Toric ideals are ideals of polynomial rings with interesting algebraic, geometric, and combinatorial properties. In this talk, we'll introduce toric ideals and give several examples. We'll show how to determine Groebner bases for them, as well as discuss their applications to algebraic statistics. If time permits, we'll also give a combinatorial method for calculating their syzygies and multigraded Betti numbers.

]]>Berkovich spaces are analogues of complex manifolds when the complex numbers are replaced by a non-Archimedean field, that is, a field satifying the strong triangle inequality.

I will discuss two instances where Berkovich spaces naturally appear within complex geometry. The first concerns the Yau--Tian--Donaldson conjecture, on the existence of Kähler--Einstein metrics on Fano manifolds. The second situation appears in the context of degenerations of Calabi--Yau manifolds, and features conjectures by Strominger--Yau--Zaslow, and Kontsevich--Soibelman.

This is based on joint work with R. Berman, S. Boucksom, J, Hultgren, E. Mazzon, and N. McCleerey.

We introduce combinatorial formulas of (double) Schubert and Grothendieck polynomials based on bumpless pipe dreams and give a combinatorial proof of Monk’s rule for Schubert and double Schubert polynomials using bumpless pipe dreams that generalizes Schensted’s insertion on semi-standard Young tableaux. We also give a bijection between pipe dreams and bumpless pipe dreams and discuss its canonical nature.

]]>We study graphon mean-field backward stochastic differential equations (BSDEs) with jumps and associated dynamic risk measures. We establish the existence, uniqueness and measurability of solutions under some regularity assumptions and provide some estimates for the solutions. We moreover prove the stability with respect to the graphon particle systems and obtain the convergence of an interacting mean-field particle system with inhomogeneous interactions to the graphon mean-field BSDE. We then provide some comparison theorems for the graphon mean-field BSDEs. As an application, we introduce the graphon dynamic risk measure induced by the solution of a graphon mean-field BSDE system and study its properties. We finally provide a dual representation theorem for the graphon dynamic risk measure in the convex case.

]]>Suppose that a group acts on a variety. When can the variety and the action be resolved so that all stabilizers are finite? Kirwan gave an answer to this question in the 1980s through an explicit blowup algorithm for smooth varieties with group actions in the context of Geometric Invariant Theory (GIT). In this talk, we will explain how to generalize Kirwan's algorithm to Artin stacks in derived algebraic geometry, which, in particular, include classical, potentially singular, quotient stacks that arise from group actions in GIT. Based on joint work with Jeroen Hekking and David Rydh.

]]>We discuss quantum ergodicity in the Benjamini-Schramm limit. This concerns equidistribution of eigenfunctions of Laplacian-like operators on sequences of spaces which ``converge'' to their common universal cover. We shall be particularly interested in the case when the universal cover is a symmetric space or an affine building (the non-archimedean analogue of a symmetric space). A result of this kind was first proven by Anantharaman-Le Masson for regular graphs and for which the underlying Laplacian-like operator is the adjacency operator. This result was reproven by Brooks-Le Masson-Lindenstrauss using a new technique which has been subsequently adapted to also work for rank one locally symmetric spaces (Le Masson-Sahlsten, Abert-Bergeron-Le Masson) and for higher rank locally symmetric spaces associated to $SL(d, R)$ (Brumley-Matz). We have obtained analogous results for Bruhat-Tits buildings associated to $SL(3, F)$ where $F$ is a non-archimedean local field. We shall discuss the strategy of proof common to all of these examples as well as discuss some of the new techniques introduced to handle the $SL(3, F)$ case.

]]>In this project, we consider a class of generalized Kyle-Back strategic insider trading models in which the insider is able to use the dynamic information obtained by observing the instantaneous movement of an underlying asset that is allowed to be influenced by its market price. Since such a model will be largely outside the Gaussian paradigm, we shall try to Markovize it by introducing an auxiliary (factor) diffusion process, in the spirit of the weighted total order process, as a part of the "pricing rule". As the main technical tool in solving the Kyle-Back equilibrium in such a setting, we study a class of Stochastic Two-Point Boundary Value Problem (STPBVP), which resembles the dynamic Markov bridge in the literature, but without insisting on its local martingale requirement. In the case when the solution of the STPBVP has an affine structure, we show that the pricing rule functions, whence the Kyle-Back equilibrium, can be determined by the decoupling field of a forward-backward SDE obtained via a non-linear filtering approach, along with a set of compatibility conditions. This is a joint work with Jin Ma.

]]>We will continue our study of continuous logic by

introducing model-theoretic notions such as theories and elementary

equivalence and the Tarski-Vaught test for continuous logic.

Define test configurations; Rees construction; Define the Futaki invariant; Intersection-theoretic formula for the Futaki invariant; Definition of K-stability; Examples.

]]>This is the second of a 2-3 part series on elementary Teichmüller theory, and on the SL(2, R) action on related moduli spaces. We will also explore links between the dynamics on Teichmüller space to the dynamics of discrete subgroups of Lie groups on homogeneous spaces.

]]>In their seminal work, Polyak and Juditsky showed that stochastic approximation algorithms for solving smooth equations enjoy a central limit theorem. Moreover, it has since been argued that the asymptotic covariance of the method is best possible among any estimation procedure in a local minimax sense of H´ajek and Le Cam. A long-standing open question in this line of work is whether similar guarantees hold for important non-smooth problems, such as stochastic nonlinear programming or stochastic variational inequalities. In this work, we show that this is indeed the case. This is joint work with Damek Davis and Liwei Jiang.

]]>This talk will discuss an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to closed manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exact topological invariants. To demonstrate the utility of the proposed method, the application is considered for predicting binding free energy (BFE) changes in protein-protein interactions (PPIs) induced by mutations with machine learning modeling. The technique has great application in studying the SARS-CoV-2 virus' infectivity, antibody resistance, and vaccine breakthrough, which will be presented in this talk.

]]>We begin by introducing left and right RSK insertions for Schubert calculus of complete flag varieties. The objects being inserted are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. Unlike in the Grassmannian case, the left and right insertions do not always commute. We give a criterion under which they do commute, which motivates the definition of associative biwords that can be used to give a positive rule for Schubert structure constants in the separated-descent case. We generalize Knuth relations to associative biwords and show some basic properties. Furthermore, we demonstrate that the associative biwords naturally admit the Demazure crystal structure and hence give the Schubert-to-key expansions. Finally, we will briefly discuss hopes and obstacles for these techniques to solve more unknown cases of Schubert product rules, as well as some relevant problems for exploration. This is joint work with Pavlo Pylyavskyy; the part relevant to crystals is also joint with Tianyi Yu.

]]>Ruled surface is one of the most concrete examples we see when studying algebraic surfaces. These are surfaces that admit a fibration by $\PP^{1}$ over a curve. However, ruled surfaces are very boring since they are too easy in various ways. For instance, they can be easily classified, there are no degenerations, the algebraic structure of the fibre does not change, and the canonical bundle is easy to describe.

On the other hand, fibration by elliptic curves is way more entertaining since there are lots of things happening! We can study how the algebraic structure varies, how the fibre degenerates to a singular one, and can describe the canonical bundle in terms of the singular fibres and the moduli.

It turns out that these phenomena for elliptic surfaces can be generalized to many deep results in algebraic geometry such as variation of Hodge structure, degeneration of Hodge structure, adjunction and subadjunction, canonical bundle formula, semipositivity theorems, volume asymptotics and so on.

Despite the fact that elliptic fibrations are related to these profound theories in algebraic geometry, the example itself is very classical and can be understood explicitly. I will talk about these phenomena for elliptic surfaces in various perspectives.

This talk focuses on q-orthogonal polynomials, orthogonal polynomials whose orthogonality condition is supported on the discrete lattice: q^k, for integer k . We investigate the behaviour of such polynomials in the case k>0 (q<1) and, if time permits, we study what happens when we relax this condition and allow k to be negative. Using the RHP framework we deduce the asymptotic behaviour of these polynomials as their degree tends to infinity, as well as other properties such as uniquness.

]]>Title: Orbital integrals for gln and smoothening

Abstract: Orbital integral is a fundamental object in the geometric side of the trace formula. A traditional method to study orbital integrals is through Bruhat-Tits building or affine Springer fiber. In this talk, we will propose another method to study orbital integrals using smoothening.

As an application, we will explain a closed formula of the orbital integral for gln with n=2,3 and a new lower bound for a general n. We also propose a conjecture about the estimation of the orbital integral for any n. Our method works for any local field of characteristic 0 or >n. This is a joint work with Sungmun Cho.

The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on $d$-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson localization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types. We will consider quasiperiodic operators

$$

(H(x)\psi)_n=\epsilon(\Delta\psi)_n+f(x+n\cdot\omega)\psi_n,\quad n\in \mathbb Z^d,

$$

where $\Delta$ is the discrete Laplacian, $\omega$ is a vector with rationally independent components, and $f$ is a $1$-periodic function on $\mathbb R$, monotone on $(0,1)$ with a positive lower bound on the derivative and some additional regularity properties. We will focus on two methods of proving Anderson localization for such operators: a perturbative method based on direct analysis of cancellations in the Rayleigh—Schr\”odinger perturbation series for arbitrary $d$, and a non—perturbative method based on the analysis of Green’s functions for $d=1$, originally developed by S. Jitomirskaya for the almost Mathieu operator.

The talk is based on joint works with S. Krymskii, L. Parnovski, and R. Shterenberg (perturbative methods) and S. Jitomirskaya (non-perturbative methods).

TBA

]]>A full exceptional collection is an important structure on a derived category with many valuable implications. For instance, such a collection produces a basis for the Grothendieck group. After reviewing the landscape of full exceptional collections on linear GIT quotients, we will discuss how to produce them using ideas from "window" categories and equivariant geometry. As an application, we will consider a large class of linear GIT quotients by a reductive group G of rank two, where this machinery produces full exceptional collections consisting of tautological vector bundles. This talk is based on joint work with Daniel Halpern-Leistner.

]]>Abstract: Two salient features of empirical temporal (i.e., time-varying) network data are the time-varying nature of network structure itself and heavy-tailed distributions of inter-contact times. Both of them can strongly impact dynamical processes occurring on networks, such as contagion processes, synchronization dynamics, and random walks. In the first part of the talk, I introduce theoretical explanation of heavy-tailed distributions of inter-contact times by state-dynamics modeling approaches in which each node is assumed to switch among a small number of discrete states in a Markovian manner and the nodes' states determine time-dependent edges. This approach is interpretable, facilitates mathematical analyses, and seeds various related mathematical modeling, algorithms, and data analysis (e.g., theorizing on epidemic thresholds, random walks on metapopulation models, inference of mixtures of exponential distributions, new Gillespie algorithms, embedding of temporal network data), some of which we will also discuss. The second part of the talk is on modeling of temporal networks by static networks that switch from one to another at regular time intervals. This approach facilitates analytical understanding of diffusive and epidemic dynamics on temporal networks as well as an efficient algorithm for containing epidemic spreading as convex optimization. Finally, I will touch upon some of my interdisciplinary collaborations including those on static networks.

Event will take place in-person in 4448 East Hall and online via Zoom.

Zoom Webinar Link:

https://umich.zoom.us/j/98734707290

Introduce the space of valuations; Central fiber of test configurations viewed as divisorial valuations.

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]]>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.

TBA

]]>I present an overview of recent research on defining projections in general positively-curved spaces (PC spaces), in particular the 2-Wasserstein space and the Gromov-Wasserstein space defined on metric measure spaces. I will highlight applications to economics and causal inference. This talk is based on joint works with Meng Hsuan Hsieh, Myung Jin Lee, and Yiman Ren.

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]]>S (volume) and T-invariants of a filtration; Define uniform K-stability.

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

]]>K-stability of canonically polarized and Calabi-Yau varieties; A necessary condition for K-stability.

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

]]>We develop different models to study the flutter of membranes (of zero bending rigidity) with vortex-sheet wakes in two- and three-dimensional inviscid flows. For 2D flows, we use a nonlinear, time-stepping method to study large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density, and stretching rigidity. With a linearized version of the membrane-vortex-sheet model we also investigate the instability of a membrane by solving a nonlinear eigenvalue problem iteratively, for three boundary conditions---both ends fixed, one end fixed and one free, and both free. We further consider a simple physical setup: a membrane held by tethers with hinged ends, that interpolates between the fixed--fixed and free--free cases. We additionally study an infinite membrane model mounted on a periodic array of Hookean springs. This model allows us to compute asymptotic scaling laws for how the frequencies, growth rates, and eigenmodes depend on membrane pretension and mass density. Finally, we develop a nonlinear model and computational method to study large-amplitude membrane flutter in 3D inviscid flow for 12 different boundary conditions.

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]]>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)

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]]>Special test configurations; $\beta$-invariant and Fujita-Li criterion; $\alpha$ and $\delta$-invariants.

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]]>Area-minimizing integral currents are natural generalization of area-minimizing oriented surfaces, which allow to get existence of minimizers even in those cases in which they cannot be smooth.

Almgren's famous Big Regularity Paper proves that the interior singular set of any $m$-dimensional area-minimizing integral current in any smooth Riemannian manifold $\mathcal{M}$ has (Hausdorff) dimension at most $m-2$. Except for the case $m=2$, when it was proved that interior singularities are isolated, little is known about the structure of the singular set. Moreover a recent theorem by Liu proves that we cannot expect it to be a $C^1$ $m-2$-dimensional submanifold (unless the ambient $\mathcal{M}$ is real-analytic) as in fact it can be a fractal set of any Hausdorff dimension $\alpha \leq m-2$. On the other hand it seems likely that it is an $(m-2)$-rectifiable set, i.e. that it can be covered by countably many $C^1$ submanifolds.

In this talk I will explain why the problem is very challenging and how it can be broken down into easier pieces following a recent joint work with Anna Skorobogatova.

TBA

]]>Given two probability measures on sequential data, we investigate the transport problem with time-inconsistent preferences under a discrete-time setting. Motivating examples include nonlinear objectives, state-dependent costs, and regularized optimal transport with general $f$-divergence. Under the bi-causal constraint, we introduce equilibrium transport and characterize it with maximum theorem and extended dynamic programming principle. We apply our framework to study the state dependence of two job markets including top-ranking executives and academia. The empirical analysis shows that a job market with a stronger state dependence is less efficient. The University of California (UC) postdoc job market has the strongest state dependence even than that of top executives, while there is no evidence of state dependence on the UC faculty job market. This is a joint work with Erhan Bayraktar.

]]>Variational approach; Valuations computing $\delta$-invariant; Finite generation.

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

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]]>Overview; Normalized volume (existence and uniqueness of the minimizer); Boundedness via normalized volume.

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

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

]]>I will talk about the following question of Gromov: what closed manifolds can be efficiently wrapped with Euclidean wrapping paper? That is, for what M is there a 1-Lipschitz map $\mathbb R^n \to M$ with positive asymptotic degree? Gromov called such manifolds elliptic. We show that, for example, the connected sum of k copies of CP^2 is elliptic if and only if k ≤ 3. I will try to explain the intuition behind this example, how it extends to a more general dichotomy governed by the de Rham cohomology of M, and why ellipticity is central to the program of understanding the relationship between topology and metric properties of maps.

If I have time, I'll also explain why for a non-elliptic M, a maximally efficient map $\mathbb R^n \to M$ must have components at many different frequencies (in a Fourier-analytic sense), and even then it's at best logarithmically far from having positive asymptotic degree. This is joint work with Sasha Berdnikov and Larry Guth.

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]]>Openness (via normalized volume); existence of a good moduli; properness and projectivity of K-moduli.

]]>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)

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Speaker(s): Jared Weinstein (Boston University)

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

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

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

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

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Speaker(s): Maggie Miller (Stanford University)

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

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

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

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

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

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