Given a class in the cohomology of a projective manifold, one can ask whether the class can be represented by an irreducible subvariety. If the class is represented by an irreducible subvariety, we say that the class is realizable. One can further ask whether the subvariety can be taken to satisfy additional properties such as smooth, nondegenerate, rational, etc. These questions are closely related to central problems in algebraic geometry such as the Hodge Conjecture or the Hartshorne Conjecture. Recently, June Huh and collaborators have made significant progress in understanding realizable classes in products of projective spaces. In this talk, I will give a survey of this circle of ideas and discuss recent joint work with Julius Ross on realizable classes in Grassmannians.

In this talk, I will present the construction of solutions to the one-dimensional Dirac equation on the half-line with Dirichlet boundary conditions. While the Dirac equation is a four-dimensional system arising in quantum field theory, I will focus on the one- dimensional initial-boundary value problem. The primary analytical tool is the unified transform method (or known as Fokas method), which provides an explicit representation of the solution. To introduce the method, I will first demonstrate it in the context of the heat equation on the half-line. I will then apply it to the Dirac equation to derive explicit solution formulas and analyze the associated boundary behavior at the origin. Furthermore, I will discuss the long-time dynamics of these solutions. If time permits, I will conclude with a discussion of Sobolev-space energy estimates for the solutions, including control of both spatial norms and time-regularity.

We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend. We will discuss an argument of Brower, proving the stable forking conjecture between a type of rank 2 and a type of finite rank in a supersimple theory.

The theory of complexes of groups allows one to study generalizations of free group actions on simply connected polyhedral complexes. In this talk, we discuss motivations of the theory, including applications to recent work by Groves—Manning and Einstein—Groves extending Agol’s theorem beyond the setting of geometric actions. We will also introduce a new notion of an immersion of complexes of groups and state our main result on locally convex immersions into non-positively curved complexes of groups.

A/B testing is an effective method to assess the potential impact of two treatments. For A/B tests conducted by IT companies like Meta and LinkedIn, the test users can be connected and form a social network. Users’ responses may be influenced by their network connections, and the quality of the treatment estimator of an A/B test depends on how the two treatments are allocated across different users in the network. In this talk, I will discuss optimal design criteria based on some commonly used outcome models, under assumptions of network-correlated outcomes or network interference. I will show that the optimal design criteria under these network assumptions depend on several key statistics of the random design vector. I will discuss a framework to develop algorithms that generate rerandomization designs meeting the required conditions of those statistics. I further talk about asymptotic distributions to guide the specification of algorithmic parameters and validate the proposed approach using both synthetic and real-world networks.

Active Learning techniques (IBL, POGIL, ...) rely on teamwork to promote student learning. Far too often, assessments don't follow suit and instead ask students to perform computational tasks without assessing whether or not students can solve problems, apply their knowledge to new settings, or think creatively. This talk provides a detailed examination of how the University of Minnesota has revised its precalculus sequence to incorporate communication skills as a significant component of the grading scheme.

Five Internship Alums will discuss their internship experiences with other MSCS Grads. Please bring your questions!

The notion of G-varieties was introduced by Manin when he studied rationality problems of surfaces. Broadly speaking, a G-variety is a variety X carrying an action of a group G. The group can act via automorphisms of X or via Galois actions if the base field is non-closed. There are close connections, as well as drastic differences between these two types of actions from the perspective of birational geometry. In this talk, I will explore these similarities and differences with a focus on equivariant unirationality of Fano threefolds. This is joint work with Yuri Tschinkel and Ivan Cheltsov.

We consider the following general question that encompasses some of the most celebrated theorems in Combinatorics. Given a small graph H and a large graph G with density x, what is the possible number of induced subgraphs of G that are isomorphic to H. A complete answer is known only in the case when H is a clique or a two edge star (and their complements). We will discuss some general theory around this problem and then focus on some specific H. This is joint work with Xizhi Liu and Christian Reiher.

Flexural waves, the propagation of waves in thin elastic sheets, arise in a number of contexts, and, particularly, in the study of ice shelves. In the frequency domain, they are commonly modeled as a fourth order PDE in two dimensions with clamped plate, free plate, or supported plate boundary conditions. Here, we review existing approaches for solving boundary value problems of this type, and discuss some limitations. Building on this, for the supported plate and free plate problems, we propose novel representations which ultimately reduce the problems to second kind integral equations. Moreover, the resulting integral equations are amenable to standard high order discretization approaches and fast algorithms. Several numerical examples will be presented which illustrate the properties of these integral equations. Finally, generalizations to other wave phenomena will be discussed.

In dynamics, the speed of mixing depends on the chaos of the map and the regularity of the observables. Notably, two classical linear models—the Bernoulli doubling map and the CAT map—exhibit double exponential mixing for analytic observables. Are linear maps the only ones with this property? In dimension one, we provide a full classification for maps from the space of finite Blaschke products acting on the circle (as well as for free semigroup actions generated by a finite collection of such maps). In higher dimensions, we identify a necessary condition for double exponential mixing and present several families of examples and non-examples. Key ideas of the proof involve the Koopman precomposition operator on spaces of hyperfunctions (elements of the dual space of analytic functions), which turns out to be non-self-adjoint, compact, and quasinilpotent, with spectrum reduced to zero. The talk is accessible to all; no background knowledge is required.

A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given bipartite graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk. So far so good, but this is a logic seminar and the title says the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions. The new results that will appear in the talk have been obtained jointly with Pantelis Eleftheriou.

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We introduce the dynamical commensurator group for a generalized odometer action, that is for minimal equicontinuous group actions on Cantor sets. We show there is a map from the pointed mapping class group of a solenoidal manifold (ie a weak solenoid) to a dynamical commensurator group, and give conditions for when this map is either surjective or an isomorphism. Odden proved that this map is an isomorphism for the mapping class of the universal hyperbolic solenoid; Bering and Studenmund proved that the mapping class group of a universal solenoid over a compact K(G,1) manifold maps onto the commensurator group of G. We extend the results of both of these papers to arbitrary solenoidal manifolds. This work is joint with Olga Lukina.

We generalize an intersection-theoretic local inequality of Fulton–Lazarsfeld to weighted blowups. Using this together with the classification of 3-dimensional divisorial contractions, we prove nonrationality of many families of terminal Fano 3-folds. This is a joint work with Igor Krylov and Takuzo Okada.

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