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 == Abstracts ==   == Abstracts == 
   
−  ===January 12: Botong Wang (Notre Dame)===  +  ===September 4: Isaac Goldbring (UIC) === 
   
−  ====Cohomology jump loci of algebraic varieties====  +  ==== ==== 
−   
−  In the moduli spaces of vector bundles (or local systems), cohomology jump loci are the algebraic sets where certain cohomology group has prescribed dimension. We will discuss some arithmetic and deformation theoretic aspects of cohomology jump loci. If time permits, we will also talk about some applications in algebraic statistics.
 
−   
−  ===January 14: Jayadev Athreya (UIUC)===
 
−   
−  ====Counting points for random (and notsorandom) geometric structures====
 
−   
−  We describe a philosophy of how certain counting problems can be studied by methods of probability theory and dynamics on appropriate moduli spaces. We focus on two particular cases:
 
−   
−  (1) Counting for RightAngled Billiards: understanding the dynamics on and volumes of moduli spaces of meromorphic quadratic differentials yields interesting universality phenomenon for billiards in polygons with interior angles integer multiples of 90 degrees. This is joint work with A. Eskin and A. Zorich
 
−   
−  (2) Counting for almost every quadratic form: understanding the geometry of a random lattice allows yields striking diophantine and counting results for typical (in the sense of measure) quadratic (and other) forms. This is joint work with G. A. Margulis.
 
−   
−  ===January 15: Chi Li (Stony Brook)===
 
− 
 
−  ====On KahlerEinstein metrics and Kstability====
 
−   
−  The existence of KahlerEinstein metrics on Kahler manifolds is a basic problem in complex differential geometry. This problem has connections to other fields: complex algebraic geometry, partial differential equations and several complex variables. I will discuss the existence of KahlerEinstein metrics on Fano manifolds and its relation to Kstability. I will mainly focus on the analytic part of the theory, discuss how to solve the related complex MongeAmpere equations and provide concrete examples in both smooth and conical settings. If time permits, I will also say something about the algebraic part of the theory, including the study of Kstability using the Minimal Model Program (joint with Chenyang Xu) and the existence of proper moduli space of smoothable Kpolystable Fano varieties (joint with Xiaowei Wang and Chenyang Xu).
 
−   
−  ===January 21: Jun Kitagawa (Toronto)===
 
−   
−  ====Regularity theory for generated Jacobian equations: from optimal transport to geometric optics====
 
−   
−  Equations of MongeAmpere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical MongeAmpere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from nearfield reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.
 
−   
−  ===January 23: Nicolas Addington (Duke)===
 
−   
−  ====Recent developments in rationality of cubic 4folds====
 
−   
−  The question of which cubic 4folds are rational is one of the foremost open problems in algebraic geometry. I'll start by explaining what this means and why it's interesting; then I'll discuss three approaches to solving it (including one developed in the last year), my own work relating the three approaches to one another, and the troubles that have befallen each approach.
 
−   
−  ===January 26: Minh Binh Tran (CAM)===
 
−   
−  ====Nonlinear approximation theory for the homogeneous Boltzmann equation====
 
−   
−  A challenging problem in solving the Boltzmann equation
 
−  numerically is that the velocity space is approximated by a finite region.
 
−  Therefore, most methods are based on a truncation technique and the
 
−  computational cost is then very high if the velocity domain is large.
 
−  Moreover, sometimes, nonphysical conditions have to be imposed on the
 
−  equation in order to keep the velocity domain bounded. In this talk, we
 
−  introduce the first nonlinear approximation theory for the Boltzmann
 
−  equation. Our nonlinear wavelet approximation is nontruncated and based on
 
−  a nonlinear, adaptive spectral method associated with a new wavelet
 
−  filtering technique and a new formulation of the equation. The
 
−  approximation is proved to converge and perfectly preserve most of the
 
−  properties of the homogeneous Boltzmann equation. It could also be
 
−  considered as a general framework for approximating kinetic integral
 
−  equations.
 
−   
−  ===February 2: Afonso Bandeira (Princeton)===
 
−   
−  ====Tightness of convex relaxations for certain inverse problems on graphs====
 
−   
−  Many maximum likelihood estimation problems are known to be
 
−  intractable in the worst case. A common approach is to consider convex
 
−  relaxations of the maximum likelihood estimator (MLE), and relaxations
 
−  based on semidefinite programming (SDP) are among the most popular. We
 
−  will focus our attention on a certain class of graphbased inverse
 
−  problems and show a couple of remarkable phenomena.
 
−   
−  In some instances of these problems (such as community detection under
 
−  the stochastic block model) the solution to the SDP matches the ground
 
−  truth parameters (i.e. achieves exact recovery) for information
 
−  theoretically optimal regimes. This is established using new
 
−  nonasymptotic bounds for the spectral norm of random matrices with
 
−  independent entries.
 
−   
−  On other instances of these problems (such as angular
 
−  synchronization), the MLE itself tends to not coincide with the ground
 
−  truth (although maintaining favorable statistical properties).
 
−  Remarkably, these relaxations are often still tight (meaning that the
 
−  solution of the SDP matches the MLE). For angular synchronization we
 
−  can understand this behavior by analyzing the solutions of certain
 
−  randomized Grothendieck problems. However, for many other problems,
 
−  such as the multireference alignment problem in signal processing,
 
−  this remains a fascinating open problem.
 
−   
−  ===February 6: Morris Hirsch (UC Berkeley and UW Madison)===
 
−   
−  ====Fixed points of Lie transformation group, and zeros of Lie algebras of vector fields====
 
−   
−  The following questions will be considered:
 
− 
 
−  When a connected Lie group G acts effectively on a manifold M, what general conditions on G, M and the action ensure that the action has a fixed point?
 
− 
 
−  If g is a Lie algebra of vector fields on M, what general conditions on g and M ensure that g has a zero?
 
− 
 
−  Old and new results will be discussed. For example:
 
− 
 
−  Theorem: If G is nilpotent and M is a compact surface of nonzero Euler characteristic, there is a fixed point.
 
− 
 
−  Theorem: Suppose G is supersoluble and M is as above. Then every analytic action of G on M has a fixed point, but this is false for continuous actions, and for groups that are merely solvable.
 
− 
 
−  Theorem: Suppose M is a real or complex manifold that is 2dimensional over the ground field, and g is a Lie algebra of analytic vector fields on M. Assume some element X in g spans a 1dimensional ideal. If the zero set K of X is compact and the Poincar'eHopf index of X at K is nonzero, then g vanishes at some point of K.
 
− 
 
−  No special knowledge of Lie groups will be assumed.
 
−   
−  ===February 13: Mihai Putinar (UC Santa Barbara)===
 
−   
−  ====Quillen’s property of real algebraic varieties====
 
−   
−  A famous observation discovered by Fejer and Riesz a century ago
 
−  is the quintessential algebraic component of every spectral decomposition
 
−  result. It asserts that every nonnegative polynomial on the unit circle is a
 
−  hermitian square. About half a century ago, Quillen proved that a positive polynomial
 
−  on an odd dimensional sphere is a sum of hermitian squares. Fact independently
 
−  rediscovered much later by D’Angelo and Catlin, respectively Athavale. The main subject of
 
−  the talk will be: on which real algebraic sub varieties of <math>\mathbb{C}^n</math> is Quillen theorem valid?
 
−  An interlace between real algebraic geometry, quantization techniques and complex
 
−  hermitian geometry will provide an answer to the above question, and more.
 
−  Based a recent work with Claus Scheiderer and John D’Angelo.
 
−   
−  ===February 20: David ZureickBrown (Emory University)===
 
−   
−  ====Diophantine and tropical geometry====
 
−   
−  Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers
 
−  <math>a,b,c \geq 2</math> satisfying <math>\tfrac1a + \tfrac1b + \tfrac1c > 1</math>, Darmon and Granville proved that the individual generalized Fermat equation <math>x^a + y^b = z^c</math> has only finitely many coprime integer solutions. Conjecturally something stronger is true: for <math>a,b,c \geq 3</math> there are no nontrivial solutions.
 
−   
−  I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.
 
−   
−  ==='''Monday''' February 23: Jayadev Athreya (UIUC)===
 
−   
−  ====The ErdosSzuszTuran distribution for equivariant point processes====
 
−   
−  We generalize a problem of ErdosSzuszTuran on diophantine approximation to a variety of contexts, and use homogeneous dynamics to compute an associated probability distribution on the integers.
 
−   
−   
−  ===February 27: Allan Greenleaf (University of Rochester)===
 
−   
−  ====ErdosFalconer Configuration problems====
 
−   
−  In discrete geometry, there is a large collection of problems due
 
−  to Erdos and various coauthors starting in the 1940s, which have the
 
−  following general form: Given a large finite set P of N points
 
−  in ddimensional Euclidean space, and a geometric configuration (a line
 
−  segment of a given length, a triangle with given angles or a given area,
 
−  etc.), is there a lower bound on how many times that configuration must
 
−  occur among the points of P? Relatedly, is there an upper bound
 
−  on the number of times any single configuration can occur? One of the most
 
−  celebrated problems of this type, the Erdos distinct distances problem
 
−  in the plane, was essentially solved in 2010 by Guth and Katz, but for many
 
−  problems of this type only partial results are known.
 
−   
−  In continuous geometry, there are analogous problems due to Falconer and
 
−  others. Here, one looks for results that say that if a set A is large enough (in
 
−  terms of a lower bound on its Hausdorff dimension, say), then the set of
 
−  configurations of a given type generated by the points of A is large (has
 
−  positive measure, say).
 
−  I will describe work on Falconertype problems using some techniques from
 
−  harmonic analysis, including estimate for multilinear operators. In some
 
−  cases, these results can be discretized to obtain at least partial results
 
−  on Erdostype problems.
 
−   
−   
−   
−  ===March 6: Larry Guth (MIT)===
 
−   
−  ====Introduction to incidence geometry====
 
−   
−  Incidence geometry is a branch of combinatorics that studies the possible intersection patterns of lines, circles, and other simple shapes. For example, suppose that we have a set of L lines in the plane. An rrich point is a point that lies in at least r of these lines. For a given L, r, how many rrich points can we make? This is a typical question in the field, and there are many variations. What if we replace lines with circles? What happens in higher dimensions? We will give an introduction to this field, describing some of the important results, tools, and open problems.
 
−   
−  We will discuss two important tools used in the area. One tool is to apply topology to the problem. This tool allows us to prove results in R^2 that are stronger than what happens over finite fields. The second tool is to look for algebraic structure in the problem by studying lowdegree polynomials that vanish on the points we are studying. We will also discuss some of the (many) open problems in the field and try to describe the nature of the difficulties in approaching them.
 
−   
−   
−   
−  ===March 13: Cameron Gordon (UTAustin)===
 
−   
−  ====Leftorderability and 3manifold groups====
 
−   
−  The fundamental group is a more or less complete invariant of a 3dimensional manifold. We will discuss how the purely algebraic property of this group being leftorderable is related to two other aspects of 3dimensional topology, one geometrictopological and the other essentially analytic.
 
−   
−  ===March 20: Aaron Naber (Northwestern)===
 
−   
−  ====Regularity and New Directions in Einstein Manifolds====
 
−   
−  In this talk we give an overview of recent developments and new directions of manifolds which satisfy the Einstein equation Rc=cg, or more generally just manifolds with bounded Ricci curvature Rc<C. We will discuss the solution of the codimension four conjecture, which roughly says that GromovHausdorff limits (M^n_i,g_i)>(X,d) of manifolds with bounded Ricci curvature are smooth away from a set of codimension four. In a very different direction, in this lecture we will also explain how Einstein manifolds may be characterized by the behavior of the analysis on path space P(M) of the manifold. That is, we will see a Riemannian manifold is Einstein if and only if certain gradient estimates for functions on P(M) hold. One can view this as an infinite dimensional generalization of the BakryEmery estimates.
 
−   
−  ===March 27 11am B239: Ilya Kossovskiy (University of Vienna)===
 
−   
−  ====On Poincare's "Probleme local"====
 
−   
−  In this talk, we describe a result giving a complete solution to the
 
−  old question of Poincare on the possible dimensions of the
 
−  automorphism group of a realanalytic hypersurface in twodimensional
 
−  complex space.
 
−  As the main tool, we introduce the socalled CR (CauchyRiemann
 
−  manifolds)  DS (Dynamical Systems) technique.
 
−  This technique suggests to replace a real hypersurface with certain
 
−  degeneracies of the CRstructure by an appropriate dynamical system,
 
−  and then study mappings and symmetries of the initial real
 
−  hypersurface accordingly.
 
−  It turns out that symmetries of the singular differential equation
 
−  associated with the initial real hypersurface are much easier to study
 
−  than that of the real hypersurface, and in this way we obtain the
 
−  solution for the problem of Poincare.
 
−   
−  This work is joint with Rasul Shafikov.
 
−   
−  ===March 27: Kent Orr (Indiana University)===
 
−   
−  ====The Isomorphism Problem for metabelian groups====
 
−   
−  Perhaps the most fundamental outstanding problem in algorithmic group theory, the Isomorphism Problem for metabelian groups remains a mystery.
 
−   
−  I present an introduction to this problem intended to be accessible to graduate students. In collaboration with Gilbert Baumslag and Roman Mikhailov, I present a new approach to this ancient problem which potentially connects to algebraic geometry, cohomology of groups, number theory, Gromov's view of groups as geometric objects, and a fundamental algebraic construction developed for and motivated by the topology of knots and links.
 
−   
−  ===April 10: Jasmine Foo (University of Minnesota)===
 
−   
−  ====Evolutionary dynamics in spatially structured populations====
 
−   
−  In this talk I will present results on a model of spatial evolution on a lattice, motivated by the process of carcinogenesis from healthy epithelial tissue. Cancer often arises through a sequence of genetic alterations. Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion. To model this we will consider a generalization of the biased voter process which incorporates successive mutations modulating fitness, which is interpreted as the bias in the classical process. Under this model we will investigate questions regarding the rate of spread and accumulation of mutations, and study the dynamics of spatial heterogeneity in these evolving populations.
 
−   
−  ===April 17: Kay Kirkpatrick (UIUC)===
 
−   
−  ====Nonnormal asymptotics of the meanfield Heisenberg model====
 
−   
−  I will discuss spin models of magnets and superconductors, with spins in the circle (XY model) and in the sphere (Heisenberg model), that exhibit interesting phase transitions. I will discuss work with Elizabeth Meckes on the meanfield Heisenberg model and its nonnormal behavior at the phase transition. There is much that is still mysterious about these models: I’ll mention work in progress with Tayyab Nawaz and Leslie Ross.
 
−   
−  ===April 24: Marianna Csornyei (University of Chicago)===
 
−   
−  ====Tangents of curves and differentiability of functions====
 
−   
−  One of the classical theorems of Lebesgue tells us that Lipschitz
 
−  functions on the real line are differentiable almost everywhere. We study
 
−  possible generalisations of this theorem and some interesting geometric
 
−  corollaries.
 
−   
−  ===May 1: Bianca Viray (University of Washington)===
 
−   
−  ====Cryptography and abelian varieties with extra endomorphisms====
 
−   
−  Cryptosystems that are based on the discrete logarithm problem require an instantiation of a group of large prime order. For example, for certain primes p, one could use a subgroup of the group of units in F_p, the F_ppoints on an elliptic curve, or the F_ppoints on the Jacobian of a genus 2 curve. (The group of integers modulo p results in an insecure cryptosystem.) This raises the question of how to construct a curve C of genus 1 or 2 over F_p whose Jacobian has N F_ppoints, where N is a large prime. Perhaps surprisingly, this problem is related to curves over the complex numbers whose Jacobian has "extra" endomorphisms, i.e., Jacobians with complex multiplication. This in turn relates to a different counting problem for curves over F_p. We will give an overview of this connection and describe joint work with Lauter which answers this counting problem. This talk will be suitable for a general mathematical audience.
 
−   
−  ===May 8: Marcus Roper (UCLA)===
 
−   
−  ====Nonlinear flow in micro and mycofluidics====
 
−   
−  Simulations and asymptotic analysis of particle movement in microfluidic channels is made easier by the fact that the equations of motion for particles moving in a microfluidic pipe are linear and reversible. We discuss two examples of interesting phenomena that that emerge from the breakdown of linearity: (1) Inertial microfluidic devices use high speed flow to focus and segregate particles of different sizes. However, there is no consensus on how focusing forces vary with particle size and flow speed, or even on the number of focusing positions that exist within channels having different shapes. I’ll explain how simulations and experiments may have been misinterpreted and develop a partly numerical, partly asymptotic theory for particle focusing. (2) Fungi form microfluidic networks to transport nutrients, fluid and nuclei. Surprisingly these networks are anticongestive – the greater the density of traffic on a fungal freeway, the faster the traffic travels. I’ll explain how this anticongestive property arises using a kinetic model of the motion of nuclei on a fungal freeway, and speculate about why the fungus might benefit from organizing its flow in this way.
 
   
 == Past Colloquia ==   == Past Colloquia == 