For random groups in the Gromov density model at $d<3/14$, we construct walls in the Cayley complex $X$ which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities $d<1/5$ and $d<5/24$, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density $d<1/4$.

I will discuss the quenched complexity of saddles in the spherical p spin-glass model. Our main result confirms the almost surely existence of a layered structure of critical points of the energy landscape. I will then relate this computation to a detailed information about the landscape around the ground state energy and the structure of the Parisi measure at zero temperature. Based on a joint work with Julian Gold (Northwestern University) and Yi Gu (Northwestern University).

Abstract : We consider a nonlinear heat equation on $R^n$ with a homogeneous, superlinear nonlinearity. We study solutions which are anti-symmetric with respect to the spatial variables. It is shown that very singular initial values, e.g. derivatives of the Dirac delta function, give rise to local (regular) solutions. These solutions exist when the homogeneity of the nonlinearity is below a value which is consistent with the scaling properties of the equation. These results enable us to obtain new finite time blowup results for certain classes of regular anti-symmetric initial values of the form $u_0 = \lambda f$. Counterintuitively, these blow-up results hold for $\lambda > 0$ sufficiently small. Blowup results of this type, where the initial value has a small coefficient, were first found by Dickstein [1]. The work to be presented is joint with S. Tayachi [2, 3]. References [1] F. Dickstein, Blowup stability of solutions of the nonlinear heat equation with a large life span, J. Dierential Equations 223 (2006), 303{328. [2] S. Tayachi and F. B. Weissler, The nonlinear heat equation with high order mixed derivatives of the Dirac Delta as initial values, Trans. Amer. Math. Soc. 366 (2014), 505-530. [3] S. Tayachi and F. B. Weissler, The nonlinear heat equation involving highly singular initial values and new blowup and life span results, Journal of Elliptic and Parabolic Equations, 4 (2018), 141-17

In this talk, I will present my work on studying a sequence of invariants, called Hodge ideals, which detect singularities of a hypersurface on a smooth complex variety and measures the Hodge theory on the complement of the hypersurface. These Hodge ideals arise naturally from Saito’s theory on the Hodge filtration of Hodge modules associated to the localization along a hypersurface and give a good generalization of multiplier ideals. I will give a general introduction to Hodge ideals for Q-divisors and show some applications in singularity theory. In particular, I will give explicit formulas of these ideals in some special cases and develop computational results of various invariants of singularities, e.g., generating level of Hodge filtration, roots of Bernstein-Sato polynomials and Hodge ideal spectrum.

There are several attempts to describe theories by Galois groups, and new notions of Galois group have been defined for this purpose (Shelah Galois group, Kim-Pillay Galois group, Lascar Galois group). My project goes in the other direction: instead of introducing new Galois groups, finding theories which are controlled by the “classical” Galois groups. In the case of the theory of fields, there is a special class of fields, pseudoalgebraically closed fields (PAC fields). PAC fields were the core of research in field theory in the second half of the 20th century. Why? Because the theory of a PAC field is controlled by its absolute Galois group, so all the machinery from Galois theory can be invoked and used with success; e.g. Nick Ramsey showed that a PAC field is NSOP1 if and only if its absolute Galois group is NSOP1. Therefore it makes sense to develop model-theoretic Galois theory in the case of PAC structures, a generalization of PAC fields. With my co-authors, I obtained recently a generalization of the Elementary Equivalence Theorem for PAC structures: two PAC structures share the same first order theory provided they have isomorphic absolute Galois groups. Also Ramsey’s result was generalized. In my talk, I will summarize the situation and explain connections between some results from my preprints, because combining them together gives us an algorithm for obtaining PAC structures with an absolute Galois group which can be “calculated”, and so there is a prospective way to generate new examples of NSOP1 structures.

Finding locally optimal solutions for max-cut and max-k-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP for max-cut has been studied in the smoothed complexity model. Etscheid and Roglin (2014) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is O(φ^5n^15.1), where φ is an upper bound on the random edge-weight density and n is the number of vertices in the input graph. In this talk, I will present an analysis technique that substantially improves the smoothed run-time bound. Our techniques provide a general framework for analyzing FLIP in the smoothed model. We illustrate this general framework by showing that the smoothed complexity of FLIP for max-3-cut in complete graphs is polynomial and for max-k-cut in arbitrary graphs is quasi-polynomial for constant k. Based on joint work with Ali Bibak and Charles Carlson.

For negatively curved surfaces, we study visual metrics on the boundary at infinity of their universal covers. We prove that the visual metric is classified up to bi-Lipschitz equivalence by its Hausdorff dimension, and we use this to construct many non-isometric negatively curved surfaces whose universal covers are almost-isometric (= quasi-isometric with multiplicative constant 1). As an application, we answer a question Alex posed in the first seminar this semester (due to him? Or Hamenstaedt?): Are there negatively curved metrics on a surface such that they are not isometric in any finite cover but almost-isometric on the universal cover? Joint work with Jean-François Lafont and Ben Schmidt.

In 1965, Paul Erdos easily proved that if $S$ is a finite set of nonzero real numbers, then there exists a sum-free subset $S' \subseteq S$ such that $|S'| \ge \frac{1}{3}|S|.$ Here, a sum-free subset $S$ is such that there is no triple of elements $a, b, c$ in $S$ for which $a + b = c.$ Eberhard, Green and Manners proved in 2013 that the same is not true for a constant bigger than $\frac{1}{3}$, i.e. $\frac{1}{3}$ is the biggest possible constant with this property. Here, we consider the analogous problem where triples $a, b, c$ in $S$ for which $a^b = c$ are forbidden. We show that $\frac{1}{8}$ is a lower bound for the optimal constant (private communication with Noga Alon), as well as that $\frac{1}{2}$ is an upper bound for it.

We present new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in the sense that it only detects the equality of means and the traces of covariance matrices in the high-dimensional setup. We propose a new class of metrics which inherit the desirable properties of the energy distance/distance covariance in the low-dimensional setting and is capable of detecting the homogeneity of/ completely characterizing independence between the low-dimensional marginal distributions in the high dimensional setup. We further propose t-tests based on the new metrics to perform high-dimensional two-sample testing/ independence testing and study its asymptotic behavior under both high dimension low sample size (HDLSS) and high dimension medium sample size (HDMSS) setups. The computational complexity of the t-tests only grows linearly with the dimension and thus is scalable to very high dimensional data. We demonstrate the superior power behavior of the proposed tests for homogeneity of distributions and independence via both simulated and real datasets.

This talk concerns a PDE system that models tumor growth. We show that a novel free boundary problem arises in the incompressible limit. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. This is joint work with Inwon Kim.

I will overview recent advances on the mathematical theory of wave propagation in fluids, especially problems involving several scales and nonlinear interactions between shocks, gravitational waves, and phase interfaces. Understanding the global dynamics of complex fluid flows requires insights and methods from the fields of continuum physics, geometric analysis (curvature operators on a Riemannian manifold), partial differential equations (Euler equations, Einstein equations), and scientific computation (finite volume methods, mesh-free algorithms). Blog: philippelefloch.org

Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the limit sets of Kleinian groups. Using these measures, they showed a close relationship between the critical exponent, $\delta(\Gamma)$, of a Kleinian group $\Gamma < \mathrm{Isom}(\mathbb{H}^n)$ and the Hausdorff dimension, $\mathrm{Hd}(\Lambda(\Gamma))$, of the limit set $\Lambda(\Gamma)$ of $\Gamma$. Intuitively, the critical exponent gives a geometric measurement of the growth rate of a(ny) $\Gamma$-orbit in $\mathbb{H}^n$ and, on the other hand, the Hausdorff dimension describes the size of the limit set $\Lambda(\Gamma)$. For instance, for convex-cocompact Kleinian groups $\Gamma$, Sullivan proved that $\delta(\Gamma) = \mathrm{Hd}(\Lambda(\Gamma))$. Anosov subgroups (or Anosov representations), introduced by Labourie and further developed by Guichard-Wienhard and Kapovich-Leeb-Porti, extend the notion of convex-cocompactness to the higher-rank. In this talk, we discuss how one can similarly understand the Hausdorff dimension of the limit sets of Anosov subgroups in terms of their appropriate critical exponents. This is a joint work with Michael Kapovich.

In the theory of dense graph limits, a graphon is a symmetric measurable function W from $[0,1]^2$ to $[0,1]$. Each graphon gives rise naturally to a random graph distribution, denoted $G(n,W)$, that can be viewed as a generalization of the Erdos-Renyi random graph. Recently, Dolezal, Hladky, and Mathe gave an asymptotic formula of order log(n) for the size of the largest clique in $G(n,W)$ when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of $G(n,W)$ will be of order $\sqrt{n}$ almost surely. We also give a family of examples with clique number of order $n^c$ for any c in $(0,1)$, and some conditions under which the clique number of $G(n,W)$ will be $o(\sqrt{n})$ or $\omega(\sqrt{n})$. This talk assumes no previous knowledge of graphons.

We develop distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called $m$-determinacy and show that theories of distality rank $m$ require certain products to be $m$-determined. Furthermore, for NIP theories, this behavior characterizes distality rank $m$.

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This paper proposes a new and effective simulation-based approach, called Repro Sampling method, to conduct statistical inference in high dimensional linear models. The Repro method creates and studies the performance of artificial samples (referred to as Repro samples) that are generated by mimicking the sampling mechanism that generated the true observed sample. By doing so, this method provides a new way to quantify model and parameter uncertainty and provide confidence sets with guaranteed coverage rates on a wide range of problems. A general theoretical framework and an effective Monte-Carlo algorithm, with supporting theories, are developed for high dimensional linear models. This method is used to jointly create confidence sets of selected models and model coefficients, with both exact and asymptotic inferences theories provided. It also provides a theoretical development to support the computational efficiency. Furthermore, this development allows us to handle inference problems involving covariates that are perfectly correlated. A new and intuitive graphical tool to present uncertainties in model selection and regression parameter estimation is also developed. We provide numerical studies to demonstrate the utility of the proposed method in a range of problems. Numerical comparisons suggest that the method is far better (in terms of improved coverage rates and significantly reduced sizes of confidence sets) than the approaches that are currently used in the literature. The development provides a simple and effective solution for the difficult post-selection inference problems.

Interpretability is the requirement that a model obtained by machine learning, beyond having predictive power, should be comprehensible for the user, or it should be possible to reason about its properties. For example, decision trees appear to be more interpretable than neural networks. We give a brief introduction to this topic, and discuss experimental results on interpretability aspects of word embeddings in natural language processing, and a theoretical approach to interpretability for Bayesian network classifiers using ordered binary decision diagrams. Joint work with Vanda Balogh, Gabor Berend, Karine Chubarian and Dimitris Diochnos.

I will describe work in progress that aims to prove a subconvex bound for the L-functions associated to a family of automorphic forms on $U(n+1) \times U(n)$, for arbitrary $n$. The methods involve period integral formulas, amplification, and insights from harmonic analysis.

This talk is part of the Midwest Dynamical Systems Conference - http://homepages.math.uic.edu/~hurder/mwds2019/ Totally geodesic submanifolds play an important role in the theory of hyperbolic manifolds. I will discuss a new rigidity theorem in this context: if a finite volume hyperbolic manifold M contains infinitely many closed totally geodesic hypersurfaces, then M is arithmetic. This answers a question asked by Reid and McMullen. I will explain why it is natural to think of arithmetic manifolds as rare or special in this context. I will also discuss a variant of the theorem for closed totally geodesic submanifolds of higher codimension and also an analogue where M is complex hyperbolic. I hope to give some ideas of the proofs. The proof in the real hyperbolic case is a combination of homogeneous dynamics with a superrigidity theorem also proven by dynamical methods. The proof in the complex hyperbolic case is more complicated. In addition to using those tools, it draws on the theory of Higgs bundles and also on a theorem about incidence geometry proven by Pozzetti in her study of maximal representations. This is joint work with Bader, Miller and Stover.

I will describe work with Derek Harland and work of my student Ragini Singhal on the deformation of instantons on nearly Kähler manifolds and nearly parallel G2 manifolds. Both type of manifolds admit Killing spinors, and this provides a framework to handle the study of the space of deformations of instantons, particularly in the homogeneous setting.

In the string trace reconstruction problem, there is an unknown binary string, and we observe noisy samples of this string after it has gone through a deletion channel. This deletion channel independently deletes each bit with constant probability q and concatenates the remaining bits. The goal is to learn the original string with high probability using as few traces as possible, where the sample complexity is characterized by the length of the string, n. String trace reconstruction is arguably the hottest problem in information theory, and researchers are incredibly stuck at bridging the gap between the exponential upper bound and n² lower bound. In the first part of this talk, I will explain the string trace reconstruction problem and the current best upper bound techniques. In the latter part, I will discuss several generalizations and variants of trace reconstruction, including our work on tree trace reconstruction. For many classes of trees, including complete trees and spiders, we provide algorithms that reconstruct the labels using only a polynomial number of traces. This exhibits a stark contrast to known results on string trace reconstruction, which require exponentially many traces, and where a central open problem is to determine whether a polynomial number of traces suffice. Based on joint work with Miklos Z. Racz and Cyrus Rashtchian

We will discuss classification results for t-structures on the bounded and unbounded derived categories of a Noetherian ring, identifying them with filtrations of subsets of SpecR. We will then use this to prove the non-existence of bounded t-structures on the category of perfect complexes and discuss the nature of the obstructions preventing t-structures on this category.

Let $C_m: y^2=x^m+c$ be a smooth projective curve defined over $\mathbb Q$. We would like to study the limiting distributions of the coefficients of the normalized L-polynomial for $C_m$. To determine the distributions, we study the Sato-Tate groups of the Jacobians of the curves. In this talk, I will give both general results and explicit examples of Sato-Tate groups for certain curves $C_m$. I will then use these groups to determine the limiting distributions of the coefficients of the normalized L-polynomial. This is joint work with M. Emory.

An Outreach Program for Chicago school students interested in Math

For a generic, rank $r$ Drinfeld module, we discuss its reductions and their endomorphism rings. We discuss the significance of the Frobenius indices and prove that they can be arbitrarily large. We also give an algorithm to compute these indices explicitly and a basis of the endomorphism ring. This is joint work with Mihran Papikian.

A Math Outreach Program for gifted Chicago school students

A Math Outreach Program for gifted Chicago school students

A Math Outreach Program for gifted Chicago school students

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We consider structures on the set of real numbers having the property that connected components of definable sets are definable. All o-minimal structures on the real line (R,<) have the property, as do all expansions of the real field that define the set N of natural numbers. Our main analytic-geometric result is that any such expansion of (R,<,+) by boolean combinations of open sets (of any arities) is either o-minimal or undecidable. We also show that expansions of (R, <, N) by subsets of N^n (n allowed to vary) have the property if and only if all arithmetic sets are definable. (Joint with A. Dolich, A. Savatovsky and A. Thamrongthanyalak.)

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