(**This is a rescheduled talk from the past week. Note the non-standard time and place**)

In this second talk of a two part sequence, we will continue discussing the geometry of the space of Kähler potentials on a compact manifold. In particular, we will see how Hamiltonian flows defined by the Cauchy data can give solutions to the geodesic equation in this space. The talk will be self-contained and we will review necessary results from the previous talk in the beginning.

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

]]>The Benjamin-Ono equation is a dispersive PDE in one space dimension introduced by T.B. Benjamin in 1967 as a model for long internal gravity waves in a two-layer fluid with infinite depth. It admits a Lax pair involving Toeplitz operators on the Hardy space. I will explain how this structure leads to an explicit formula for the general solution in terms of its initial datum, in both cases of periodic boundary conditions and of solutions on the line decaying at infinity. Applications to low regularity wellposedness and to the zero-dispersion limit will be discussed.

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

Abstract: I will present an optimal stopping problem related to choosing one of two products. The decision maker does not have complete information about one of them, and pays information acquisition costs to learn its quality. The choice action may be reversible or irreversible. This model gives rise to a new family of combined optimal stopping and filtering problems (joint work with R. Xu and L. Zhang).

In this talk, we will discuss the asymptotic behavior of the solutions of differential inclusions governed by maximally monotone operators. In the case where the LaSalle’s invariance principle is inconclusive, we provide a refined version of the invariance principle theorem. This result derives from the problem of locating the ω-limit set of a bounded solution of the dynamic. In addition, we propose an extension of LaSalle’s invariance principle, which allows us to give a sharper location of the ω-limit set. The provided results are given in terms of nonsmooth Lyapunov pair-type functions. We will conclude this presentation by applying our results to an important second-order gradient-like dissipative dynamical system with Hessian-driven damping called (DIN) by Alvarez et and his collaborators.

]]>Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

In many applications the unknown is an interface: a curve or a surface that evolves in time. I will describe numerical methods that have been developed to simulate these motions, focusing on a particular application from material science: evolution of microstructure in polycrystalline materials (e.g. metals, ceramics).

]]>Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

We study an inverse problem for the wave equation, concerned with estimating the wave speed aka the velocity, from data gathered by an array of sources and receivers that emit probing signals and measure the resulting waves. The typical mathematical formulation of velocity estimation is a nonlinear least squares minimization of the data misfit, over a search velocity space. There are two main impediments to this approach, which manifest as multiple local minima of the objective function: The nonlinearity of the mapping from the velocity to the data, which accounts for multiple scattering effects, and poor knowledge of the kinematics (smooth part of the wave speed) which causes "cycle-skipping". We introduce a novel approach to waveform inversion, which is based on a reduced order model that describes the wave propagation. This model is data driven and we show how it can be used to facilitate the inversion.

]]>The mathematical analysis of global properties of polynomial dynamical systems can be very challenging (for example: the second part of Hilbert’s 16th problem, or the analysis of chaotic dynamics in the Lorenz system).

On the other hand, any dynamical system with polynomial right-hand side can essentially be regarded as a model of a reaction network. Key properties of reaction systems are closely related to fundamental results about global stability in classical thermodynamics. For example, the Global Attractor Conjecture can be regarded as a finite dimensional version of Boltzmann’s H-theorem. We will discuss some of these connections, as well as the introduction of toric differential inclusions as a tool for proving the Global Attractor Conjecture.

We will also discuss some implications for the more general Persistence Conjecture (which says that solutions of any weakly reversible system cannot "go extinct"), as well as some applications to biochemical mechanisms that implement noise filtering and cellular homeostasis.

Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

The Schubert polynomials of the Symmetric group of n form a basis of the vector space they span. This space is well-studied with a dimension of n! and its Hilbert series being the q-analogue of n!. Key polynomials, which are characters of the Demazure modules, also form a basis for this space along with Schubert polynomials. Schubert and key polynomials are the ``bottom layers’’ of Grothendieck and Lascoux polynomials, respectively, which are two inhomogeneous polynomials. In this talk, we will focus on the space spanned by their ``top layers’’. We construct two bases using the top layer of Grothendieck and the top layer of Lascoux. We will also introduce a diagrammatic method to compute the degrees of these polynomials, involving drawing dark clouds and snowflakes. Finally, we will describe the Hilbert series of this space using a classical q-analogue of the Bell numbers.

]]>The Hodge-Tate decomposition for an abelian variety is a p-adic analogue of the Hodge decomposition of a complex algebraic variety which allows us to relate the étale cohomology of a variety to its Hodge cohomology groups. In this talk, we sketch a proof of this decomposition for H^1 of an abelian variety over a p-adic field with good reduction. The only prerequisites are the basic facts about p-adic fields and abelian varieties.

]]>Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

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]]>Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

Department of Mathematics Recruitment Weekend

March 24 – March 25, 2023

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Abstract: Since their inception perfectoid spaces have catalyzed a revolution in p-adic geometry. We redevelop the foundations of perfectoid spaces from the point of view of Berkovich Spaces, where the underlying topological space of an affinoid perfectoid space is a compact Hausdorff space — closely resembling the situation in complex geometry. The key technical ingredient in our construction is arc_pi descent for perfectoid Banach algebras.

Hybrid Defense:

Room: 2437 Mason Hall

Join Zoom Meeting: https://umich.zoom.us/j/95138305325

Passcode: 648229

TBA

]]>Title: Progenerators of Bernstein blocks

Abstract: Let F be a non-archimedean local field and G be a connected reductive group over F. For a Bernstein block in the category of smooth complex representations of G(F), we have two kinds of progenerators: the compactly induced representation ind_K^G(rho) of a type (K, rho), and the parabolically induced representation I_P^G(Pi^M) of a progenerator Pi^M of a Bernstein block for a Levi subgroup M of G. In this talk, we construct an explicit isomorphism of these two progenerators. We also explain that the induced isomorphism between the endomorphism algebras is compatible with their descriptions in terms of affine Hecke algebras.

Given a Riemannian manifold (M,g), it was realized a long time ago that special maps from (M,g) to some metric spaces give important insights on the geometry of (M,g). Here “special” means isometric, harmonic or area minimizing etc. I will give a survey of such special mappings and some applications to systolic geometry, inverse problems, dynamical systems. I will also discuss new applications to stability problems for geometric inequalities, and to the construction of canonical shapes for manifolds.

]]>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.

The Deligne-Mumford-Knudsen moduli space of n-pointed stable curves of genus 0 is a smooth projective variety of dimension n - 3. We construct an integral isomorphism from its Grothendieck ring of vector bundles to its Chow cohomology ring which satisfies a Hirzebruch-Riemann-Roch-type formula. This isomorphism, which is unrelated to the Chern character, can be used to reduce many K-theoretic computations to easier problems in intersection theory. Joint with Shiyue Li, Sam Payne, and Nick Proudfoot.

]]>Abstract: The configuration space of n ordered unit squares sliding in a p by q rectangle is a subspace of the configuration space of n ordered points in the plane, which is the classifying space of the pure braid group. The homology of the pure braid group is generated by cycles that have a nice geometric description. To what extent is this true of squares in a rectangle? Small computations suggest that some more intricate generators may be needed.

]]>I will present an upcoming work with J. Luk (Stanford), where we develop a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions, which is applicable to nonlinear problems on dynamical backgrounds. In addition to its inherent interest, such information is crucial for studying problems involving the interaction of waves with a spatially localized object; indeed, our motivation for developing this method comes from the Strong Cosmic Censorship Conjecture. I will explain how our method recovers and refines Price's law for linear problems on stationary backgrounds, and also how it shows that the late time tails are in general different(!) from the linear stationary case in the presence of nonlinearity and/or a dynamical background.

]]>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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]]>Abstract: We study the convergence of certain classes of complex geometric measures to certain non-Archimedean measures. This convergence takes place on the non-Archimedean hybrid space introduced by Boucksom and Jonsson. Given a family X of complex analytic spaces parametrized by the punctured unit complex disk, the hybrid space associated to this family is a partial compactification of this family obtained by filling in the puncture with the Berkovich analytification of X. Furthermore, if each of the complex analytic spaces in the family carry a natural measure, we can think of these measures as being supported on the hybrid space, then their weak limit is a measure supported on the Berkovich space.

First, we study the convergence of volume forms on a degenerating holomorphic family of log Calabi–Yau varieties, extending a result of Boucksom and Jonsson. Secondly, we prove a folklore conjecture that the Bergman measures along a holomorphic family of curves parametrized by the punctured unit disk weakly converge to the Zhang measure on the associated Berkovich space.

Hybrid Defense:

271 Weiser Hall

https://umich.zoom.us/j/93673413700?pwd=clQ0eVBFZDI4eXBXTVBtOHFPbTNmdz09

Meeting ID: 936 7341 3700

Passcode: thesis

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TBA

]]>Title: Higher Modularity of Elliptic Curves

Abstract: Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), but one can also extend the idea further and ask whether E is "r-modular" for r=2,3.... To define this generalization, the modular curve gets replaced with Drinfeld's concept of a "shtuka space". The r-modularity of E is predicted by Tate's conjecture. In joint work with Adam Logan, we give some classes of elliptic curves E which are 2- and 3-modular.

Professors Michele Intermont, Gavin LaRose, Stephen Oloo, Amy Shell-Gellasch, and Nina White will explain the process of finding a teaching-focused job in academia, as well as answer questions from the audience.

Location: B860 Math Lab East Hall

In the classification of unitary representations of the Lie group SL_2(R), the "discrete series" representations can be modeled on a space of L^2 functions on the upper half plane, and there's essentially one for each integer. What if R is replaced with the field Q_p of p-adic numbers? The group SL_2(Q_p) still has a discrete series, but it is much richer than that of SL_2(R). To study it, one can consider a p-adic version of the upper half plane. But strangely, the p-adic upper half plane isn't simply connected, and to unlock the full discrete series, one needs to study some of its finite covering spaces (the Lubin-Tate tower).

And for p-adic groups other than SL_2(Q_p)? This talk will be an invitation to the chain of beautiful ideas leading from the Lubin-Tate tower to a much more general notion of "local Shimura varieties" which are implicated in the Langlands program for p-adic fields.

Projective spaces contain lots of genus 1 curves. Does every twisted form of projective space contain a genus 1 curve? This basic arithmetic question was asked by David Saltman and Pete Clark and answered in low dimensions by Johan de Jong and Wei Ho using geometric methods. I will explain the history of the problem as well as recent joint work with Asher Auel where we show using arithmetic geometry, and the help of an observation of David Saltman, that the answer is `yes’ in small dimensions over global fields.

]]>Abstract: We investigate the relation between boundary data of a compact manifold and its interior geometry. A compact Riemannian manifold $D$ with smooth boundary $\del D$ is boundary rigid if its interior geometry is uniquely determined by $\del D$ and distances between points on $\del D$. $D$ is a minimal filling if for any $D'$ with $\del D'=\del D$, having larger distances between points on $\del D$ implies $\vol(D')\geq\vol(D)$.

In this thesis, we generalize D. Burago and S. Ivanov's work \cite{Burago2} on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of real, complex, quaternionic and Cayley hyperbolic metrics.

Hybrid Defense:

Angell Hall 2163

Zoom link: https://umich.zoom.us/j/97495692365?pwd=RG5BSTJyb3Z4QzlWWGxicmU4bTJwZz09

Meeting ID: 974 9569 2365

Passcode: ruanypdef

Abstract: TBA

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

Zoom Webinar Link:

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

In this paper, we develop the theory of functional generation of portfolios in an equity market of a changing dimension. By introducing dimensional jumps in the market, as well as jumps in stock capitalization between the dimensional jumps, we construct different types of self-financing stock portfolios (additive, multiplicative, and rank-based) in the most general setting. Our study explains how a dimensional change caused by a listing or delisting event of a stock and unexpected shocks in the market affect portfolio return. We also provide empirical analyses of some classical portfolios, quantifying the impact of dimensional change in relative portfolio performance with respect to the market.

]]>Given a closed Riemannian manifold M, the length of the shortest geodesic for each free homotopy class of loops on M is called the (minimal) length of the class. This gives a map called marked length spectrum. It is conjectured that the fundamental group and marked length spectrum together determine the isometric type of negatively curved manifolds. This conjecture has been verified for surfaces and locally symmetric spaces. In this talk, we show that for negatively curved arithmetic manifolds, the fundamental group with all pairs of different equal length classes, i.e., marked length pattern, is enough to recover the metric up to scaling. The central idea is marked length spectrum rigidity of cocycles.

]]>Professors Charlotte Chan, Alex Perry, and Alex Wright will explain the process of finding a research-focused job in academia as well as answer questions from the audience.

Location: B860 Math Lab East Hall

Speaker: Marius Beceanu (SUNY, Albany)

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TBA

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]]>Abstract: In these lectures, I will discuss results, conjectures, and counterexamples related to the cohomology and algebraic cycle theory of three fundamental moduli spaces in algebraic geometry: the moduli of curves, the moduli of K3 surfaces, and the moduli of abelian varieties. The lectures will emphasize various beautiful connections between these spaces. The goal will be to present an up-to-date view of the structure of the tautological classes without assuming any previous knowledge of the study of these moduli spaces.

A reception for Professor Pandharipande will be held Tuesday, April 11, at 5:00 p.m. in the Mathematics Upper Atrium, East Hall

This presentation will provide graduate students with library resources available to help with research, dissertation writing, and job searches for mathematicians.

]]>Market-wide trading halts, also called circuit breakers, have been widely adopted as part of the stock market architecture, in the hope of stabilizing the market during dramatic price declines. We develop an intertemporal equilibrium model to examine how circuit breakers impact market behavior and welfare. We show that a circuit breaker tends to lower the overall level of the stock price and significantly alters its dynamics. In particular, as the price approaches the circuit breaker, its volatility rises drastically, accelerating the chance of triggering the circuit breaker -- the so-called ``magnet effect''; in addition, returns exhibit increasing negative skewness and positive drift, while trading activity spikes up. Our empirical analysis finds supportive evidence for the model's predictions. Moreover, we show that a circuit breaker can affect the overall welfare either negatively or positively, depending on the relative significance of investors' trading motives for risk sharing vs. irrational speculation. This is a joint work with Hui Chen, Anton Petukhov, and Jiang Wang.

]]>Abstract: Calderon's inverse problem asks whether can one determine the conductivity of a medium by making voltage and current measurements at the boundary. This question arises in several areas of applications including medical imaging and geophysics. I will report on some of the progress that has been made on this problem since Calderon proposed it, including recent developments on similar problems for nonlinear equations and nonlocal operators.

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

Zoom Webinar Link:

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

Speaker(s): Rahul Pandharipande (ETH Zürich)

Abstract: In these lectures, I will discuss results, conjectures, and counterexamples related to the cohomology and algebraic cycle theory of three fundamental moduli spaces in algebraic geometry: the moduli of curves, the moduli of K3 surfaces, and the moduli of abelian varieties. The lectures will emphasize various beautiful connections between these spaces. The goal will be to present an up-to-date view of the structure of the tautological classes without assuming any previous knowledge of the study of these moduli spaces.

TBA

]]>In this workshop, Professors Wolfe and Bennett will guide graduate students with writing a teaching statement for the academic job search.

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

Abstract: In these lectures, I will discuss results, conjectures, and counterexamples related to the cohomology and algebraic cycle theory of three fundamental moduli spaces in algebraic geometry: the moduli of curves, the moduli of K3 surfaces, and the moduli of abelian varieties. The lectures will emphasize various beautiful connections between these spaces. The goal will be to present an up-to-date view of the structure of the tautological classes without assuming any previous knowledge of the study of these moduli spaces.

The presentation deals with the Nonlinear Evolution Equations Integrable by the Inverse Scattering Method. Several self-consistent sources for these Nonlinear Evolution Equations are presented.

]]>I will sketch three projects in opinion dynamics. (1) The popular Hegselmann-Krause model of opinion dynamics assumes that you are persuaded to change your views only by those whose views are not too far from yours to start with. The model predicts the emergence of “echo chambers” --- groups of people with identical views --- with different echo chambers holding starkly different views. Alternatively, we might assume that you are persuaded by your neighbors. This model would predict the emergence of “red states” and “blue states” --- large-scale geographic regions in which one or the other view dominates --- with milder differences in views. This can be seen as akin to the well-known fact that iterative methods for Laplace’s equation smooth rapidly but converge slowly. In a computational study, we combined the two ideas, and in some instances found geographically coherent echo chambers --- large-scale geographic regions with views starkly different from those in other regions. We applied our model to attitudes about COVID vaccines in the United States, reproducing some aspects of reality. (2) The Hegselmann-Krause model is discrete. We propose ODE and PDE analogues and analyze some of their properties. (3) We model two candidates maneuvering opportunistically to maximize their share of the vote while opinions in the electorate evolve as well. The optimal candidate strategy (not just the outcome) can depend discontinuously on voter behavior. The underlying mathematical mechanism is a saddle-node bifurcation.

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In this workshop, hosts from the local AWM will guide graduate students through the process of creating a professional webpage to highlight during their job search.

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Abstract: TBA

TBD.

The talk will be virtual, but we will have a watch party in our usual room. If you would like the zoom info to attend virtually instead, email an organizer or join our mailing list.

We say a knot in the 3-sphere is slice if it bounds a smooth disk into the 4-ball. If the disk can be obtained from a special kind of immersion into S^3 (pushed into B^4), then we say the knot is ribbon. A long-standing open conjecture of Fox posits that every slice knot is ribbon. I will talk about this open problem and discuss why it would be so difficult to disprove this conjecture, if it is false (and one obstruction that might work). I’ll relate sliceness of knots to lower bounds on the genus of some surfaces embedded in nontrivial 4-manifolds, and use some old (difficult) such lower bounds to recover a recent interesting result of Dai—Kang—Mallick—Park—Stoffregen that a certain knot is not slice (which is unfortunate, because it’s much easier to prove that it isn’t ribbon). This is joint work with Paolo Aceto, Nickolas A. Castro, JungHwan Park and András Stipsicz.

]]>In this workshop, graduate students will learn how to write a research statement designed to find a research based post-doctoral position.

]]>This presentation will give an overview of post-doc positions and the application process required to find one.

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]]>This workshop is designed to highlight the important aspects to include on a CV and cover letter for mathematicians looking for a job in academia.

]]>Increasingly, hiring committees are interested in how prospective faculty job candidates will contribute to diversity, equity, and inclusion. As a result, many academic employers have begun to request a diversity statement as part of the faculty job application process. In this interactive session, we will discuss best practices for writing diversity statements, examine sample statements, and work through activities designed to help participants start writing their own statement.

Learning objectives:

Reflect on ways you are committed to diversity, equity, and inclusion in your research, teaching, engagement, leadership, or other areas

Identify resources that allow you to participate and contribute to DEI initiatives, opportunities, projects, and research

Review best practices to write a diversity statement and learn how to critically evaluate diversity statements

Abstract: Log Fano cone singularities are generalizations of cones over log Fano varieties. There is a local K-stability theory for log Fano cone singularities, originally developed by Collins-Sz\'ekelyhidi,

generalizing the one for log Fano varieties. Similar to its global counterpart, local K-stability serves as the obstruction to the existence of a Ricci-flat K\"ahler cone metric on log Fano cone singularities.

We study local K-stability from a non-Archimedean point of view. We first introduce the local Monge-Amp\`ere energy, generalizing a valuative invariant studied by Blum-Jonsson in the log Fano case. We then move on to developing a non-Archimedean pluripotential theory, generalizing the global theory developed by Boucksom-Jonsson. Finally, as an application of the local pluripotential theory, we give a non-Archimedean characterization of local K-stability.

Hybrid Defense:

1084 East Hall

Zoom link: https://umich.zoom.us/j/94776920765?pwd=Z1craTUzb3c1a2QvWHZFWmZRRW5Ldz09

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

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]]>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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