Joint Mathematics Meetings 2022

Invited Speakers - A Closer Look

Jeremy Avigad Carnegie Mellon University	ASL Invited Address The promise of formal mathematics Friday January 7, 2022, 10:00 a.m.-10:50 a.m. Since the early twentieth century, it has been understood that mathematical state- ments can be expressed in formal languages and that mathematical proofs can be represented in formal deductive systems with precise rules and semantics, at least in principle. Remarkably, the development of computational proof assistants over the last few decades has made it possible to do this in practice. The technology is firmly based on the methods and concepts of modern logic, and in many ways represents the contemporary embodiment of the foundational tradition. I will give a brief overview of formally verified mathematics and the state of the field today. I will discuss a particular theorem prover, Lean, and its formal library, mathlib, which are attracting a growing community of users. I will explain why the technology is likely to have a transformative effect on mathematics, and I will explain why mathematical logicians might be interested in it.
Omer Ben-Neria Einstein Institute of Mathematics, Jerusalem	ASL Invited Address Diamonds compactness and ultrafilters in set theory Friday January 7, 2022, 2:00 p.m.-2:50 p.m. Guessing and compactness principles are two of the most fertile tools in set theory, which play a central role in the construction of many infinite objects (e.g., groups, graphs, topological spaces, etc.) with various desirable properties. The goal of this talk is to discuss a long ongoing research in set theory which studies the interaction between the two. The most well-known guessing principle is the Diamond principle, which was introduced by Ronald Jensen in the 1970s in his seminal study of the constructible universe. Compactness principles in set theory can be viewed as strong extensions to the compactness theorem in first-order logic, and are closely related to large cardinal axioms and the existence of various types of ultrafilters. It is well-known that certain compactness principles imply the existence of diamond sequences, however, the extent to which reflection principles assert the existence of a diamond sequence remains quite mysterious. After introducing the two principles, we will discuss the history of this problem and new results from joint work with Jing Zhang.
Robert Q. Berry, III University of Virginia	NAM Cox-Talbot Address Interest Convergence: An analytical viewpoint for examining how power dictates policies and reforms in mathematics Friday January 7, 2022, 7:45 p.m.-8:35 p.m. This Cox-Talbot talk uses a hybrid policy analysis-critical race theory lens informed largely by legal scholars like Derrick Bell to make the case that policies and reforms in mathematics education failed to address the needs of historically excluded learners. Rather, these policies and reforms are often designed and enacted to protect those in power's economic, technological, and social interests. This talk offers contrasting narratives between policy intentions and policy enactment, highlighting how the language of mathematics policies positions historically excluded learners as deficient within their cultures and communities. Finally, this talk considers features necessary in mathematics policies and reform documents when discussing the historically excluded learners.
Peter Cholak University of Notre Dame	ASL Invited Address Ramsey like theorems on the rationals Saturday January 8, 2022, 10:00 a.m.-10:50 a.m. Let’s color subsets of rationals of size 2 with 496 different colors. Then, it is known, that we can find an isomorphic substructure of the rationals where only 2 colors appear (among these pairs of rationals). In fact, if we color subsets of the rationals of size n with 496 colors we can find isomorphic substructures of the rationals where the number of colors appear is exactly the nth odd tangent number. So the rationals have finite big Ramsey degree. We will explore some other structures which have finite big Ramsey degree. Milliken's tree theorem plays a large role in showing these results.
Karl-Dieter Crisman Gordon College	ACMS Guest Speaker Mersenne Matters: Mathematics, Music, Monotheism, and More Thursday January 6, 2022, 7:00 p.m. - 7:20 p.m. Marin Mersenne is usually considered, when he is considered at all in mathematics, in one of two ways. Either he is the inspiration for the latest newly discovered (and enormous) “Mersenne Prime,” or he is the interlocutor who helped induce Fermat and Descartes to properly, and “publically” discuss their methods of tangents. But who was Mersenne, what did he do, and why does he matter? This talk will give an overview of his life and the important roles he played in the history of science and music, with many examples from his own writings. We’ll especially look into why a monk, from an order devoted to being the least of all, saw his explorations of things like pure mathematics and practical acoustics as being so closely related to his writings in defense of the faith.
Marianna Csörnyei University of Chicago	AWM-AMS Noether Lecture The Kakeya needle problem for rectifiable sets Thursday January 6, 2022, 10:05 a.m.-10:55 a.m. A planar set admits the “Kakeya property” if it can be moved continuously to any other position covering arbitrary small area during the movement. It was known for more than 100 years that line segments have this property, but until recently there were only very few other known examples. In the talk we will study two variants of this problem, the geometric and the analytic version. In the classical, geometric version, we find all connected closed sets with the Kakeya property. In the analytic version, where we are allowed to delete a null set at each time moment, we will show that every rectifiable set admits the Kakeya property, moreover, they can be moved to any other position covering not only arbitrary small but zero area.
Qiang Du Columbia University	SIAM Invited Address Analysis and applications of nonlocal models Thursday January 6, 2022, 11:10 a.m.-12:00 p.m. Nonlocality has become increasingly noticeable in nature. The modeling and simulation of its presence and impact motivate new development of mathematical theory. In this lecture, we focus on nonlocal models with a finite horizon of interactions, and illustrate their roles in the understanding of various phenomena involving anomalies, singularities and other effects due to nonlocal interactions. We also present some recent analytical studies concerning nonlocal operators and nonlocal function spaces. The theoretical advances are making nonlocal modeling and simulations more reliable, effective and robust for applications ranging from classical mechanics to traffic flows of autonomous and connected vehicles.
Elamin Elbasha Merck & Co., Inc.	Current Events Bulletin Session - Lecture IV Supported by a generous donation from Salilesh Mukhopadhyay, in honor of Satyendra Nath Bose, Mahadev Dutta, and Pranab K. Sarkar, to bring appreciation for mathematics to a broader audience Mathematics and the quest for vaccine-induced herd immunity threshold Friday January 7, 2022, 5:00 p.m.-6:00 p.m. Mathematics plays a major role in providing realistic insights into the spread and control of infectious diseases, dating back to the pioneering works of the likes of Daniel Bernoulli (on smallpox immunization modeling) in the 1870s. For example, mathematics provides answers to pertinent questions relating to the control and mitigation of vaccine-preventable diseases such as: what is the minimum fraction of the unvaccinated susceptible population that needs to be vaccinated to achieve disease elimination (in a local setting) or end pandemics (globally)? This minimum fraction is called herd immunity threshold. In this talk, I will discuss the mathematical theories and modeling methodologies associated with the derivation of vaccine-induced herd immunity thresholds for eliminating or eradicating infections, highlight the properties and assumptions behind the derived thresholds, discuss common misconceptions, and outline areas for future research. Examples of a few vaccine-preventable diseases, such as the COVID-19 pandemic, will be used for illustrative purposes.
Nicolas Fillion Simon Fraser University	POM SIGMAA Guest Speaker Trust but Verify: What Can We Know About the Reliability of a Computer-Generated Result? Friday January 7, 2022, 5:30 p.m.-6:30 p.m. Since the Second World War, science has become increasingly reliant on the use of computers to perform mathematical work. Today, computers have justifiably become a trusted ally of scientists and mathematicians. At the same time, there is a panoply of cases in which computers generate demonstrably incorrect results; and there is currently no reason to expect that this situation will change. This prompts the careful user to verify computer-generated results, but it is clear that we are often not in a position to review the work of computers as we would traditionally review a putative derivation or calculation. In this sense, computational processes are epistemically opaque. Since Humphreys introduced the phrase `epistemic opacity' in the philosophical literature in 2004, the concept of opacity has been developed along different lines; furthermore, many incompatible claims have been advanced---be they about what opacity is or about whether we should worry about it---leaving this field of the philosophy of computing in a state of confusion. In this paper, we propose a framework that disentangles three core questions (1. What kinds of epistemic opacity are there in scientific computing? 2. Should we worry about epistemic opacity? 3. Should we seek greater transparency whenever possible?) and systematically survey how their answers inter-relate.
Elena Giorgi Columbia University	Current Events Bulletin Session - Lecture II The stability of black holes with matter Friday January 7, 2022, 3:00 p.m.-4:00 p.m. Black holes are fundamental objects in our understanding of the universe. The mathematics behind them has surprising geometric properties, and their dynamics is governed by hyperbolic PDEs. A basic question one may ask is whether these solutions to the Einstein equation are stable under small perturbations, which is a typical requirement to be physically meaningful. We will see how the dispersion of gravitational waves plays a key role in the stability problem, illustrating the main conjectures and some recent theorems regarding the evolution of black holes and their interaction with matter fields.
Anna Gilbert Yale University	von Neumann Lecture Metric representations: Algorithms and Geometry Saturday January 8, 2022, 9:00 a.m.-9:50 a.m. Given a set of distances amongst points, determining what metric representation is most “consistent” with the input distances or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. In this talk, we discuss a number of variants of this problem, from convex optimization problems with metric constraints to sparse metric repair.
Edray Herber Goins Pomona College	MAA Project NExT Lecture on Teaching and Learning Addressing Anti-Black Racism in Our Departments Thursday January 6, 2022, 11:10 a.m.-12:00 p.m. In April 2021, the PBS Newshour ran a story with the headline “Even as colleges pledge to improve, share of engineering graduates who are Black declines”. Indeed, there is a dearth of Black students in our mathematics classrooms. A 2018 study by the Pew Research Center found that Black students earned just 7 percent of STEM bachelor’s degrees. Unfortunately, this is an issue for our faculty as well. A 2017 report in Inside Higher Ed states that there has been an increase over time in the diversity of senior and junior faculty members in the STEM fields — except black faculty. A New York Times article, titled “For a Black Mathematician, What It’s Like to Be the ‘Only One’”, quoted that there are just a dozen black mathematicians among nearly 2,000 tenured faculty members in the nation’s top 50 math departments. What can we as faculty members do to make our mathematics departments more welcoming and diverse for Black students and faculty alike? These are daunting problems, and many with an interest in presenting solutions do not even have tenure! In this interactive presentation, we present some practices that even tenure-track faculty can engage in to showcase how #BlackLivesMatter — from increasing the number of pathways for majors, to building community by conducting research with students, and having hard conversations within hiring committees.
Monica Jackson American University	NAM Claytor-Woodard Lecture Spatial Data Analysis for Public Health Data Thursday January 6, 2022, 2:40 p.m.-3:30 p.m. Spatial data analysis concerns data that are correlated by location, and relies upon the assumption that objects closer together in space (e.g. geographical location) will most likely have similar responses. This talk provides an overview of graphical and quantitative methods I developed for the analysis of spatial data. Emphasis is on lattice data (also known as areal data or aggregated data) however modeling of geostatistical data and point patterns will be discussed. I will apply these methods to public health data with applications to cancer trends, maternal mortality in the Dominican republic, and COVID-19 disease surveillance.
Franziska Jahnke University of Münster	ASL Invited Address Decidability and definability in unramified henselian valued fields Friday January 7, 2022, 1:00 p.m.-1:50 p.m. Unramified and finitely ramified henselian valued fields are central to studying model-theoretic phenomena in mixed characteristic. Decidability and definability in unramified henselian valued fields with perfect residue field are well understood, starting with the seminal work of Ax, Kochen, and Ershov. In this talk, we present recent developments in unramified henselian valued fields with imperfect residue field, and also comment on what changes in the case of finite ramification. Joint work with Sylvy Anscombe and Philip Dittmann.
Tyler J. Jarvis Brigham Young University	AMS Lecture on Education Supported by a generous donation from JMM partner COMAP, Inc. in memory of Bob Moses Restoring confidence in the value of mathematics Saturday January 8, 2022, 11:10 a.m.-12:00 p.m. Ten years ago a group of my department’s math majors told my colleague, Jeff Humpherys, and me, “We majored in math because we like it, but we know it won’t get us a job unless we want to teach.” That comment motivated us to create an entirely new program in applied and computational mathematics (ACME) at BYU—a program to teach students mathematics that is deep and beautiful and that employers are also eager to pay for, mathematics that students can use on the job to solve the problems of the 21st century. Since we started the ACME program eight years ago, the number of majors in our department has almost doubled, ACME students account for two-thirds of all our majors, and resources have flowed to our department. Our graduates’ starting salaries are substantially higher, and many of them are turning those big offers down to go to top graduate programs, where they are flourishing. Our alumni are fiercely loyal to ACME and eager to help the students that follow them. In this presentation I’ll talk about some of the problems we had to overcome to get ACME started, how we made ACME successful, and what we have learned along the way to help those of you wanting to do something similar for your students.
Autumn Kent University of Wisconsin - Madison	Spectra Lavender Lecture Families Thursday January 6, 2022, 11:05 a.m.-11:55 a.m. We'll talk about the ubiquity of family in low-dimensional topology and geometry.
Daniel Reuben Krashen Rutgers University	AMS Invited Address Title TBA Wednesday January 5, 2022, 10:05 a.m.-10:50 a.m.
Dave Kung Charles A. Dana Center, The University of Texas at Austin	MAA-SIAM-AMS Hrabowski-Gates-Tapia-McBay Lecture Why the Math Community Struggles with Equity & Diversity - and Why There’s Reason for Hope Friday January 7, 2022, 9:00 a.m.-9:50 a.m. Numbers don’t lie – we in the mathematical sciences are bad at equity and diversity. Our majors, graduate students, and faculty don’t come close to reflecting our population – especially when it comes to gender, race, ethnicity, and disability status. And our community is far from creating equitable opportunities, despite the good will and extensive efforts of many. While we aren’t the only field that struggles, some deeply-held beliefs, pervasive in the mathematical sciences, hold us back. From thinking that success requires “genius,” to viewing our beloved subject as “pure,” to being blind to structures that hold others back, aspects of our culture must be confronted–and changed–if we are to create a more just community. We must do better. We can do better. And thankfully, there are many people, programs and projects that are pushing us in exactly that direction.
Xihong Lin Harvard University, Broad Institute of MIT and Harvard	ASA Committee of Presidents of Statistical Societies Lecture Learning from COVID-19 Data on Transmission, Health Outcomes, Interventions and Vaccination Thursday January 6, 2022, 3:50 p.m.-4:40 p.m. COVID-19 is an emerging respiratory infectious disease that has become a pandemic. In this talk, I will first provide a historical overview of the epidemic in Wuhan. I will then provide the analysis results of 32,000 lab-confirmed COVID-19 cases in Wuhan to estimate the transmission rates using Poisson Partial Differential Equation based transmission dynamic models. This model is also used to evaluate the effects of different public health interventions on controlling the COVID-19 outbreak. I will next present transmission dynamic regression models for estimating transmission rates in USA and other countries, as well as factors including intervention effects that affect transmission rates. I will discuss estimation of the proportion of undetected cases and the disease prevalence. I will also present the analysis results of >500,000 participants of the HowWeFeel project on health outcomes and behaviors in US, and discuss the factors associated with infection, behavior, and vaccine hesitancy. To help plan safely reopen schools. I will discuss efficient pooled testing design using hypergraph factorization
Dan Margalit Georgia Institute of Technology	AMS Maryam Mirzakhani Lecture Mixing surfaces, algebra, and geometry Thursday January 6, 2022, 9:00 a.m.-9:50 a.m. Taffy pullers, lab stirrers, and paint mixers are complicated dynamical systems. To any such system we can ascribe a real number, called the entropy, which describes the amount of mixing being achieved. Which real numbers arise, and what do they say about the dynamics of the system? We will explore this question through the lens of topological surfaces, making unexpected connections to algebra and number theory. Our tour will take us from the work of Max Dehn and Jakob Nielsen a century ago, to the revelations of the Fields medalist William Thurston in the 1970s, to the breakthroughs of Fields medalist Maryam Mirzakhani in the 21st century.
Sandra Müller Technical University of Vienna	ASL Invited Address Lower Bounds in Set Theory Saturday January 8, 2022, 1:00 p.m.-1:50 p.m. Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by G\”odel's analysis of the constructible universe $L$. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced and studied by Jensen, Mitchell, Steel, Woodin, Sargsyan, and others. We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals $\kappa$. The second result studies the strength of a model of determinacy in which all sets of reals are universally Baire. Sargsyan conjectured that the existence of such a model is as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin's derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan's conjecture.
Gaston Mandata N'Guerekata Morgan State University	AMS Invited Address An invitation to periodicity Wednesday January 5, 2022, 2:15 p.m.-3:05 p.m. Periodicity is everywhere, every day. Considering some periodic phenomena, we will revisit the mathematical concept of periodicity and its recent generalizations up to almost automorphy. We will study their applications to some differential equations. An elementary proof of the celebrated Massera Theorem will be presented. We will also show that an almost periodic second order semilinear elliptic equation may not have almost periodic solutions, but many almost automorphic solutions in the envelop of the equation. An application to almost periodically forced pendulum will be given.
Hee Oh Yale University	AMS Erdős Lecture for Students Euclidean lines on hyperbolic manifolds Wednesday January 5, 2022, 11:10 a.m.-12:00 p.m. A classical theorem says that the closure of a line on the torus is always a subtorus, depending on the direction of the line. A similar rigidity phenomenon persists for a Euclidean line living in the hyperbolic world, called a horocycle. On closed hyperbolic manifolds, the closure of any Euclidean line is a hyperbolic submanifold, up to translation. This was proved by Ratner (and by Shah independently) more than 30 years ago. What happens if we now venture into hyperbolic worlds of infinite volume? We now need to choose carefully which hyperbolic worlds are safe for rigidity, as not every world is safe. We present an infinite family of hyperbolic manifolds of infinite volume of any given dimension, where every Euclidean line is dense in some translate of a hyperbolic submanifold (based on joint works with McMullen and Mohammadi for dimension three and with Lee for higher dimensions).
Jill Pipher Brown University	AMS Retiring Presidential Address Regularity of Solutions to Elliptic Operators and Elliptic Systems Wednesday January 5, 2022, 3:20 p.m.-4:10 p.m. The celebrated De Giorgi-Nash-Moser theory, developed in the middle of the last century, showed that a structural condition on the matrix of coefficients of a second order PDE implied the Holder continuous regularity of its solutions, even for rough (measurable, bounded) coefficients. The structural condition is called ellipticity. This theory had a big impact for the study of non-linear equations and opened the door to a better understanding of how to quantify the connection between regularity (smoothness) of the coefficients and that of solutions. In this talk, we will review progress towards that understanding, and introduce a recently discovered structural condition, generalizing ellipticity, that has sparked new results for complex coefficient operators and real/complex systems.
Heather Price North Seattle College	SIGMAA EM Guest Speaker Climate Justice Integrated Learning in STEM Thursday January 6, 2022, 7:30 p.m.-8:20 p.m. Our students learn about climate change from the news and in many of our classes, and they are hungry for what to do with that knowledge and how to connect it within their careers and communities. Climate touches and belongs in every subject we teach, from Humanities, business, and health sciences, to all areas of STEM, including mathematics and statistics. Dr. Price will share her work leading the Climate Justice Project at North Seattle College. This initiative seeks to build bridges between disciplines to help faculty incorporate climate justice and civic engagement into their core curriculum, in ways that empower students and encourage student retention and success. In today’s talk Dr. Price will share ideas of how and why to integrate climate justice and civic engagement into STEM, with examples from mathematics courses.
Kavita Ramanan Brown University	AAAS-AMS Invited Address Title TBA Friday January 7, 2022, 11:10 a.m.-12:00 p.m.
Anup Rao University of Washington	Current Events Bulletin Session - Lecture III Sunflowers: from soil to oil Friday January 7, 2022, 4:00 p.m.-5:00 p.m. A sunflower is a collection of sets whose pairwise intersections are all the same. Erdos and Rado showed that any large family of sets of size k must contain a large sunflower, and made a conjecture about the dependence of the size of sunflower on the size of the family of sets. Very recently, Alweiss, Lovett, Wu and Zhang made significant progress towards proving their conjecture. I discuss the key ideas involved in this line of work, and show how this problem is connected to a diverse array of applications in mathematics and computer science.
Adrian Rice Randolph-Macon College	HOM SIGMAA Speaker Beyond the strength of a woman's physical power: Mathematics, Machines, and the Mind of Ada Lovelace Wednesday January 5, 2022, 5:00 p.m.-5:50 p.m. Ada Lovelace is widely regarded as an early pioneer of computer science, due to an 1843 paper about Charles Babbage's Analytical Engine, which, had it been built, would have been a general-purpose computer. Her paper contains an account of the principles of the machine, along with a table often described as 'the first computer program'. However, over the years there has been considerable disagreement among scholars as to her mathematical proficiency, with opinions ranging from 'genius' to 'charlatan'. This talk presents an analysis of Lovelace's extant mathematical writings and will attempt to convey a more nuanced assessment of her mathematical abilities than has hitherto been the case.
Tom Scanlon University of California, Berkeley	Current Events Bulletin Session - Lecture I Tame Geometry for Hodge theory Friday January 7, 2022, 2:00 p.m.-3:00 p.m. Hodge theory brings the methods of complex analysis and differential geometry to algebraic geometry. As such, highly transcendental constructions, such as those of period mappings produced through integration, are used to study problems of an algebraic nature. Some fundamental conjectures in the subject, most notably the Hodge Conjecture itself, predict that certain objects defined using these transcendental methods are in fact algebraic. In 1994, Cattani, Deligne, and Kaplan proved one of the strongest theorems in this vein on the algebraicity of the so-called Hodge locus. In a paper published in 2020, Bakker, Klingler, and Tsimerman gave a simplified proof of the Cattani-Deligne-Kaplan theorem by showing that the period mappings appearing in that theorem are definable in an o-minimal structure. Here, “definable” carries its precise meaning in the sense of first-order logic and o-minimality is a technical, tameness condition on structures (again in the sense of first-order logic) on the real numbers. The Bakker-Klingler-Tsimerman theorem and a string of subsequent results tying o-minimal to Hodge theory exhibit once more that o-minimality may serve as tame geometry. In this lecture, I will discuss o-minimality in concrete terms, recall some of the basics of Hodge theory, state the Bakker-Klingler-Tsimerman theorem in a simplified form, and explain the relevance of o-minimality to this theorem and its generalizations.
Lynn Scow California State San Bernardino	ASL Invited Address Semi-retractions and the Ramsey Property Friday January 7, 2022, 9:00 a.m.-9:50 a.m. Say that an injection $f : A \to B$ is quantifier-free type-respecting infinite tuples from $A$ that share the same quantifier-free type in $A$ are mapped by $f$ to tuples in $B$ that share the same quantifier-free type in $B$. For structures $A$ and $B$ in possibly different languages we say that $A$ is a semi-retraction of $B$ if there are quantifier-free type-respecting injections $g: A \to B$ and $f: B \to A$ such that $f \circ g: A \to A$ is an embedding. Given finite structures $A\subseteq C$, define $\left(\begin{array}{c}C\\A\end{array}\right)$ to be all substructures of $C$ isomorphic to $A$. We say that an age $K$ of finite structures has the Ramsey property $(RP)$ if for all $A: B\in K$ and integers $k\geq 2$ there exists $C\in K$ such that for any $k$-coloring $c : \left(\begin{array}{c}C\\A\end{array}\right)\to k$, there is $B'\in\left(\begin{array}{c}C\\B\end{array}\right)$ such that for any $A', A''\in\left(\begin{array}{c}B\\A\end{array}\right), c(A')=c(A'')$. In [1], it was shown that if $A$ and $B$ are locally finite ordered structures, then if the age of $B$ has RP, the age of $A$ has RP. In this talk we will present some improvements on this result and comment on the connection to categorical notions in Ramsey theory. [1] L. Scow, Ramsey transfer to semi-retractions, *Annals of Pure and Applied Logic*, vol. 172 (2021), no. 3, Paper no. 102891,18.
Karen E. Smith University of Michigan Photo courtesy of University of Michigan Photography	AMS Colloquium Lectures Understanding and Measuring Singularities in Algebraic Geometry In these three talks, I hope to share some of the beauty of an ancient field of mathematics called Algebraic Geometry, and some of the excitement of modern techniques used to investigate it. The talks do build on each other, but each can stand alone as well. Algebraic geometry is the study of algebraic varieties, or geometric shapes described by polynomial equations. You already know many examples, such as the circle, whose polynomial equation is $x^2 + y^2 = 1$; or a sphere. Algebraic varieties are ubiquitous throughout mathematics and its applications to science and engineering. Not only do they naturally arise in important contexts---the set of all rigid transformations of space, for example, can be given the structure of an algebraic variety---but often complicated behavior can be described (or approximated) by polynomials. Because polynomials are relatively easy to manipulate by hand or by machine, algebraic geometry is a tool for scientists, engineers and even artists, as well as a rich source of examples throughout mathematics. Of course, algebraic geometry is also beautiful theoretical subject in its own right, and it is from this perspective that the talks approach the subject. AMS Colloquium Lecture I Resolutions of Singularities and Rational Singularities Wednesday January 5, 2022, 1:00 p.m.-1:50 p.m. While conic sections and spheres are smooth varieties, in general, a variety can have singular points---places where it is pinched or intersects itself. In the first talk, we discuss Hironaka's famous theorem on resolution of Singularities---a technique to “get rid" of the singular points. We introduce a class of singular varieties called rational singularities that are important because they are well-approximated by their resolutions. While it can be difficult to prove a given variety has rational singularities, we explain a remarkably checkable way to characterize rational singularities using “reduction modulo p". AMS Colloquium Lecture II Measuring Singularities Thursday January 6, 2022, 1:00 p.m.-1:50 p.m. In the second lecture, we discuss ways to quantify “how singular" a particular singular point on a particular variety might be. For complex varieties, we define a numerical measure of the singularity (called the log canonical threshold) analytically---in terms of the integrability of a natural real-valued function in a neighborhood of the singular point. For varieties defined over a field of characteristic $p$, however, a completely different approach is needed: here, we show how to iterate the $p$-th power (or Frobenius) map to produce a numerical measure of singularities called the $F$-pure threshold, which has beautiful fractal-like properties. Remarkably, these two approaches turn out to be closely related: for a point on a complex variety $X$, we explain how to “reduce modulo p" to get a point on a variety $X_p$ over a field of prime characteristic $p$. Amazingly, as we let $p$ approach infinity, the limit of the F-pure thresholds in characteristic $p$ approaches the log canonical threshold of the original singular point on $X$. AMS Colloquium Lecture III Extremal Singularities Friday January 7, 2022, 1:00 p.m.-1:50 p.m. It is natural to inquire what might be the most singular singularity of all. Is there a point on a variety which is “more singular"---measured using our invariants from the second lecture---than any other point on any other variety? This seemingly naive question has an interesting answer in prime characteristic: for homogeneous hypersurfaces---meaning (affine) varieties defined by a single homogeneous polynomial---we discuss a sharp lower bound on the F-pure threshold in terms of the degree. We can argue that the hypersurfaces achieving this smallest possible F-pure threshold are the most singular ones among all degree d hypersurfaces. Indeed, we can completely classify these maximally singular hypersurfaces, and prove that they have remarkable algebraic and geometric properties. For example, among the homogeneous equations of smooth projective surfaces, those with minimal F-pure threshold define surfaces containing a radical abundance of lines (approximately $d^4$ where $d$ is the degree of the surface---an impossibility for complex surfaces), and moreover, these lines never intersect in a triangle. This connects extremal singularities to several disparate areas, including finite geometries, rational points on varieties, algebraic coding theory, and more.
Eitan Tadmor University of Maryland	AMS Josiah Willard Gibbs Lecture Emergent Behavior in Collective Dynamics Thursday January 6, 2022, 5:00 p.m.-6:00 p.m. A fascinating aspect of collective dynamics is self-organization, where small scale interactions lead to the emergence of high-order structures with larger-scale patterns. It is a characteristic feature in collective dynamics of “social particles” which actively probe the environment and aggregate into various forms of clusters. In different contexts these take the form of flocks, swarms, consensus, synchronized states etc. In this talk I will survey recent mathematical developments in collective dynamics, starting with the influential works of Reynolds, Krause, Vicsek and Cucker & Smale. The dynamics is governed by different protocols of pairwise interactions, quantified in terms of proper communication kernels. Collisions are avoided. A main question of interest is how different classes of such kernels affect the large-time large-crowd dynamics. We will ask how short-range interactions can affect the emergence of large-scale patterns, what is the role of repulsion away thermal equilibrium, and how graph connectivity dictates the emergent behavior of multi-species dynamics.
Pauline van den Driessche University of Victoria, B.C., Canada	ILAS Invited Address Sign Patterns Meet Dynamical Systems Wednesday January 5, 2022, 9:00 a.m.-9:50 a.m. Biological systems, including those for predator-prey and disease transmission models, often give rise to systems of first order ordinary differential equations (ODEs). Linearization then yields a system $\dot x= Ax$ where $A$ is the community matrix. By contrast, mechanical and electrical systems often give rise to a second order ODE system $\"{x}= A\dot{x}+ Bx$, which is equivalent to a first order system with coefficient matrix $C =\begin{bmatrix}A&B\\I&0\end{bmatrix}$. In cases for which the signs rather than the magnitudes of matrix entries are known, the matrices become sign patterns with entries $\in\{+,-,0\}$. What can be determined about the behavior of a dynamical system governed by such a sign pattern matrix? This general question is addressed by developing results on sign patterns. Some answers in special cases are given that determine stability and inertia properties, which are important for the underlying dynamical systems. Joint work with Adam H. Berliner, Minerva Catral, D.D. Olesky.
Erik Walsberg University of California Irvine	ASL Invited Address Model theory of large fields Saturday January 8, 2022, 9:00 a.m.-9:50 a.m. In the words of Pop, large fields are ``the right class of fields over which one can do a lot of interesting mathematics''. All known infinite fields with well behaved first order theories are large, and the main examples of logically wild fields (number fields and function fields) are the main examples of non-large infinite fields. This suggests that largeness should have a central place in the model theory of fields. I have begun to explore the \'etale open topology in recent joint work with (various subsets of) Minh Chieu Tran, Jinhe Ye, Will Johnson, Anand Pillay, Sylvy Anscombe, and Philip Dittmann. This topology is defined on the $K$-points of a $K$-variety (so in particular is defined on $K^n$ for each $n$). This topology is nontrivial iff $K$ is large. When $K$ is algebraically, real, $p$-adically closed the \'etale open topology agrees with the Zariski, order, $p$-adic topology, respectively. We get an entirely novel topology over other large fields such as psuedofinite fields. The \'etale open topology provides a useful tool for handling definable sets in large fields, for example it guided us towards a proof of the large case of the stable fields conjecture. I will discuss this subject, assuming minimal background in algebra and algebraic geometry.
Lauren K. Williams Harvard University	MAA-AMS-SIAM Gerald and Judith Porter Public Lecture The positive Grassmannian and amplituhedron Saturday January 8, 2022, 3:00 p.m.-4:00 p.m. The positive Grassmannian is the subset of the real Grassmannian where all Plucker coordinates are nonnegative. It has a beautiful combinatorial structure as well as connections to statistical physics, integrable systems, and scattering amplitudes. The amplituhedron is the image of the positive Grassmannian under a positive linear map. I'll survey some of the beautiful results and conjectures about these objects, and explain how ideas from oriented matroids, tropical geometry, and cluster algebras can be used to understand them.
Talithia Williams Harvey Mudd College	JPBM Communications Award Lecture The Power of Talk: Engaging the Public in Mathematics Saturday January 8, 2022, 1:30 p.m.-2:30 p.m. When it comes to inspiring the future productivity and innovation of our nation, mathematicians are the on the front lines. In this talk, I will discuss the importance of engaging a wide range of audiences in conversations about the nature of our work and of scientific discovery. As we change the way readers, viewers, and audience members think about the natural world and the STEM disciplines, we can begin conversations that improve public perception of science and bring people from all backgrounds into this important work.