Stanley-Reisner theory provides a link between commutative algebra and combinatorics. We will define simplicial complexes and give the Stanley-Reisner correspondence between square-free monomial ideals in a polynomial ring and simplicial complexes. We will discuss the Alexander dual of simplicial complexes and of square-free monomial ideals. Finally, we present various criterions for Cohen-Macaulayness of a square-free monomial ideal quotient in terms of combinatorial and geometrical information on the corresponding simplicial complex, such as shellability, pureness, and homology. Speaker(s): Dawei Shen (UM)

]]>Speaker(s): Zachary Deiman (University of Michigan)

]]>I will talk about the spectral transform, which is the torus-analogue of the boundary measurement map to the Grassmannian, and how it identifies the cluster integrable system with the Beauville integrable system. Speaker(s): Terrence George (University of Michigan)

]]>The theory of Eisenstein series has played an important role in the study of automorphic forms. In this talk, we will define a generalization of the Eisenstein series, and introduce Langlands' work on spectral decomposition of the L^2 space of the automorphic quotient of a reductive group using the Eisenstein series. Speaker(s): Guanjie Huang

]]>Speaker(s): Ruijie Yang (Max Plank Institute)

]]>The middle cohomology of hyperkahler fourfolds of Kummer type was studied by Hassett and Tschinkel, who showed that a large portion is generated by cycle classes of fixed-point loci of symplectic involutions. In recent joint work with Katrina Honigs, we study symplectic fourfolds over arbitrary fields which are constructed as fibers of the Albanese map on moduli spaces of stable sheaves on an abelian surface. We have extended the results of Hassett and Tschinkel and characterized the Galois action on the cohomology. We do this by giving an explicit description of the symplectic involutions on the fourfolds. This has natural consequences for derived equivalences between Kummer fourfolds. Speaker(s): Sarah Frei (Dartmouth)

]]>Mean field control (MFC) theory allows us to conclude that certain high-dimensional control problems are "approximately distributed", in the sense that (i) we can construct a "distributed control" (a feedback whose ith component depends only on the position of the ith particle) which is approximately optimal and (ii) the law of the optimally controlled state process is approximately a product measure. Of course, this analysis applies only when the controller's cost functional is symmetric. Nevertheless, it makes sense to ask when we can expect non-symmetric control problems to be approximately distributed as well. In an ongoing joint work with Daniel Lacker, we provide an answer to this question through several explicit estimates. When specialized to the mean field setting, our estimates give a new approach to the (quantitative) convergence problem for MFC which does not require an analysis of the relevant HJB equation on the space of measures. Speaker(s): Joseph Jackson (UT Austin)

]]>: The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric. Speaker(s): Karen Butt (U Michigan)

]]>Monitoring plant biodiversity and its threats are critical for maintaining Earth's functions and services. First, this talk will discuss recent approaches for sensing different components of plant biodiversity, like species presence, abundance, and functions of native and invasive species, using various remote sensing data sources. Secondly, we will discuss alternative approaches to upscale these biodiversity and invasion patterns to the landscape level using a combination of drones and satellites with Machine and Deep Learning algorithms. Finally, we will present the development of our current project, where we are sensing the relationship between plant functional diversity and drought adaptations in Mediterranean environments. Speaker(s): Javier Lopatin (Universidad Adolfo Ibáñez)

]]>This talk will review the classical tree augmented naive Bayes classifier (TAN) and then present two alternative learning approaches. The first approach automatically controls the number of edges supported by the training examples in the Bayesian network classifier by adopting a Bayes factor strategy, yielding more realistic network structures. In the second approach, we construct TAN classifiers without estimating conditional mutual information. Instead, the model learns the weights from the data using an evolution strategy to obtain a good classification performance. Applications of these learning approaches will be presented for Twitter sentiment analysis and Orthodontics. Speaker(s): Gonzalo Ruz (Universidad Adolfo Ibáñez)

]]>Let R be a regular local ring with residue field k and I a perfect ideal of R of grade 3. In 1978, Buchsbaum and Eisenbud showed that a minimal free resolution of R/I has a differential graded (DG) algebra structure, which induces a structure on the Tor algebra. By independent results of Weyman and of Avramov, Kustin, and Miller, this graded algebra structure may be classified into different classes. The classification is incomplete in the sense that it remains open which algebra structures actually occur; this realizability question was formally raised by Avramov in 2012. We survey which classes have been realized in the literature and detail the presenter's contributions to further answer the realizability question. Speaker(s): Alexis Hardesty (Texas Tech)

]]>Speaker(s): Nattalie Tamam (U Michigan)

]]>We've seen in the seminar that while the BMO is explicitly computable, showing that it is the only obstruction can be a tricky issue. Indeed, one of the most successful strategies is to rely on a very difficult conjecture about prime values of polynomials. However, such number theoretic input is a little easier to prove "on average". We'll discuss what this means for the study of rational points and survey some recent results in this direction. Speaker(s): Nick Rome (UM)

]]>In this talk, we first discuss a connection between the Halpern fixed-point iteration and Nesterov's accelerated schemes for a root-finding problem involving a co-coercive operator. We also study this connection for different recent schemes, including extra anchored gradient method and its variants. We show how the convergence results from one scheme can be transferred to another. Next, we develop a randomized block-coordinate algorithm for solving the above root-finding problem, which is different from existing randomized coordinate methods in optimization. Finally, we consider the applications of this randomized coordinate scheme to monotone inclusions and finite-sum monotone inclusions. The latter one can be applied to a federated learning setting. Speaker(s): Quoc Tran-Dinh (University of North Carolina at Chapel Hill)

]]>Geometric Data Science develops continuous parameterizations on moduli spaces of data objects up to important equivalences. The key example is a finite or periodic set of unlabeled points considered up to rigid motion or isometry preserving inter-point distances. Periodic point sets model all solid crystalline materials (periodic crystals) with zero-size points at all atomic centers. A periodic point set is usually given by a finite motif of points (atoms or ions) in a unit cell (parallelepiped) spanned by a linear basis. The underlying lattice can be generated by infinitely many bases. Even worse, the set of possible motifs for any periodic set is continuously infinite.

This typical ambiguity of data representation was recently resolved by generically complete and continuous isometry invariants: Pointwise Distance Distributions (PDD) of periodic point sets. The near-linear time algorithm for PDD invariants was tested on more than 200 billion pairwise comparisons of all 660K+ periodic crystals in the world's largest collection of real materials: the Cambridge Structural Database.

The huge experiment above took only two days on a modest desktop and detected five pairs of isometric duplicates. In each pair, the crystals are truly isometric to each other but one atom is replaced with a different atom type, which seems physically impossible without perturbing distances to atomic neighbors. Five journals are now investigating the integrity of the underlying publications that claimed these crystals.

The more important conclusion is the Crystal Isometry Principle meaning that all real periodic crystals have unique geographic-style locations in a common continuous Crystal Isometry Space (CRISP). This space is parameterized by complete isometry invariants and continuously extends Mendeleev's table of elements to all crystals.

The relevant publications are in NeirIPS 2022, MATCH 2022, SoCG 2021. The latest paper in arxiv:2207.08502 defined complete isometry invariants with continuous computable metrics on any finite sets of unlabeled points in a Euclidean space. Many papers are co-authored with colleagues at Liverpool Materials Innovation Factory and inked at http://kurlin.org/research-papers.php#Geometric-Data-Science.

Speaker(s): Vitaliy Kurkin (University of Liverpool)

For a smooth projective variety X, the derived category is a natural object to study, containing the data of complexes of (quasi-)coherent sheaves with morphisms being maps of complexes up to a weak notion of equivalence. This turns out to be not only a natural bookkeeping device with which to define derived functors in algebraic geometry and commutative algebra, but an interesting geometric invariant in its own right. I will present the definition of the derived category in this setting and some preliminary results, before describing how they classify curves up to isomorphism with some explanation of how the theory extends to higher-dimensional settings. Some exposure to homological algebra and (quasi-)coherent will be useful, but not necessary. Speaker(s): Saket Shah (Michigan)

]]>In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s Speaker(s): Lei Xue (University of Michigan)

]]>https://arxiv.org/abs/2201.11283 Speaker(s): Stephen Pietromonaco

]]>Trapped surfaces are a central topic of study in mathematical general relativity. Penrose's incompleteness theorem (1965) tells us that the presence of these surfaces in a suitable class of spacetimes implies that the spacetime is geodesically incomplete, thus tying the concept of trapped surfaces to the study of incompleteness and singularity formation in GR. In this talk, we will introduce the concept of trapped surfaces and discuss Christodoulou's breakthrough result from 2009 showing that closed trapped surfaces can form in vacuum via the focusing of incoming gravitational radiation. Speaker(s): Chris Stith (University of Michigan)

]]>Hyperbolic structures on 3-manifolds tend to be rigid, relative to certain boundary data. Families of such structures, with varying boundary data, can degenerate to other types of geometric structures that are much more flexible. The particular limit structures typically are solutions to extremal problems. This lecture will discuss several examples of this phenomenon.

Speaker(s): Steven Kerckhoff (Stanford)

TBA

]]>Speaker(s): Elad Zelingher (UM)

]]>Speaker(s): Maria Ntekoume (Rice University)

]]>Speaker(s): Andrea Dotto (University of Chicago)

]]>Speaker(s): Calvin Yost-Wolff (UM)

]]>Researchers from different areas have independently defined extensions of the usual weak topology between laws of stochastic processes. This includes Aldous' extended weak convergence, Hellwig's information topology and convergence in adapted distribution in the sense of Hoover-Keisler. In this talk, we show that on the set of continuous processes with canonical filtration these topologies coincide and are metrized by a suitable adapted Wasserstein distance. Moreover we show that the resulting topology is the weakest topology that guarantees continuity of optimal stopping.

While the set of canonical processes is not complete, we establish that its completion is the space of filtered processes. We also observe that this complete space is Polish, Martingales form a closed subset and approximation results like Donsker's theorem extend to the adapted Wasserstein distance. This talk is based on the joint work with Daniel Bartl, Mathias Beiglböck, Gudmund Pammer and Stefan Schrott.

Speaker(s): Zhang Xin (University of Vienna)

Speaker(s): Zakaria Zerrouki (University of Michigan)

]]>TBA Speaker(s): Davi Obata

]]>Compact Hyperkähler manifolds are one of the building blocks of Kähler manifolds with trivial first chern class, but very few examples are known. One strategy for potentially finding new examples is to look at finite groups of symplectic automorphisms of the known examples, and study the fixed loci or quotient. In this talk, we will obtain a classification of birational symplectic involutions of manifolds of OG10 type. We do this from two vantage points: firstly following classical techniques relating birational transformations to automorphisms of the Leech lattice. Secondly, we look at involutions that are obtained from cubic fourfolds via the compactified intermediate Jacobian construction. In this way, we obtain new involutions that could potentially give rise to new holomorphic symplectic varieties. If time permits, we will mention ongoing work to identify the fixed loci in one of these examples. Speaker(s): Lisa Marquand (Stony Brook University)

]]>We propose a new approach to deriving quantitative mean field approximations for any strongly log-concave probability measure. The main application discussed in this talk is to a class of stochastic control problems in which a large number of players cooperatively choose their drifts to maximize an expected reward minus a quadratic running cost. For a broad class of potentially asymmetric rewards, we show that there exist approximately optimal controls which are decentralized, in the sense that each player's control depends only on its own state and not the states of the other players. Moreover, the optimal decentralized controls can be constructed non-asymptotically, without reference to any mean field limit. Our framework is inspired by the recent theory of nonlinear large deviations of Chatterjee-Dembo, for which we offer an efficient non-asymptotic perspective in log-concave settings based on functional inequalities. If time allows, we discuss additional implications for continuous Gibbs measures on large graphs. Joint work with Daniel Lacker and Sumit Mukherjee. Speaker(s): Lane Yeung (Columbia University)

]]>Speaker(s): Laurel Ohm (Princeton University)

]]>Speaker(s): Riccardo Montalto (University of Milan)

]]>In this talk, we discuss the role that subgradients play in various second-order variational analysis constructions and its consequences. Focusing mainly on the behavior of the second subderivative and subgradient proto-derivative of certain composite functions, we demonstrate that choosing the underlying subgradient, utilized in the definitions of these concepts, from the relative interior of the subdifferential mapping ensures stronger second-order variational properties such as strict twice epi-differentiability and strict subgradient proto-differentiability. Using this observation, we provide a simple characterization of continuous differentiability of the proximal mapping of our composite functions. As another application, we discuss the equivalence of metric regularity and strong metric regularity of a class of generalized equations at their nondegenerate solutions. This talk is based on joint works with Nguyen T. V. Hang. Speaker(s): Ebrahim Sarabi (Miami University)

]]>Speaker(s): Qianyu Chen (Michigan)

]]>Speaker(s): Nikolaos Evangelou (Johns Hopkins University)

]]>In this talk we introduce a notion of set valued PDEs. The set values have been introduced for many applications, such as time inconsistent stochastic optimization problems, multivariate dynamic risk measures, and nonzero sum games with multiple equilibria. One crucial property they enjoy is the dynamic programming principle (DPP). Together with the set valued Itô formula, which is a key component, DPP induces the PDE. In the context of multivariate optimization problems, we introduce the set valued Hamilton-Jacobi-Bellman equations and established its wellposedness. In the standard scalar case, our set valued PDE reduces back to the standard HJB equation.

Our approach is intrinsically connected to the existing theory of surface evolution equations, where a well-known example is mean curvature flows. Roughly speaking, those equations can be viewed as first order set valued ODEs, and we extend them to second order PDEs. Another difference is that, due to different applications, those equations are forward in time (with initial conditions), while we consider backward equations (with terminal conditions). The talk is based on a joint work with Prof. Jianfeng Zhang.

Speaker(s): Melih Iseri (USC)

Speaker(s): Dima Drusvyatskiy (University of Washington)

]]>Speaker(s): Ilia Gaiur (University of Toronto)

]]>Speaker(s): Naomi Sweeting (Harvard University)

]]>Speaker(s): Yunqing Tang (UC Berkeley)

]]>Speaker(s): Jackson Morrow (UC Berkeley)

]]>Speaker(s): Olivier Lafitte (Paris 13)

]]>Speaker(s): Hung Phan

]]>TBA

]]>Speaker(s): Anna Bot (University of Basel)

]]>Speaker(s): Lei Chen (University of Maryland)

]]>Speaker(s): Daniel Litt (University of Toronto)

]]>Speaker(s): Lenya Ryzhik (Stanford University)

]]>Speaker(s): Patrick L. Combettes (North Carolina State University)

]]>TBA

]]>Speaker(s): Joseph Cummings (University of Notre Dame)

]]>Speaker(s): Bruno Klingler (Humboldt-Universität zu Berlin)

]]>Speaker(s): Bruno Klingler (Humboldt University)

]]>Speaker(s): Marius Beceanu (SUNY, Albany)

]]>Please see attached file for the abstract. Speaker(s): Christiane Tammer (Martin-Luther-University Halle-Wittenberg, Institute of Mathematics, Germany)

]]>TBA

]]>Speaker(s): Michail Savvas (University of Texas at Austin)

]]>TBA

]]>Speaker(s): Yuchan Lee (Postech, Korea)

]]>Speaker(s): Ilya Kachkovskiy (Michigan State University)

]]>Speaker(s): Kimoi Kemboi (Cornell University)

]]>Speaker(s): Nghia Tran (Oakland University)

]]>TBA

]]>In 1870 Jordan explained how Galois theory can be applied

to problems from enumerative geometry, with the group encoding

intrinsic structure of the problem. Earlier Hermite showed

the equivalence of Galois groups with geometric monodromy

groups, and in 1979 Harris initiated the modern study of

Galois groups of enumerative problems. He posited that

a Galois group should be `as large as possible' in that it

will be the largest group preserving internal symmetry in

the geometric problem.

I will describe this background and discuss some work

in a long-term project to compute, study, and use Galois

groups of geometric problems, including those that arise

in applications of algebraic geometry. A main focus is

to understand Galois groups in the Schubert calculus, a

well-understood class of geometric problems that has long

served as a laboratory for testing new ideas in enumerative geometry. Speaker(s): Frank Sottile (Texas A & M University)

Speaker(s): Florian Gunsulius (UM Econ)

]]>TBA

]]>Speaker(s): Jack Carlisle (University of Notre Dame)

]]>TBA

]]>Speaker(s): Winter Break (University of Michigan)

]]>Speaker(s): Mau Nam Nguyen (Portland State University)

]]>Speaker(s): Brian Hall (University of Notre Dame)

]]>Speaker(s): Cameron Gordon (University of Texas at Austin)

]]>Speaker(s): Martin Larsson (Carnegie Mellon)

]]>TBA

]]>Speaker(s): Camillo De Lellis (Institute for Advanced Study)

]]>Speaker(s): Asher Auel (Dartmouth College)

]]>TBA

]]>Speaker(s): Thaleia Zariphopoulou (Presidential Chair in Mathematics) (University of Texas at Austin)

]]>Speaker(s): Thaleia Zariphopoulou (University of Texas at Austin)

]]>Speaker(s): Sung Gi Park (Harvard University)

]]>Speaker(s): Thaleia Zariphopolou (University of Texas at Austin)

]]>Speaker(s): Gheorghe Craciun (University of Wisconsin)

]]>Speaker(s): Antoine Song (Caltech)

]]>Impairments in retinal blood flow and oxygenation have been shown to contribute to the progression of glaucoma. In this study, a theoretical model of the human retina is used to predict blood flow and tissue oxygenation in retinal vessels and tissue for varied levels of intraocular pressure and in the presence or absence of blood flow regulation. The model includes a heterogeneous representation of retinal arterioles and a compartmental representation of capillaries and venules. A Greenâ€™s function method is used to model oxygen transport in the arterioles, and a Krogh cylinder model is used in the capillaries and venules. Model results predict that both increased intraocular pressure and impaired blood flow regulation can cause decreased tissue oxygenation. Results also indicate that a conducted metabolic response mechanism reduces the fraction of poorly oxygenated tissue but that pressure- and shear stress-dependent response mechanisms may hinder the vascular response to changes in oxygenation. Importantly, the heterogeneity of the vascular network demonstrates that average values of tissue oxygen levels hide significant localized defects in tissue oxygenation that may be involved in glaucoma. Ultimately, the model framework presented in this study will allow for future comparisons to sectorial-specific clinical data to help assess the potential role of impaired blood flow regulation in ocular disease. Speaker(s): Julia Arciero (IUPUI)

]]>Speaker(s): Benjamin Antieau (Northwestern University)

]]>Speaker(s): Jared Weinstein (Boston University)

]]>Speaker(s): Jared Weinstein (Boston University)

]]>Speaker(s): Benjamin Antieau (Northwestern)

]]>TBA

]]>TBA Speaker(s): Rahul Pandharipande (ETH Zürich)

]]>Speaker(s): Rahul Pandharipande (ETH Zurich)

]]>Speaker(s): Hao Xing (Boston University)

]]>Speaker(s): Maggie Miller (Stanford University)

]]>Speaker(s): Yvain Bruned (University of Lorraine)

]]>Speaker(s): Andy Zimmer (University of Wisconsin)

]]>Speaker(s): Michael Groechenig (University of Toronto)

]]>Speaker(s): Bianca Viray (University of Washington)

]]>