Differential graded algebra techniques have played a crucial role in the
development of homological algebra, especially in the study of
homological properties of commutative rings carried out by Serre, Tate,
Gulliksen, Avramov, and others. In our work, we extend the construction
of the Koszul complex and acyclic closure to a more general setting. As
an application of our constructions, we show that the Ext algebra of
quotients of skew polynomial rings by ideals generated by normal
elements is the universal enveloping algebra of a color Lie algebra, and
therefore a color Hopf algebra. As a consequence, we give a presentation
of the Ext algebra when the elements generating the ideal form a regular
sequence, this generalizes a theorem of Bergh and Oppermann. It follows
that in this case the Ext algebra is noetherian, providing a partial
answer to a question of Kirkman, Kuzmanovich and Zhang.

Minimal surfaces (critical points of the area functional) have a rich and successful history in the study of the interaction between geometry and topology that goes back to the 1960s. In practice, the presence and properties of minimal surfaces inside a Riemannian manifold profoundly influences the ambient geometry. In this talk, we will discuss how one can use the Allen--Cahn equation to guarantee the existence of a rich class of geometrically and topologically distinct minimal surfaces inside a generic Riemannian 3-manifold. As a byproduct, one obtains a pure PDE resolution of a number of previously unapproachable questions in minimal surface theory, which parallels recent simultaneous advances that instead use geometric measure theory.

In order to approximate the solution of a PDE by the finite element method, the problem domain is generally first subdivided into a collection of cells, like quadrilaterals or triangles. This collection of cells, called a mesh, has a direct impact on the accuracy of the numerical solution, and the properties of 2D and 3D meshes are very well studied. However, some new schemes for solving numerical PDEs require 4D meshes. These schemes, called Space-Time Finite Element Methods (STFEMs), treat time and the spatial variables in the same way when approximating PDEs. As a result, time-dependent problems in three spatial variables are considered in a four-dimensional space-time domain. This talk introduces techniques for creating and manipulating four-dimensional conforming simplicial meshes for use with STFEMs. The approach relies on the theory of combinatorial maps, which will be introduced and considered from the computational perspective. After a discussion of the
challenges and benefits of this approach, we present some initial results in creating space-time meshes.

n 1974, Erd\H{o}s and Rothchild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. Since then, a new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, Boolean lattices and sets. In this talk, I will first give an overview of some previous results on this topic. The second half of this talk is to explore the rainbow variants of the Erd\H{o}s-Rothschild problem. With Jozsef Balogh, we confirm conjectures of Benevides, Hoppen and Sampaio, and Hoppen, Lefmann, and Odermann, and complete the characterization of the extremal graphs for the edge-colorings without rainbow triangles. Next, we study a similar question on sum-free sets, where we describe the extremal configurations for integer colorings with forbidden rainbow sums. The latter is joint work with Yangyang Cheng, Yifan Jing, Wenling Zhou and Guanghui Wang.

Modern machine learning algorithms have achieved remarkable performance in a myriad of applications, and are increasingly used to make impactful decisions in the hiring process, criminal sentencing, healthcare diagnostics and even to make new scientific discoveries. The use of data-driven algorithms in high-stakes applications is exciting yet alarming: these methods are extremely complex, often brittle, notoriously hard to analyze and interpret. Naturally, concerns have raised about the reliability, fairness, and reproducibility of the output of such algorithms. This talk introduces statistical tools that can be wrapped around any ``black-box'' algorithm to provide valid inferential results while taking advantage of their impressive performance. We present novel developments in conformal prediction and quantile regression, which rigorously guarantee the reliability of complex predictive models, and show how these methodologies can be used to treat individuals equitably. Next, we focus on reproducibility and introduce an operational selective inference tool that builds upon the knockoff framework and leverages recent progress in deep generative models. This methodology allows for reliable identification of a subset of important features that is likely to explain a phenomenon under-study in a challenging setting where the data distribution is unknown, e.g., mutations that are truly linked to changes in drug resistance.

The cellular cytoskeleton is essential in proper cell function as well as in organism development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss several examples where questions about filament-motor protein interactions require the development of novel mathematical modeling, analysis, and simulation.

In the development of egg cells into embryos, RNA molecules bind to and unbind from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters. We consider these models in the framework of partial differential equations as well as stochastic processes and derive large time properties of cargo movement for a general class of problems. The proposed methods have applications to macroscopic models of protein localization and microscopic models of cargo movement by teams of motor proteins. I will also discuss an agent-based modeling and data analysis framework for understanding how actin filaments and myosin motors interact to form contractile ring channels essential in development. In particular, we propose tools drawing from topological data analysis to analyze time-series data of filament network interactions and illustrate the impact of key parameters on significant ring emergence, thus giving insight into formation and maintenance of these biological channels.

The homology cobordism group of integer homology three-spheres is a natural invariant of interest to four-dimensional topologists. In this talk, we recall its definition and give a short introduction to involutive Floer homology, As an application, we see that there is an infinite-rank summand of the homology cobordism group. This includes joint work with Irving Dai, Jen Hom, and Linh Truong.

The Stiefel manifold is the set of orthonormal bases for $k$-planes in an $n$-dimensional space. We compute its degree as an algebraic variety in the set of $k$-by-$n$ matrices using techniques from classical algebraic geometry, representation theory, and combinatorics. We give an interpretation of this degree in terms of non-intersecting lattice paths. This is joint work with Fulvio Gesmundo.

The homology cobordism group of integer homology three-spheres is a natural invariant of interest to four-dimensional topologists. In this talk, we recall its definition and give a short introduction to involutive Floer homology, As an application, we see that there is an infinite-rank summand of the homology cobordism group. This includes joint work with Irving Dai, Jen Hom, and Linh Truong.

The Landau equation is a mesoscopic model in plasma physics that describes the evolution in phase-space of the density of colliding particles. Due to the non-local, non-linear terms in the equation, an understanding of the existence, uniqueness, and qualitative behavior of solutions has remained elusive except in some simplified settings (e.g., homogeneous or perturbative). In this talk, I will report on recent progress in the application of ideas of parabolic regularity theory to this kinetic equation. Using these ideas we can, in contrast to previous results requiring boundedness of fourth derivatives of the initial data, construct solutions with low initial regularity (just $L^\infty$) and show they are smooth and bounded for all time as long as the mass and energy densities remain bounded. This is a joint work with S. Snelson and A. Tarfulea.

Let $C$ be a curve over a finite field and let $\rho$ be a
nontrivial representation of $\pi_1(C)$. By the Weil conjectures, the
Artin $L$-function associated to $\rho$ is a polynomial with algebraic
coefficients. Furthermore, the roots of this polynomial are
$\ell$-adic units for $\ell \neq p$ and have Archemedian absolute
value $\sqrt{q}$. Much less is known about the $p$-adic properties of
these roots, except in the case where the image of $\rho$ has order
$p$. We prove a lower bound on the $p$-adic Newton polygon of the
Artin $L$-function for any representation in terms of local monodromy
decompositions. If time permits, we will discuss how this result
suggests the existence of a category of wild Hodge modules on Riemann
surfaces, whose cohomology is naturally endowed with an irregular
Hodge filtration.

The presentation will start by reviewing data on the current status of gender equity in science, technology, engineering and mathematics (STEM) disciplines and summarizing social science research that illuminates some causes of gender disparities in STEM. With this context established, the focus will shift to how women enter into leadership roles in academic settings, what they experience and how gender impacts the way they exercise their authority. The final part of the talk will discuss how we can all contribute to changing the face of leadership for the future, to the benefit of all of us in STEM.

The Fontaine-Wintenberger theorem relates the absolute Galois group
of $\mathbb{Q}_p(\mu_{p^{\infty}})$ to the absolute Galois group of a
certain local field over $\mathbb{F}_p$. Such an equivalence enables us to
understand the Galois representation of $\mathbb{Q}_p$ with some extra
work. In this talk I will explain several key ideas behind its classical
proof and its generalization via Faltings's almost purity theorem if time
permits.

The McKay correspondence takes many guises but at its core connects the
geometry of minimal resolutions for quotient singularities $C^n / G$ to
the representation theory of the group $G$. When $G$ is an abelian subgroup
of $SL(3)$, Craw-Ishii showed that every minimal resolution can be
realised as a moduli space of stable quiver representations naturally
associated to $G$, although the chamber structure for the stability
parameter and associated wall-crossing behaviour is poorly understood. I
will describe my recent work giving explicit representation-theoretic
descriptions of the walls and wall-crossing behaviour for the chamber
corresponding to a particular minimal resolution called the G-Hilbert
scheme. Time permitting, I will also discuss ongoing work with Yukari
Ito (IPMU) and Tom Ducat (Bristol) to better understand the geometry,
chambers, and corresponding representation theory for other minimal
resolutions.

Causal inference from observational data is a vital problem, but it
comes with strong assumptions. Most methods assume that we observe all
confounders, variables that affect both the causal variables and the
outcome variables. But whether we have observed all confounders is a
famously untestable assumption. We describe the deconfounder, a way to
do causal inference from observational data allowing for unobserved
confounding.

How does the deconfounder work? The deconfounder is designed for
problems of multiple causal inferences: scientific studies that
involve many causes whose effects are simultaneously of interest. The
deconfounder uses the correlation among causes as evidence for
unobserved confounders, combining unsupervised machine learning and
predictive model checking to perform causal inference. We study the
theoretical requirements for the deconfounder to provide unbiased
causal estimates, along with its limitations and tradeoffs. We
demonstrate the deconfounder on real-world data and simulation
studies.