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 __NOTOC__
 

 
 = Mathematics Colloquium =   = Mathematics Colloquium = 

 

 All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.   All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''. 

 

 <! ==[[Tentative ColloquiaTentative schedule for next semester]] == >
  The calendar for spring 2019 can be found [[Colloquia/Spring2019here]]. 

 

 ==Fall 2017==   ==Spring 2019== 

 

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 !align="left"  host(s)   !align="left"  host(s) 
     
 September 8   Jan 25 
  Tess Anderson (Madison)    [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW 
 [[# TBA TBA ]]   [[#Beata Randrianantoanina (Miami University Ohio)  Some nonlinear problems in the geometry of Banach spaces and their applications ]] 
  Yang    Tullia Dymarz 
     
     
 September 15   Jan 30 '''Wednesday''' 
     [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University) 
 [[# TBA TBA ]]   [[#Lillian Pierce (Duke University)  Short character sums ]] 
   Boston and Street 
     
   
  Jan 31 '''Thursday''' 
   [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M) 
  [[#Dean Baskin (Texas A&M)  Radiation fields for wave equations ]] 
   Street 
     
     
  '''Wednesday, September 20, LAA lecture   Feb 1 
  Andrew Stuart (Caltech)    [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
  Jin    Qin 
     
     
 September 22   Feb 5 '''Tuesday''' 
     [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
   Denisov 
     
   
  Feb 8 
   [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern) 
  [[#Aaron Naber (Northwestern)  A structure theory for spaces with lower Ricci curvature bounds ]] 
   Street 
     
     
 September 29   Feb 15 
     
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     
     
     
 October 6   Feb 22 
  [http://www3.nd.edu/~jhauenst/ Jonathan Hauenstein] (Notre Dame)    [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
  Boston    Erman and Corey 
     
     
 October 13   March 4 
     [http://wwwusers.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     Kim 
     
     
 October 20   March 8 
  [http://cims.nyu.edu/~pgermain/ Pierre Germain] (Courant, NYU)    [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
  MinhBinh Tran    Erman 
     
     
 October 27   March 15 
     Maksym Radziwill (Caltech) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     Marshall 
     
     
 November 3   March 29 
     Jennifer Park (OSU) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     Marshall 
     
     
 November 10   April 5 
  Reserved for possible job talks    JuLee Kim (MIT) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     Gurevich 
     
     
 November 17   April 12 
  Reserved for possible job talks    Evitar Procaccia (TAMU) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     Gurevich 
     
     
 November 24   April 19 
 '''Thanksgiving break'''    [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     JeanLuc 
     
     
 December 1   April 26 
  Reserved for possible job talks    [https://www.brown.edu/academics/appliedmathematics/faculty/kavitaramanan/home Kavita Ramanan] (Brown University) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
     WIMAW 
     
     
 December 8   May 3 
  Reserved for possible job talks    Tomasz Przebinda (Oklahoma) 
 [[# TBA TBA ]]   [[# TBA TBA ]] 
   Gurevich 
     
 
 
 
 

 
 }   } 

 

 == Abstracts ==   == Abstracts == 
 === September 16: PoShen Loh (CMU) ===
 
 Title: Directed paths: from Ramsey to Pseudorandomness
 

 

 Abstract: Starting from an innocent Ramseytheoretic question regarding directed
  ===Beata Randrianantoanina (Miami University Ohio)=== 
 paths in graphs, we discover a series of rich and surprising connections
 
 that lead into the theory around a fundamental result in Combinatorics:
 
 Szemeredi's Regularity Lemma, which roughly states that every graph (no
 
 matter how large) can be wellapproximated by a boundedcomplexity
 
 pseudorandom object. Using these relationships, we prove that every
 
 coloring of the edges of the transitive Nvertex tournament using three
 
 colors contains a directed path of length at least sqrt(N) e^{log^* N}
 
 which entirely avoids some color. The unusual function log^* is the
 
 inverse function of the tower function (iterated exponentiation).
 

 

 === September 23: Gheorghe Craciun (UWMadison) ===
  Title: Some nonlinear problems in the geometry of Banach spaces and their applications. 
 Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture  

 

 Abstract: The Global Attractor Conjecture says that a large class of polynomial dynamical systems, called toric dynamical systems, have a globally attracting point within each linear invariant space. In particular, these polynomial dynamical systems never exhibit multistability, oscillations or chaotic dynamics.   Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics. 

 

 The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's Htheorem.
  ===Lillian Pierce (Duke University)=== 

 

 We discuss the history of this problem, including the connection between this conjecture and the Boltzmann equation. Then, we introduce toric differential inclusions, and describe how they can be used to prove this conjecture in full generality.
  Title: Short character sums 

 

 === September 30: Akos Magyar (University of Georgia) ===
  Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a socalled character sum. For example, both understanding the Riemann zeta function or Dirichlet Lfunctions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations. 
 Title: Geometric Ramsey theory
 

 

 Abstract: Initiated by Erdos, Graham, Montgomery and others in the 1970's, geometric Ramsey theory studies geometric configurations, determined up to translations, rotations and possibly dilations, which cannot be destroyed by finite partitions of Euclidean spaces. Later it was shown by ergodic and Fourier analytic methods that such results are also possible in the context of sets of positive upper density in Euclidean spaces or the integer lattice. We present a new approach, motivated by developments in arithmetic combinatorics, which provide new results as well new proofs of some classical results in this area.
  ===Dean Baskin (Texas A&M)=== 

 

 === October 14: Ling Long (LSU) ===
  Title: Radiation fields for wave equations 
 Title: Hypergeometric functions over finite fields  

 

 Abstract: Hypergeometric functions are special functions with lot of   Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space. 
 symmetries. In this talk, we will introduce hypergeometric functions over finite
 
 fields, originally due to Greene, Katz and McCarthy, in a way that is  
 parallel to the classical hypergeometric functions, and discuss their
 
 properties and applications to character sums and the arithmetic of
 
 hypergeometric abelian varieties.
 
 This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher, and FangTing Tu.
 

 

 === Tuesday, October 25, 9th floor: Stefan Steinerberger (Yale) ===   ===Aaron Naber (Northwestern)=== 
 Title: Three Miracles in Analysis
 

 

 Abstract: I plan to tell three stories: all deal with new points of view on very classical objects and have in common that there is a miracle somewhere. Miracles are nice but difficult to reproduce, so in all three cases the full extent of the underlying theory is not clear and many interesting open problems await. (1) An improvement of the Poincare inequality on the Torus that encodes a lot of classical Number Theory. (2) If the HardyLittlewood maximal function is easy to compute, then the function is sin(x). (Here, the miracle is both in the statement and in the proof). (3) Bounding classical integral operators (Hilbert/Laplace/Fouriertransforms) in L^2  but this time from below (this problem originally arose in medical imaging). Here, the miracle is also known as 'Slepian's miracle' (this part is joint work with Rima Alaifari, Lillian Pierce and Roy Lederman).
  Title: A structure theory for spaces with lower Ricci curvature bounds. 

 

 === October 28: Linda Reichl (UT Austin) ===
  Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li. 
 Title: Microscopic hydrodynamic modes in a binary mixture
 

 

 Abstract: Expressions for propagation speeds and decay rates of hydrodynamic modes in a binary mixture can be obtained directly from spectral properties of the Boltzmann equations describing the mixture. The derivation of hydrodynamic behavior from the spectral properties of the kinetic equation provides an alternative to ChapmanEnskog theory, and removes the need for lengthy calculations of transport coefficients in the mixture. It also provides a sensitive test of the completeness of kinetic equations describing the mixture. We apply the method to a hardsphere binary mixture and show that it gives excellent agreement with light scattering experiments on noble gas mixtures.
 

 

 ===Monday, October 31: Kathryn Mann (Berkeley) ===   == Past Colloquia == 
 Title: Groups acting on the circle
 
  
 Abstract: Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic and natural question from geometric topology, but also a very difficult one  even in the case where M is the circle, and G is a familiar, finitely generated group.
 
  
 In this talk, I’ll introduce you to the theory of groups acting on the circle, building on the perspectives of Ghys, Calegari, Goldman and others. We'll see some tools, old and new, some open problems, and some connections between this theory and themes in topology (like foliated bundles) and dynamics.
 
  
 ===November 7: Gaven Martin (New Zealand Institute for Advanced Study) ===
 
 Title: Siegel's problem on small volume lattices
 
  
 Abstract: We outline in very general terms the history and the proof of the identification
 
 of the minimal covolume lattice of hyperbolic 3space as the 353
 
 Coxeter group extended by the involution preserving the symmetry of this
 
 diagram. This gives us the smallest regular tessellation of hyperbolic 3space.
 
 This solves (in three dimensions) a problem posed by Siegel in 1945. Siegel solved this problem in two dimensions by deriving the
 
 signature formula identifying the (2,3,7)triangle group as having minimal
 
 coarea.
 

 
 There are strong connections with arithmetic hyperbolic geometry in
 
 the proof, and the result has applications in the maximal symmetry groups
 
 of hyperbolic 3manifolds in much the same way that Hurwitz's 84g84 theorem
 
 and Siegel's result do.
 
  
 ===Wednesday, November 16 (9th floor): Kathryn Lindsey (U Chicago) ===
 
 Title: Shapes of Julia Sets
 
  
 Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. William Thurston asked "What are the possible shapes of polynomial Julia sets?" For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is "yes!" I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.
 
  
 ===November 18: Andrew Snowden (University of Michigan)===
 
 Title: Recent progress in representation stability
 
  
 Abstract: Representation stability is a relatively new field that studies
 
 somewhat exotic algebraic structures and exploits their properties to
 
 prove results (often asymptotic in nature) about objects of interest.
 
 I will describe some of the algebraic structures that appear (and
 
 state some important results about them), give a sampling of some
 
 notable applications (in group theory, topology, and algebraic
 
 geometry), and mention some open problems in the area.
 
  
 ===Monday, November 21: Mariya Soskova (University of WisconsinMadison)===
 
 Title: Definability in degree structures
 
  
 Abstract: Some incomputable sets are more incomputable than others. We use
 
 Turing reducibility and enumeration reducibility to measure the
 
 relative complexity of incomputable sets. By identifying sets of the
 
 same complexity, we can associate to each reducibility a degree
 
 structure: the partial order of the Turing degrees and the partial
 
 order of the enumeration degrees. The two structures are related in
 
 nontrivial ways. The first has an isomorphic copy in the second and
 
 this isomorphic copy is an automorphism base. In 1969, Rogers asked a
 
 series of questions about the two degree structures with a common
 
 theme: definability. In this talk I will introduce the main concepts
 
 and describe the work that was motivated by these questions.
 
  
 ===Friday, December 2: Hao Shen (Columbia)===
 
 Title: Singular Stochastic Partial Differential Equations  How do they arise and what do they mean?
 
  
 Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
 
  
 ===Monday, December 5: Botong Wang (UWMadison)===
 
 Title: Enumeration of points, lines, planes, etc.
 
  
 Abstract: It is a theorem of de Bruijn and Erdos that n points in the plane determine at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization of this theorem, which confirms a “topheavy” conjecture of Dowling and Wilson in 1975. I will give a sketch of the key ideas of the proof, which are the hard Lefschetz theorem and the decomposition theorem in algebraic geometry. I will also talk about a logconcave conjecture on the number of independent sets. These are joint works with June Huh.
 
  
 === Friday, December 9: Aaron Brown (U Chicago) ===
 
 ''Lattice actions and recent progress in the Zimmer program''
 
  
 Abstract: The Zimmer Program is a collection of conjectures and questions regarding actions of lattices in higherrank simple Lie groups on compact manifolds. For instance, it is conjectured that all nontrivial volumepreserving actions are built from algebraic examples using standard constructions. In particular—on manifolds whose dimension is below the dimension of all algebraic examples—Zimmer’s conjecture asserts that every action is finite.
 
  
 I will present some background, motivation, and selected previous results in the Zimmer program. I will then explain two of my results within the Zimmer program:
 
 (1) a solution to Zimmer’s conjecture for actions of cocompact lattices in SL(n,R) (joint with D. Fisher and S. Hurtado);
 
 (2) a classification (up to topological semiconjugacy) of all actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).
 
  
 === Monday, December 19: Andrew Zimmer (U Chicago) ===
 
 ''Metric spaces of nonpositive curvature and applications in several complex variables''
 
  
 Abstract: In this talk I will discuss how to use ideas from the theory of metric spaces of nonpositive curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance nonincreasing with respect to holomorphic maps. Moreover, this metric often satisfies wellknown nonpositive curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use these conditions to understand the behavior of holomorphic maps. Some of what I will talk about is joint work with Gautam Bharali.
 
  
 === Monday, January 9: Miklos Racz (Microsoft) ===
 
 ''Statistical inference in networks and genomics''
 
  
 Abstract: From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.
 
  
 I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are rootfinding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying highdimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a highdensity, durable, and easytomanipulate storage medium of digital data.
 
  
 === Friday, January 13: Mihaela Ifrim (Berkeley) ===
 
 ''Two dimensional water waves''
 
  
 The classical waterwave problem consists of solving the Euler equations in the presence of a free fluid surface (e.g the waterair interface). This talk will provide an overview of recent developments concerning the motion of a two dimensional incompressible fluid with a free surface. There is a wide range of problems that fall under the heading of water waves, depending on a number of assumptions that can be applied: surface tension, gravity, finite bottom, infinite bottom, rough bottom, etc., and combinations thereof. We will present the physical motivation for studying such problems, followed by the discussion of several interesting mathematical questions related to them. The first step in the analysis is the choice of coordinates, where multiple choices are available. Once the equations are derived we will discuss the main issues arising when analysing local wellposedness, as well as the long time behaviour of solutions with small, or small and localized data. In the last part of the talk we will introduce a new, very robust method which allows one to obtain enhanced lifespan bounds for the solutions. If time permits we will also introduce an alternative method to the scattering theory, which in some cases yields a straightforward route to proving global existence results and obtaining an asymptotic description of solutions. This is joint work with Daniel Tataru, and in part with John Hunter.
 
  
 === Tuesday, January 17: Fabio Pusateri (Princeton) ===
 
 ''The Water Waves problem''
 
  
 We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results  obtained in collaboration with Ionescu and DengIonescuPausader  and sketch some of the main ideas.
 
  
 === Friday, January 20: Sam Raskin (MIT) ===
 
 ''Tempered local geometric Langlands ''
 
  
 The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain "local terms," and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local.
 
  
 Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately explored.
 
  
 The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on "temperedness"), and provide evidence for this conjecture.
 
  
 === Monday, January 23: Tamas Darvas (Maryland) ===
 
 ''Geometry on the space of Kahler metrics and applications to canonical metrics''
 
  
 A basic problem in Kahler geometry, going back to Calabi in the 50's, is to find Kahler
 
 metrics with the best curvature properties, e.g., Einstein metrics. Such special metrics are
 
 minimizers of well known functionals on the space of all Kahler metrics H. However these
 
 functionals become convex only if an adequate geometry is chosen on H. One such choice of
 
 Riemannian geometry was proposed by Mabuchi in the 80's, and was used to address a number of
 
 uniqueness questions in the theory. In this talk I will present more general Finsler geometries on
 
 H, that still enjoy many of the properties that Mabuchi's geometry has, and I will give
 
 applications related to existence of special Kahler metrics, including the recent resolution of
 
 Tian's related properness conjectures.
 
  
  
 === Friday, February 3: Melanie Matchett Wood (UWMadison) ===
 
 ''Random groups from generators and relations''
 
  
 We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations. We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the CohenLenstra heuristics, which predict the distribution of class groups of number fields. We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen. Finally, we discuss joint work with Yuan Liu on the nonabelian random groups built in this way, including the existence of a limit of the random groups as n goes to infinity.
 
  
 === Monday, February 6: Benoit Perthame (University of Paris VI) ===
 
 ''Models for neural networks; analysis, simulations and behaviour''
 
  
 Neurons exchange informations via discharges, propagated
 
 by membrane potential, which trigger firing of the many connected
 
 neurons. How to describe large networks of such neurons? What are the properties of these meanfield equations?
 
 How can such a network generate a spontaneous activity?
 
 Such questions can be tackled using nonlinear integrodifferential
 
 equations. These are now classically used in the neuroscience community to describe
 
 neuronal networks or neural assemblies. Among them, the best known is certainly
 
 WilsonCowan's equation which
 
 describe spiking rates arising in different brain locations.
 
  
 Another classical model is the integrateandfire equation that describes
 
 neurons through their voltage using a particular type of FokkerPlanck equations. Several mathematical results will be presented concerning existence, blowup, convergence to steady state,
 
 for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed.
 
  
 One can also describe directly the spike time
 
 distribution which seems to encode more directly the neuronal information.
 
 This leads to a structured population equation that describes
 
 at time $t$ the probability to find a neuron with time $s$
 
 elapsed since its last discharge. Here, we can
 
 show that small or large connectivity
 
 leads to desynchronization. For intermediate regimes, sustained
 
 periodic activity occurs.
 
 A common mathematical tool is the use of the relative entropy method.
 
  
 This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets.
 
  
 === February 10: Alina Chertock (NC State Univ.) ===
 
 ''Numerical Method for Chemotaxis and Related Models''
 
  
 Chemotaxis is a movement of microorganisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convectiondiffusion equation for the cell density coupled with a reaction diffusion equation for the chemoattractant concentration. It is wellknown that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxis systems extremely delicate and challenging task.
 
  
 In this talk, I will present a family of highorder numerical methods for the KellerSegel chemotaxis system and several related models. Applications of the proposed methods to to multiscale and coupled chemotaxis–fluid system and will also be discussed.
 
  
  
 === Friday, February 17: Gustavo Ponce(UCSB) ===
 
  
 ''The Kortewegde Vries equation vs. the BenjaminOno equation''
 
  
 In this talk we shall study the <math>k</math>generalized Kortewegde Vries <math>(k</math>KdV) equation
 
  
 <math>\partial_t u + \partial_x^3u +u^k\,\partial_xu=0,\;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+, </math>
 
  
 and the <math>k</math>generalized BenjaminOno (<math>k</math>BO) equation
 
  
 <math>\partial_t u\partial_x^2\mathcal {H} u+u^k\,\partial_x u=0, \;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+,</math>
 
  
 where <math>\mathcal {H}</math> denotes the Hilbert transform,
 
  
 <math>\mathcal {H} f(x)=\frac{1}{\pi}\, {p.v.}\big(\frac{1}{x}\ast f\big)(x)=(i\,sgn(\xi) \widehat{f}(\xi))^{\vee}(x).</math>
 
  
 The goal is to review and analyze results concerning solutions of the initial value properties associated to these equations.
 

 
 These include a comparison of the local and global wellposedness and unique continuation properties
 
 as well as special features of the special solutions of these models.
 
  
 === Monday, February 20, Amy Cochran (Michigan) ===
 
 ''Mathematical Classification of Bipolar Disorder''
 
  
 Bipolar disorder is a chronic disease of mood instability. Longitudinal patterns of mood are central to any patient description, but are condensed into simple attributes and categories. Although these provide a common language for clinicians, they are not supported by empirical evidence. In this talk, I present patientspecific models of mood in bipolar disorder that incorporate existing longitudinal data. In the first part, I will describe mood as a Bayesian nonparametric hierarchical model that includes latent classes and patientspecific mood dynamics given by discretetime Markov chains. These models are fit to weekly mood data, revealing three patient classes that differ significantly in attempted suicide rates, disability, and symptom chronicity. In the second part of the talk, I discuss how combined statistical inferences from a population do not support widely held assumptions (e.g. mood is onedimensional, rhythmic, and/or multistable). I then present a stochastic differential equation model that does not make any of these assumptions. I show that this model accurately describes the data and that it can be personalized to an individual. Taken together, this work moves forward datadriven modeling approaches that can guide future research into precise clinical care and disease causes.
 
  
 === Friday, March 3, Ken Bromberg (Utah)===
 
 "Renormalized volume for hyperbolic 3manifolds"
 
  
 Motivated by ideas in physics Krasnov and Schlenker defined the renormalized volume of a hyperbolic 3manifold. This is a way of assigning a finite volume to a hyperbolic 3manifold that has infinite volume in the usual sense. We will begin with some basic background on hyperbolic geometry and hyperbolic 3manifolds before defining renormalized volume with the aim of explaining why this is a natural quantity to study from a mathematician’s perspective. At the end will discuss some joint results with M. Bridgeman and J. Brock.
 
  
 === Tuesday, March 7: Roger Temam (Indiana University) ===
 
 ''On the mathematical modeling of the humid atmosphere''
 
  
 The humid atmosphere is a multiphase system, made of air, water vapor, cloudcondensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, noncontinuous (and nonmonotone) in the framework of nonlinear partial differential equations.
 
 We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasivariational inequalities.
 
  
 === Wednesday, March 8: Roger Temam (Indiana University) ===
 
 ''Weak solutions of the ShigesadaKawasakiTeramoto system''
 
  
 We will present a result of existence of weak solutions to the ShigesadaKawasakiTeramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely selfcontained and does not rely on any earlier result.
 
 Based on an article with Du Pham, to appear in Nonlinear Analysis.
 
  
 === Wednesday, March 15: Enrique Zuazua (Universidad Autónoma de Madrid) ===
 
 ''Control and numerics: Recent progress and challenges''
 
  
 In most real life applications Mathematics not only face the challenge of modelling (typically by means of ODE and/or PDE), analysis and computer simulations but also the need control and design.
 
  
 And the successful development of the needed computational tools for control and design cannot be achieved by simply superposing the state of the art on Mathematical and Numerical Analysis. Rather, it requires specific tools, adapted to the very features of the problems under consideration, since stable numerical methods for the forward resolution of a given model, do not necessarily lead to stable solvers of control and design problems.
 
  
 In this lecture we will summarize some of the recent work developed in our group, motivated by different applications, that have led to different analytical and numerical methodologies to circumvent these difficulties.
 
  
 The examples we shall consider are motivated by problems of different nature and lead to various new mathematical developments. We shall mainly focus on the following three topics:
 
  
  Inverse design for hyperbolic conservation laws,
 
  
  The turnpike property: control in long time intervals,
 
  
  Collective behavior: guidance by repulsion.
 
  
 We shall also briefly discuss the convenience of using greedy algorithms when facing parameterdependence problems.
 

 

 This lecture has been conceived for a broad audience. Accordingly, unnecessary technicalities will be avoided.
  [[Colloquia/BlankBlank]] 

 

  [[Colloquia/Fall2018Fall 2018]] 

 

 === Friday, March 17: Lillian Pierce (Duke University) ===
  [[Colloquia/Spring2018Spring 2018]] 
 ''Ptorsion in class groups of number fields of arbitrary degree''
 

 

 Abstract: Fix a number field K of degree n over the rationals, and a prime p, and consider the ptorsion subgroup of the class group of K. How big is it? It is conjectured that this ptorsion subgroup should be very small (in an appropriate sense), relative to the absolute discriminant of the field; this relates to the CohenLenstra heuristics and various other arithmetic problems. So far it has proved extremely difficult even to beat the trivial bound, that is, to show that the ptorsion subgroup is noticeably smaller than the full class group. In 2007, Ellenberg and Venkatesh shaved a power off the trivial bound by assuming GRH. This talk will discuss several new, contrasting, methods that recover or improve on this bound for almost all members of certain infinite families of fields, without assuming GRH.
  [[Colloquia/Fall2017Fall 2017]] 
  
 === Wednesday, March 29: Sylvia Serfaty (NYU) ===
 
 ''Microscopic description of Coulombtype systems''
 
  
 We are interested in systems of points with Coulomb, logarithmic
 
 or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in various settings: an instance is the classical Coulomb gas which in some cases happens
 
 to be a random matrix ensemble, another is vortices in the GinzburgLandau
 
 model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named
 
 Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. After reviewing the motivations, we will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes are given.
 
 This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions.
 
 The main results are joint with Thomas Leblé and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache.
 
  
  
 === Friday, April 7: Hal Schenck (UIUC) ===
 
 ''Hyperplane Arrangements: Algebra, Combinatorics, Topology''
 
  
 A hyperplane arrangement is a collection of hyperplanes in affine space, usually the real or complex numbers.
 
 The complement X of the hypersurfaces has
 
 very interesting topology. In 1980 Orlik and Solomon determined
 
 that the cohomology ring is a quotient of an
 
 exterior algebra, with a generator for each hyperplane.
 
 Surprisingly, all relations are determined by the combinatorics
 
 of the arrangement. Nevertheless, there remain many interesting
 
 open questions, which involve a beautiful interplay of algebra,
 
 combinatorics, geometry, and topology. I'll spend much of the
 
 talk discussing this interplay, and close by discussing several
 
 conjectures in the field, along with recent progress on those
 
 conjectures, where the BernsteinGelfandGelfand correspondence
 
 plays a key role. Joint work with Dan Cohen (LSU) and Alex
 
 Suciu (Northeastern).
 
  
  
 === Friday, April 14: Wilfrid Gangbo (UCLA) ===
 
 ''On intrinsic differentiability in the Wasserstein space <math> P_2(R^d) </math>''
 
  
 We elucidate the connection between different notions of differentiability in <math>P_2(R^d) </math>: some have been introduced intrinsically by AmbrosioGigliSavare, the other notion due to Lions, is extrinsic and arises from the identification of <math> P_2(R^d) </math> with the Hilbert space of squareintegrable random variables. We mention potential applications such as uniqueness of viscosity solutions for HamiltonJacobi equations in <math> P_2(R^d) </math>, the latter not known to satisfy the Radon–Nikodym property. (This talk is based on a work in progress with A Tudorascu).
 
  
  
 === Monday, April 17: Ravi Vakil (Stanford) ===
 
 ''The Mathematics of Doodling''
 
  
 Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I'll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we'll have seen some important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today.
 
  
  
 === Tuesday, April 18: Ravi Vakil (Stanford) ===
 
 ''Cutting and pasting in (algebraic) geometry''
 
  
 Given some class of "geometric space", we can make a ring as follows.
 
  
 <b> Additive Structure:</b> When U is an open subset of X we set [X] = [U] + [U \ X].
 
  
 <b> Multiplicative Structure:</b> [X x Y] = [X][Y]
 
  
 In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising "stabilization" structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. This talk is intended for a broad audience. This is joint work with Melanie Matchett Wood.
 
  
  
 === Friday, April 28: Thomas Yizhao Hou (Caltech) ===
 
 ''The interplay between theory and computation in the study of 3D Euler equations''
 
  
 Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D NavierStokes Equations. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities. To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes with a maximum effective resolution of order $10^12$ in each direction. A careful local analysis also suggests that the blowingup solution is highly anisotropic and is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. Very recently we prove rigorously that the 1D model develops finite time singularity. Using a very delicate method of analysis which involves computer assisted proof, we prove the existence of a discrete family of selfsimilar profiles for a variant of this model. Moreover, we show that the selfsimilar profile enjoys some stability property.
 
  
 == Past Colloquia ==
 

 

 [[Colloquia/Spring2017Spring 2017]]   [[Colloquia/Spring2017Spring 2017]] 