers on Graphs"

Abstract: Cops and Robbers is a graph game where some number of cops and robbers take turns moving along the edges of a graph. In the traditional game, the cops wish to capture a single robber as quickly as possible and the robber wishes to evade capture for as long as possible. Many variants of the game exist and some of the questions we can ask include: How many cops are needed to guarantee capture of the robber on a certain graph? How long will it take the cops to capture the robber? Can we be more efficient by using more than the minimum number of cops and capturing the robber more quickly? What if rather than capturing the robber, the cops want to prevent them from damaging vertices? In this talk, we will explore these questions and more.

“Mathematical Modeling of Cell Volume Control and Electrolyte Balance”

Abstract: Electrolyte and cell volume regulation is essential in physiological systems. Biophysical modeling in this area, however, has been relatively sparse. After a brief introduction to cell volume control and electrophysiology, I will discuss the classical pump-leak model of electrolyte and cell volume control. It will be shown that thermodynamic considerations lead to a new perspective of cell volume control. This classical model will then be generalized to a model with spatial extent (a system of partial differential equations) modeling cell-level electrodiffusive and osmotic phenomena. A simplified version of this model will then be applied to study osmosis-driven cell movement. I will also touch upon tissue-level models of ionic electrodiffusion and osmotic water flow which we have developed to study certain pathophysiologies of the central nervous system.

"From Unknotted Curves on Seifert Surfaces to Contractible 4-manifolds"

Abstract: It is a fundamental result in geometric topology that every closed 3-dimensional manifold can be realized as the boundary of a 4-dimensional manifold, and many 4-dimensional manifolds with different topology can have the same boundary. A difficult problem, for a given closed 3- dimensional manifold, is to find a 4-dimensional manifold with the smallest possible topology (e.g. contractible) that the 3-manifold bounds. This fits under the problem of embedding 3- dimensional manifolds into 4-dimensional Euclidean space which has a rich history and proved to be tremendously important for the development of geometric topology since the 1950s. In this talk, I will provide further context and motivations for the problem above. Next, I will introduce a seemingly different problem where we will try to understand curves on Seifert surfaces of knots in three-dimensional sphere (lots of pictures, lots of fun). Finally, we will see how the progress in the previous step will help us to construct many 3-manifolds that bound contractible 4-manifolds. The talk will feature some recent results which were obtained with REU students at the speaker's home institution.

Apr. 15: Abdullah Malik, Florida State University (HC)

Apr. 22: Richard Green, University of Colorado, Boulder (HC)

“Canonical Forms of Neural Ideals”

Abstract: The neural ideal was introduced by Curto, Itskov, et al in 2013 to study the firing patterns of a set of neurons (called a neural code), turning problems in neuroscience and coding theory into algebraic questions. They also introduced the canonical form of a neural ideal, a set of generators uniquely tied to the original neural code. In this talk I will give an overview of neural ideals, describe a simple criterion for determining whether a neural ideal is in canonical form, and give an improved algorithm for computing the canonical form of a neural ideal. This work is joint with Hugh Geller.

"Olympic Games: Three Impartial Games with Infinite Octal Codes"

Abstract: A Combinatorial Game is a two-player game of pure strategy with no random elements. Players alternate moves and each player has complete information about the state of the game throughout the entire game. Each combinatorial game ends after a finite number of moves and the last move determines the winner (in Normal play, the last player to move wins, and in Misère play, the last player to move loses). An impartial game is a combinatorial game in which each player has the same set of moves available to them at each stage of the game. In this talk, we will look at the most famous impartial game of all, Nim. We will see how a related game, Wythoff, incorporates the concept of a Beatty sequence into this framework. From there, we will introduce the notion of octal game and analyze three subtraction games with infinite octal codes, showing how to play each game to win. This work began as an undergraduate research project and should be accessible to third or fourth year undergraduate mathematics majors.

"Agent-Based and Continuous Models of Locust Swarms"

Abstract: Locust swarms pose a major threat to agriculture, notably in northern Africa, the Middle East, and Australia. In the early stages of aggregation, juvenile locusts form hopper bands. These are coordinated groups that march in columnar structures that are often kilometers long and may contain millions of individuals. In later stages locusts swarms become airborne and can decimate crops over hundreds of kilometers potentially leading to famine and widespread ecological disruption. In this talk we will discuss two strategies for modeling locust swarms. Agent-based models (ABMs) yield ordinary differential equations for groups of interacting individuals and are easy to implement but challenging to analyze. Homogenizing these models replaces the individuals with population densities that are governed by partial differential equations (PDEs) which are more difficult to simulate but which can be analyzed via dynamical system methods. Finally, we will discuss the challenges of informing models with experimental observations and report on an ongoing study that uses motion tracking of tens of thousands of locusts in the field to shed light on how behavior is influenced by social interactions.

"Human-Centered Machine Intelligence"

Abstract: Building machine intelligence from a human-centered perspective is increasingly urgent, as large-scale machine learning systems ranging from personalized recommender systems to language and image generative models are deployed to interact with people daily. In this talk, Liu will give an overview of the field of human-centered machine intelligence. Using examples ranging from decision-support systems in healthcare to personalized recommender systems in online platforms to large-language models, Liu will showcase how we can account for complex human preferences in designing algorithms for building these systems.

FALL BREAK - NO COLLOQUIUM

"Mathematics and Machine Learning"

Abstract: Machine learning - or more colloquially AI - is found today in almost all areas of modern technology, science and society. While many people now have at least a vague idea of what machine learning is, and there are now many applied machine learning specialists in the world, a rigorous overview of the field and its key challenges and successes is not always available to mathematicians curious about the field. In this talk I will give a mathematical survey of some historical and current developments in AI. I will, in particular, offer high-level descriptions of some current paradigms in the field and discuss how mathematics offers insight into these. Finally, if time permits, I will discuss the prospect of AI being used someday to assist mathematicians.

"An Invitation to Mapping Class Groups and Low-dimensional Topology and Geometry"

Abstract: In this talk we'll give an introduction to low-dimensional topology and geometry, and the theory of mapping class groups. We'll explore how to use surfaces to construct three-manifolds, and how the symmetries of the surface determine geometric information of the manifold. If time allows, we'll discuss how these ideas extend to the theory of big mapping class groups and infinite-type surfaces. The talk is intended for a wide audience, particularly undergraduate students, and will be filled with pictures to illustrate the theory.

“The Redei--Berge Symmetric Function of a Directed Graph” 

Abstract: In 1934, Laszlo Redei observed a peculiar property of tournaments (directed graphs that have an arc between every pair of distinct vertices): Each tournament has an odd number of Hamiltonian paths. In 1996, Chow introduced the ``path-cycle symmetric function'' of a directed graph, a symmetric function in two sets of arguments, which was later used in rook theory. We study Chow's symmetric function in the case when the y-variables are 0. In this case, we give new nontrivial expansions of the function in terms of the power-sum basis; in particular, we find that it is p-positive as long as the directed graph has no 2-cycles. We use our expansions to reprove Redei's theorem and refine it to a mod-4 congruence. This is joint work with Richard P. Stanley.

“From Dimension 2 to 3 and Back Again”

Abstract: In this talk I’ll begin by telling a little bit of Thurston’s beautiful story connecting the dynamics of finite-type surface homeomorphisms with the geometry of 3- manifolds. I will then share some more recent work of myself and others which connect the dynamics of infinite-type surface homeomorphisms with the geometry of 3-manifolds. My aim is for the talk to be accessible to a broad audience with many illustrations to help us build our intuition without getting too far into the technical weeds.

Thanksgiving Week - No Colloquium

“Using Mathematics to Unravel the Complexities Associated to Malaria Transmission Success, following a Human-Parasite-Mosquito Interaction in a Built Environment”

Abstract: Plasmodium falciparum parasites are the causative agents of human malaria disease, while female Anopheles mosquitoes are the transmitting agents of the parasites. Part of the parasite’s life cycle resides in humans and the other part in female mosquitoes. Transmission of the parasite from an infectious human to a susceptible feeding mosquito is plausible when the mosquito successfully draws blood from the infectious human, with success if the drawn blood meal contains the transmissible parasite forms (gametocytes) from humans to mosquitoes. On the other hand, transmission from an infectious mosquito to a susceptible human is plausible following a successful feeding encounter between the mosquito and the human, with success if the feeding encounter results in the injection of the transmissible parasite forms (sporozoites) from mosquitoes to humans. Notably, the process is not always successful; the quest to draw blood is costly and may fail resulting in the mosquito's demise. Moreover, even when blood is successfully drawn from a human, parasite transmission may fail. In fact, a successful parasite transmission, and hence malaria transmission, requires two sequentially distinct successful feeding episodes by a susceptible feeding mosquito, the first from an infected human with the transmissible parasite forms followed by one from a susceptible human, with the parasite being in its transmissible state in the feeding mosquito at the latter feeding. Thus humans, parasites and female mosquitoes must interact synergistically in order for the transmission cycle to be successfully completed. The bottlenecks involved illuminate how the human-mosquito interaction enhances the parasites' exploitation of the evolutionary and reproductive needs of mosquitoes to ensure the parasites success and survivability. Therefore, understanding this complex process, viewed from the lens of transmitting mosquitoes, also driven by their evolutionary need to survive, is essential. This is situated in a built environment that showcases fluctuations in temperature which affects various aspects in the malaria problem. In this talk, I will illustrate the role of mathematics in aiding our understanding of the malaria problem.

“Mathematical Modeling and Uncertainty Quantification of Biofilm Structure”

Abstract: Mathematical tools such as partial and differential equations, data assimilation, and topological data analysis techniques can be used to describe the complex structure of biological systems. In this talk, we will discuss the use of these mathematical tools to capture the physical aspects and spatiotemporal organization of bacterial communities and biofilm component across different scales. Additionally, we will explore how Bayesian statistics can help quantify uncertainties in model parameters that are hard to measure in experiments.