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May 24 - Robin Tucker-Drob
An application of locally finite groups to ergodic theory
I will present a proof that shows that any mixing measure preserving action of a countable group is almost free. Surprisingly, the proof relies on a deep theorem of M. Kargapolov and also P. Hall and C.R. Kulatilaka about locally finite groups (a group is called locally finite if all of its finitely generated subgroups are finite): Every infinite locally finite group contains an infinite abelian subgroup.
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May 10 - Laura Peskin
Chambers, apartments, and buildings
Last week, we started with a root system and constructed a Chevalley group G over a p-adic field. Then we defined the spherical apartment (with an action, by reflections over the chamber walls, by the Weyl group of G), as well as the affine apartment (which sees the action of both the Weyl group and a fixed maximal torus of G). We also assigned a parabolic subgroup of G to each spherical chamber, and a "parahoric" subgroup to each affine chamber. This week we'll glue together the affine apartments to form the Bruhat-Tits building of G, and look at a couple of its applications to representations of p-adic groups: Moy and Prasad's classification of so-called supercuspidal representations in characteristic 0, and Barthel and Livne's study of representations of GL_2 in characteristic p.
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May 3 - Laura Peskin
Chambers, apartments, and buildings
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April 26 - Gjergji Zaimi
Some combinatorial aspects of Macdonald polynomials
This talk will be a brief overview of some important conjectures in algebraic combinatorics. We will start with some basic examples from invariant theory and commutative algebra to motivate some recent results such as the n! conjecture and the Macdonald positivity conjecture. The rest of the talk will be devoted to the combinatorial interpretation of the coefficients of Macdonald polynomials.
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April 19 - Branimir Cacic
Action functionals in mathematics and physics, part 2
This week, I conclude my survey of variational principles in mathematics and physics by discussion of Lagrangian mechanics, classical field theory, and the spectral action of noncommutative geometry.
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April 12 - Branimir Cacic
Action functionals in mathematics and physics
A recurring motif in analysis and differential geometry, and especially in mathematical physics, is the variational principle, namely, that the object of study (e.g., solutions of a certain [partial] differential equation), can be most naturally viewed as an extremiser of a certain scalar-valued function called the energy or action functional. In this talk, I will give a brief introduction to the calculus of variations and its applications to analysis and differential geometry, and then survey some of the action functionals appearing in contemporary research, such as the spectral action of noncommutative geometry.
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March 8 - Melissa Yeung
Braids, Train Tracks, and Fluid Dynamics
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February 23 - Robin Tucker-Drob
An application of the probabilistic method to Bernoulli shifts
I will give a short introduction to the probabilistic method. I will then give an application to measure preserving actions of countably infinite groups, using random partitions to establish a certain minimality property of Bernoulli shift actions.
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February 16 - Maria Nastasescu
The number of ovals of a random real plane curve
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February 9 - Victor Kasatkin
C*-algebras of singular integral operators on a real line with symbols having discontinuities in coordinates and momenta
Talk is devoted to presentation of the main ideas of the paper here. We will start from basic definitions. Then we discuss 2 algebras A and B. The C*-algebra B is generated in L^2(R) by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. Definition of A is similar. It is generated by multiplications and one convolution operator. The algebra B is (up the ideal of compact operators) represented as the restricted direct sum A1+_C A2 (where A_1 and A_2 are slightly modified versions of A). We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra B in terms of the spectra of A1 and A2. We describe all equivalence classes of irreducible representations of the algebra B. If time remains, some consequences of this computation will be described.
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February 2 - Dan Betea
Representation theory, etc, continued
This week, Dan will finish up his talk and maybe show some pictures.
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January 26 - Dan Betea
Representation theory, random matrices, statistical mechanics and combinatorics via Schur functions and RSK
We will talk about the Robinson-Schensted-Knuth correspondence and Schur functions in combinatorics. Will make brief connections to representation theory of S_n and U_n (really GL_n), and try to spend more time on recent developments in random combinatorics/statistical mechanics (random partitions and tiling problems). Random matrices will make a cameo appearance once or twice during the talk.
We will aim at proving some basic results, and state some harder ones without proofs. The main references followed are Fulton's "Young tableaux" and Okounkov's "The uses of random partitions" (and references therein).
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January 19 - Daiqi Linghu
A description of cyclic vectors in L^2(T,m)
By Spectral Theorem we know any normal operator (with simple spectrum) on a Hilbert space is unitarily equivalent to the multiplication operator: f->z*f in the space L^2(mu), where mu is a compactly supported Borel measure in the complex plane. This time I'll prove some results of canonical Hardy-Nevanlinna Theory which will be used to give a sufficient and necessary condition for the cyclic vectors (w.r.t. the shift operator) in L^2(T,m).
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January 12 - Padraic Bartlett
Random and Quasirandom Graphs
In the 1940's, Erdös pioneered the use of the probabilistic method in combinatorics, a nonconstructive set of techniques that allowed him to resolve a number of outstanding open questions. One of these concepts was the idea of examining a ``random graph,'' where information about the expected values of certain graph properties was used to show that graphs with various characteristics exist, even without an explicit construction.
In this talk, we will introduce the notion of a random graph as a probability space and quickly calculate the expected values of some properties of a random graph. From there, we will use these calculations to motivate the definition of a quasirandom graph and discuss a few famous results of Chung, Graham, and Wilson, wrapping up our talk with some examples and current research questions.
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December 1 - Brian Hwang
A glimpse into p-adic analysis
Sometimes p-adic analysis is called the "Funhouse of Mathematics" because it is a subfield where so many amazing theorems are true, and where many of the tricky problems seen in real or complex analysis become trivial. Furthermore, much like an actual funhouse, p-adic analysis puts certain things into a forced perspective that is highly nonintuitive at first, but gives you a great deal of freedom once you learn to "think p-adically." Aside from its better-known connections to arithmetic algebraic geometry and number theory, there has been a lot of recent work tying various versions of p-adic analysis to fields such as dynamical systems, convex geometry, and tropical geometry.
I'll discuss some interesting classical results in p-adic analysis, the tradeoffs involved in applying p-adic analysis versus their real and complex versions, how to identify a situation where p-adic analysis might be fruitful, and how it can be applied to solve otherwise intractable problems.
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November 17 - Brian Simanek
Logarithmic Potential Theory
In two dimensions, the electrostatic interaction is given by a logarithmic repulsion. I will introduce the basic concepts of potential theory in this setting such as equilibrium measures and Green functions. I'll then introduce the quantities called capacity, transfinite diameter, and Chebyshev constant of a set and conclude with a theorem that relates them all to each other.
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November 10 - Branimir Cacic
Differential geometry for inveterate algebraists
It is well known that a manifold $M$ gives rise to a commutative associative algebra $C^\infty(M)$, the algebra of smooth scalar functions on $M$. What is less well known is that one can go the other way, and build a manifold $M$ from a suitable commutative associative algebra. Following the monograph "Smooth manifolds and observables" by "Jet Nestruev," I'll sketch out how one can carry this out, as well as give the highlights of the remarkable dictionary that exists between the differential geometry of a manifold $M$ and commutative algebra over $C^\infty(M)$.
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November 3 - Kevin Teh
The functional calculus
The functional calculi are powerful tools used to define functions of linear operators and construct new operators with specified properties. Functions with a higher degree of regularity may be applied in more general situations. In this talk we will discuss the holomorphic functional calculus, the continuous functional calculus, and the Borel functional calculus.
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October 27 - Alden Walker
Rotation numbers of positive words
Given a homeomorphism f of the circle S^1, we can lift it to a homeomorphism f~ of R. Then we compute lim_{n\to\infty} (f~)^n(0)/n. This is well-defined in R/Z, and it's called the rotation number rot(f). It measures how much f rotates S^1, on average. Now suppose that we are given a word w in x and y, like w=xyxyyx, where x and y are homeomorphisms of S^1. We want to find sup_{rot(x)=r, rot(y)=s} rot(w) That is, we want to maximize rot(w) over all homeomorphisms x and y with rot(x)=r and rot(y)=s. It turns out that we can do this, and we get some cool pictures.
This is a general-audience talk adapted from my paper Ziggurats and rotation numbers with Danny.
We meet Thursdays at noon in Sloan 159. Pizza and drinks will be provided!
I'm ordering them in descending order chronologically now, because that makes way more sense.