Graduate Student Colloquium : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 9, 2005
David Marker :
3 p.m. in SEO 636
Abstract
In his celebrated Incompleteness Theorem, Godel showed
that there is no algorithm to decide which statements about
the natural numbers are true. A surprising counterpoint, was
provided by Tarski, who proved that there is such an algorithm
for the complex field and the real field. Moreover,
the methods from Tarski's proof provide interesting geometric
and topological information about real algebraic varieties.
In recent years, these methods have been extended to allow
certain classes of analytic functions, including exponentiation.
Sept. 23, 2005
Ian Agol :
3 p.m. in SEO 636
Abstract
According to Einstein's theory of general
relativity, our universe is a (Lorentzian) Einstein 4-manifold.
Einstein manifolds are Lorentzian manifolds whose
local structure is determined by the Einstein's field
equations, but the global structure is not determined.
This leaves open the possibility that the universe has
non-trivial topology. We'll discuss the socalled Friedman
universes, which are topologically a homogeneous 3-manifold
cross R, and seem to be the most prevalent model used by
cosmologists for the global structure of the universe. We'll then
discuss how cosmologists hope to discover
the topology of the universe, and focus on some of the
simplified mathematical ideas that have arisen in their
work.
Sept. 30, 2005
Steve Smith :
3 p.m. in SEO 636
Abstract
Historically in finite group theory, the topic of group cohomology
has been more developed in the area of algebraic topology.
Some further connections were developed starting around 1975 by
Brown, Quillen, Webb, and others: using partially ordered subsets
of a finite group, to define simplicial complexes whose topological
properties cast further light on such topics as the group cohomology.
Now in the last 5 years or so, topologists such as Levi, Oliver, Grodal
and others are developing new topological structures such as
"p-local finite groups"---which are spaces which record information
about p-group and conjugacy within them: information which might,
but also might not, arise from a finite group.
The talk will be an exposition, from my viewpoint of finite groups,
of these new developments in topology that many of us are trying to learn
about.
Oct. 7, 2005
Maxim Laurentiu :
3 p.m. in SEO 636
Abstract
Manifolds -- spaces that locally look uniform, at each point and in each
direction -- have an amazing hidden symmetry, called Poincaré duality. This
reflects the existence of a non-degenerate, symmetric bilinear form on their
rational homology groups, and turns out to be a very powerful tool for the study
of manifolds.
Singular spaces, for example a pinched torus, have no such structure on their
homology groups. Surprisingly though, much of the manifold theory (including
Poincaré duality) can be recovered for a large class of singular spaces if we
consider not the usual homology, but instead intersection homology.
I will give an elementary introduction to intersection homology, following the
original approach of Goresky-MacPherson.
April 21, 2006
Brooke Shipley :
11 a.m. in LH 301
Abstract
This talk is an advertisement for Math 568: Topics in
Algebraic Topology. This talk will be in the style
of a graduate student colloquium (but I'll assume you've
seen the fundamental group).
Math 568 this fall is currently scheduled at 9am,
but I may need to reschedule it. Please register for
this class and then we will find a time which will
work for everyone. (Likely topics for this course:
homotopy groups, spectral sequences, classifying spaces,
fibrations and vector bundles.)
Oct. 6, 2006
Kevin Whyte :
3 p.m. in SEO 636
Oct. 20, 2006
David Nicholls :
3 p.m. in SEO 636
Abstract
Free boundary and boundary value problems arise in many areas of
science and engineering. From free-surface fluid mechanics, to brain
tumor evolution, to electromagnetics and acoustics, the applications
are everywhere. In many cases such phenomena can be accurately
modeled by partial differential equations (PDE) posed on irregular
and/or moving domains. In this talk we will discuss some of these
applications, and a class of techniques (Boundary Perturbation
Methods) for analyzing these PDE, both theoretically and numerically,
based upon perturbations of the boundary domain. The talk will be
aimed at a general audience of graduate students and advanced
undergraduates.
Nov. 3, 2006
Wai Yan Pong :
3 p.m. in SEO 636
Abstract
The matching game is a two-person game. One player is the "match" player and the
other is the "no-match" player. Each player draws a ball at random from a bag
containing balls of two different colors. The match player wins the game if the
two balls drawn are of the same color, otherwise the no-match player wins. A
matching game of $n$ colors, or simply an $n$-color matching game, is played the
same way, except the balls are of $n$ different colors. We will discuss a
generalization of the matching game and show how to find the fair games.
Feb. 9, 2007
Brayton Gray :
3 p.m. in SEO 636
Abstract
In ordinary decimal arithmetic one has a finite string of digits followed by a
decimal point followed by a possibly infinite string of digits. What happens if
we reverse this? An infinite string of digits going off to the left followed by
a decimal point followed by a finite string of digits.
.......66666667 x 3 = 1
Sept. 7, 2007
Robert L. Grossman :
3 p.m. in SEO 636
Abstract
Trees have been an important tool in computation since at
least the work of Cayley. Less well known are how algebras of trees can
also be used as a computational tool. In this talk, we survey some of
the uses of algebras of trees in applied mathematics and computer
science.
Sept. 21, 2007
Alex Furman :
3 p.m. in SEO 636
Abstract
One of the most accessible general properties of a finitely
generated infinite groups is its growth function. In this talk
we will discuss some old and very recent results pertaining
to growth.
Sept. 28, 2007
Roman Shvydkoy :
3 p.m. in SEO 636
Abstract
A conservation law is a physical quantity that remains constant in
time. In the theory of fluid dynamics there are laws that play a
significant role in the study of geometry, stability and regularity of
flows. For a fluid without internal friction of particles we know
several conserved quantities such as energy, helicity, momenta, and
enstrophy in 2D. In this talk we will discuss how these laws are
derived, conditions under which they are valid, and
physical phenomena in which they are implicated.
Oct. 19, 2007
Chun-Hsiung Hsia :
3 p.m. in SEO 636
Abstract
In the talk, we introduce the concepts and the machinery of
bifurcation theory. Bifurcation theory studies the qualitative changes of
the solutions for a system of equations while the control parameters of
the equations vary. In this talk, we start with algebraic equations and
ordinary differential equations to demonstrate the basic idea of
bifurcation theory. We present different types of bifurcations including
steady-state bifurcation, Hopf bifurcation and attractor bifurcation. We
will conclude this talk by showing applications.
Nov. 2, 2007
Jie Yang :
3 p.m. in SEO 636
Abstract
This talk will start with several typical statistical learning
problems, including handwritten recognition, DNA microarray analysis, and
linear regression problem. Basic types of learning problems and relevant
terminology will be described. Two classical approaches, nearest neighbor
method and linear discriminant analysis, and their variants as well with
applications to supervised learning problems are introduced. To overcome
the curse of dimensionality, a newly developed statistical method based on
permanent process will be discussed.
Feb. 15, 2008
Ramin Takloo-Bighash :
3 p.m. in SEO 636
Abstract
Let G be a finite group. A classical theorem of Cayley asserts that one can
always find an injective homomorphism of G into a permutation group S_n. A
minimal n for which such an injection exists is called the degree of G, and
the associated homomorphism a minimal permutation representation. In this
talk, after explaining a number of classical and new results about minimal
representations, I will explain a new polynomial time algorithm to find
minimal representations of a p-group given the subgroup lattice of the group
- the naive algorithm is exponential time. I will end by describing some
(surprising) applications of the algorithm to the structure theory of
p-groups and a conjecture. This is joint with Ben Elias and Lior Silberman.
The talk will be accessible to anyone with one semester of undergraduate
algebra.
Feb. 22, 2008
Stephen Smith :
3 p.m. in SEO 636
Abstract
Abstract:
The classification of the finite simple groups was a project
that involved hundreds of mathematicians and thousands of
journal pages. It appeared to be complete by about 1981---except
for the non-publication by Mason of his work on quasithin groups.
The situation was finally fixed by Aschbacher and Smith in 2004
with the publication of their 2-volume work on the quasithin problem.
The talk will be at an expository level, introducing the simple
groups, along with the place of the quasithin groups in the
overall classification.
I will also mention at the end a more recently begun project
(joint with Aschbacher, Lyons, and Solomon) to provide
a detailed "outline" of the classification---with some new
insights and connections that may illuminate future work.
(I will continue this latter theme in my algebra seminar talk
on Feb 25, which will also be at an expository level)
Feb. 29, 2008
Christian Haesemeyer :
3 p.m. in SEO 636
Abstract
In this talk, I will try to give an introduction to algebraic K-theory
and an explanation of the different behavior this complicated invariant
exhibits for singularities, as opposed to smooth objects.
April 4, 2008
Rafail Abramov :
3 p.m. in SEO 636
Abstract
Modern computer video cards nowadays are used primarily for real-time 3D
graphic applications, where fast 3D image rendering is a computationally
intensive problem, naturally suitable for parallel processing. As a
result, graphic processing units (GPU) eventually became powerful
multiprocessors, tailored for massively parallel floating point
computations with multiple (as much as 128) floating point arithmetic
units on a single chip. Recently, one of the video card manufacturers,
NVIDIA, started offering a common purpose software interface to program
a GPU for general parallel floating point computations, allowing to
obtain a speedup of 1-2 orders of magnitude in comparison with a modern
CPU for essentially same price. I will try to give an overview of this
new computational approach, as well as show a working program example
and compare the GPU performance with a similar CPU program.
April 25, 2008
Samad Hedayat :
3 p.m. in SEO 636
Sept. 5, 2008
Alex Furman :
3 p.m. in SEO 636
Abstract
This is a graduate student colloquium on some topics in probability,
as a tribute to an extraodinary mathematician, widely considered
as the most influential probabilist, Oded Schramm. Oded has
tragically died in a climbing accident on Labor Day.
In the talk I will (briefly) discuss some topics in random processes in the plane,
such as random walks and Brownian motion, percolation, loop erased random walks,
and hope to indicate Oded's contribution to proving important conjectures
in this field via so called Schramm-Lowner Evolution.
Sept. 25, 2009
Louis H. Kauffman :
3 p.m. in SEO 636
Abstract
A classical knot or link is an embedding of a circle or disjoint union of circles in three dimensional space. Classical knot theory is the study of such embeddings up to ambient isotopy, and forms one of the key sources of problems and ideas in geometric and algebraic topology. Mikhail Khovanov in 1999 discovered a remarkable way to associate a chain complex with a diagram of a classical knot or link so that a (graded) Euler characteristic of this homology for the knot K gives the Jones polynomial for K. This Khovanov homology is often referred to as a categorification of the Jones polynomial. Categorification itself refers to creations of larger categories from given categories by taking equalities in a source category and letting them become transformations in a new larger category whose objects are the morphisms in the source category. Part of this talk will be devoted to discussing the meaning of categorification, and part in explaining how the construction of the Khovanov homology works, and how it has been used to give (via Rasmussen's work) a new proof of Milnor's conjecture for the 4-ball genus of torus knots. This talk will be pictorial and self-contained. It helps to have been exposed to homology and the definition of a category before entering the room.
http://en.wikipedia.org/wiki/Homology_(mathematics)
http://en.wikipedia.org/wiki/Category_theory
Oct. 30, 2009
Irina Nenciu :
3 p.m. in SEO 636
Abstract
In this talk, I will give a brief overview of both classical and modern ideas related to random matrix theory, with emphasis on beta-ensembles for general beta positive. These random matrices have eigenvalues following the Gibbs distribution for n particles of the Coulomb gas on the unit circle or the real line, at inverse temperature beta. In particular, I will highlight the change of perspective, from the classical unitary ensembles, which incorporate a lot of symmetry, to these new ensembles, which are very sparse, and some of the methods we use in our construction, inspired by orthogonal polynomial theory. Finally, I will present some of the consequences of these constructions, for example their relation to stochastic PDEs.
Nov. 20, 2009
Laura DeMarco :
3 p.m. in SEO 636
Abstract
A classification of the dynamics of polynomials in one complex variable has remained elusive, even when considering only the simpler "structurally stable" polynomials. In this talk, I will describe the basics of polynomial iteration, leading up to recent results in the direction of a complete classification. In particular, I will describe a (singular) metric on the complex plane induced by the iteration of a polynomial. I will explain how this geometric structure relates to topological conjugacy classes within the moduli space of polynomials.
Jan. 29, 2010
David Dumas :
3 p.m. in SEO 636
Nov. 14, 2011
Cesar Lozano Huerta :
4:15 p.m. in SEO 636
Jan. 23, 2012
Michael Siler :
4:15 p.m. in SEO 636
Abstract
I will introduce carrier graphs, which are a simple tool for using the
geometry of a hyperbolic 3-manifold to study generating sets of its
fundamental group. I will describe how they have been used to make
progress on the long standing rank versus genus conjecture and give
connections between carrier graphs and geometric group theory.
Though all are welcome to attend, this talk is intended to be accessible to
graduate students of all levels, and no background knowledge of geometric group
theory or hyperbolic geometry is expected.
Jan. 30, 2012
Jim Freitag :
4:15 p.m. in SEO 636
Abstract
We will discuss recent developments in differential algebraic geometry. Particular emphasis will be placed on the interplay of this subject with stability theory.
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Though all are welcome to attend, this talk is intended to be accessible for graduate students of all levels; no background knowledge of algebraic geometry is expected.
Feb. 6, 2012
Luigi Lombardi :
4:15 p.m. in SEO 636
Abstract
The Hodge numbers represent the dimensions of certain cohomology spaces (the Dolbeault cohomology) and
are often used in the classification theory of complex manifolds. In this talk we will review inequalities
involving Hodge numbers, starting from the classical ones and continuing through to the most recent ones. We will introduce all the
notions necessary to the understanding of the problem and plenty of time will be devoted to examples.
Though all are welcome to attend, this talk is intended to be accessible for graduate students of all levels; no background knowledge of Hodge theory is expected.
Feb. 13, 2012
Hao Liang :
4:15 p.m. in SEO 612
Abstract
I will introduce the Cayley graph, which is a geometric tool for studying
groups. I will give an explicit example showing a connection between the
coarse geometric properties of the Cayley graph and the algebraic properties
of the group.
Though all are welcome to attend, this talk is intended to be accessible for graduate students of all levels; no background knowledge of geometric group theory is expected.
Feb. 20, 2012
Holly Krieger :
4:15 p.m. in SEO 636
Abstract
I will give an introduction to the study of arithmetic properties of sequences associated to iteration of a rational function defined over a number field, focusing on the notion of the Zsigmondy set associated to such a sequence. I will also give an overview of some of the methods used to prove effective Zsigmondy results, particularly in the case of the critical orbit of $z^d+c$.
Though all are welcome to attend, this talk is intended to be accessible for graduate students of all levels; no background knowledge of arithmetic dynamics is expected.
Feb. 27, 2012
Cesar Lozano Huerta :
4:15 p.m. in SEO 636
Abstract
We will analyze a very specific way of deforming an algebraic variety (birational
models). This will be the core of the talk. If time permits, we will see that despite the fact that such deformations are badly behaved from the topological point of view, it seems that moduli spaces under such transformations remain moduli spaces. In this talk we will give evidence (mainly due to I. Coskun) suggesting that this phenomenon warrants further exploration.
This talk is meant to be accessible for graduate students of all levels; no technical knowledge of algebraic geometry will be assumed.
March 5, 2012
Hexi Ye :
4:15 p.m. in SEO 636
Abstract
Let $f$ be a generic rational function with degree $d \geq 3$. We
are going to show that for any rational function $g$ with $\mu_f=\mu_g$
(the same measure of maximal entropy), $f$ and $g$ have some common
iterate--i.e., $f^n=g^m$. Moreover, we are going to give non-exceptional functions $f$
and $g$, with the same degree and measure, such that there is no $\sigma$ in
$PSL_2(\mathbb{C})$ and no $n \geq 1$ with $f^n=\sigma \circ g^n$ (which cannot happen in the polynomial case).
This talk is meant to be accessible for graduate students of all levels; no technical knowledge of complex dynamics is expected.
March 12, 2012
Chih-Chi Chou :
4:15 p.m. in SEO 636
Abstract
I will give an introduction to the minimal model program, which is a long and
still ongoing journey of algebraic geometers trying to find relative simple representatives of algebraic varieties. I will also talk about theorems of singularities
occurring in the minimal model program.
This talk is meant to be accessible for graduate students of all levels; no technical knowledge of algebraic geometry is expected.
March 26, 2012
Paul Reschke :
4:15 p.m. in SEO 636
Abstract
The entropy of a dynamical system is an indicator of the richness of its dynamics. While automorphisms of complex curves cannot have positive entropy, certain complex surfaces do admit automorphisms with positive entropy. Because they preserve many cohomological structures, automorphisms with positive entropy offer opportunities to study rich dynamical systems in particularly great detail. We will explore these ideas by looking at explicit examples of automorphisms of abelian surfaces and projective K3 surfaces.
This talk is intended to be accessible to graduate students of all levels; no technical knowledge of complex geometry or complex dynamics is expected.
April 2, 2012
Brad Groff :
4:15 p.m. in SEO 636
Abstract
In recent years, increasing attention has been given to the idea that discrete groups can be analyzed by the geometric information contained within their Cayley graphs. I will informally introduce two classes of groups which have been fruitfully investigated from this perspective, as well as the proper notion of rigidity in this context and some modern results in that direction.
Though all are welcome to attend, this talk is intended to accessible to graduate students of all levels; no technical knowledge of geometric group theory is expected.
April 9, 2012
Andrew Brasile :
4:15 p.m. in SEO 636
Abstract
The set of boundary slopes of essential surfaces of a knot complement has been
the object of study for several decades. It is an important invariant of the knot and has
applications to Dehn fillings and the Neuwirth Conjecture. Studying this set makes use
of theory from a surprising variety of fields including complex analysis, tropical geometry,
linear algebra and group theory. We will establish these applications and connections to
other fields and finally discuss an approach for finding the set of boundary slopes of essential
surfaces of knot complements.
This talk is intended to be accessible for graduate students of all levels; no technical knowledge of knot theory is expected.
April 16, 2012
Matt Wechter :
4:15 p.m. in SEO 636
Abstract
The Fundamental Theorem of Galois Theory gives correspondences
between intermediate subfields of a Galois (separable, splitting)
extension and subgroups of automorphisms. I will talk about Jacobson
Galois Theory detailing similar correspondences for purely inseparable
field extensions of exponent one. Further, I will describe more recent
Galois-type correspondences for purely inseparable fields of higher
exponent and my attempts to bridge the two approaches using differential
algebra.
This talk is intended to be accessible to graduate students of all levels; no technical knowledge of differential algebra is expected.
April 23, 2012
Jonah Gaster :
4:15 p.m. in SEO 636
Abstract
In his Geometrization Theorem for Haken 3-Manifolds, Thurston showed that the study of Riemann surfaces could be used powerfully towards understanding hyperbolic 3-manifolds. We will introduce some of the classical aspects (Teichmuller theory, quasi-Fuchsian space, Ahlfors-Bers correspondences), and then discuss Thurston's skinning map. Given an open hyperbolic 3-manifold, with Riemann surface boundary, one can use the classical theory to 'reflect' the geometry on the boundary at infinity through the 3-manifold, and make sense of what comes out the other side. After a brief survey of relevant results, we will describe an example (work in progress) that seems to indicate that the skinning map can have a critical point.
Though all are welcome to attend, this talk is intended to be accessible for graduate students of all levels; no background knowledge of Teichmuller theory is expected.
Aug. 27, 2012
Holly, Cesar, Paul, Jonah :
4 p.m. in SEO 636
Sept. 10, 2012
Paul Reschke :
4:15 p.m. in SEO 636
Abstract
We will discuss several topics in complex analysis and complex dynamics. We will give particular attention to hyperbolic components of the Mandelbrot set, conformal isomorphisms, and finite Blaschke products on the unit disk. On September 17, Holly Krieger will present applications of the material covered in this talk to current research.
This talk is intended to be accessible to graduate students of all levels; no technical knowledge of complex analysis or complex dynamics is expected.
Sept. 17, 2012
Holly Krieger :
4:15 p.m. in SEO 636
Abstract
We will discuss the hyperbolic components of the Mandelbrot set, and bound the recurrence of the critical orbit for parameters which are not too close to centers of hyperbolic components. If time permits, we will discuss the application of these bounds to a primitive prime divisors problem in arithmetic dynamics. This is Part 2 of the talk begun by Paul Reschke on September 10.
This talk is intended to be accessible to graduate students of all levels; no technical knowledge of complex analysis or complex dynamics is expected.
Sept. 24, 2012
Chih-Chi Chou :
4:15 p.m. in SEO 636
Abstract
We will give an introduction to birational geometry and moduli spaces. Besides definitions and basic properties, we will discuss examples including the moduli space of conics and, if time permits, some geometry of it.
This talk is intended to be accessible to graduate students of all levels; no technical knowledge of algebraic geometry is expected.
Oct. 1, 2012
Cesar Lozano Huerta :
4:15 p.m. in SEO 636
Abstract
This is the continuation to last week's talk. This time, I will discuss the history of the Minimal Model Program (MMP) and describe what happens when we use it for moduli spaces. A fair part of the talk will be devoted to defining the language required to understand the Minimal Model Program and Moduli Spaces. Depending on time, I aim to finish the presentation with an exposition of results from people closely related to our department, such as I. Coskun, D. Chen, and J. Huizenga.
Oct. 8, 2012
Alex Austin :
4:15 p.m. in SEO 636
Abstract
I will introduce hyperbolic 3-manifolds, discuss the interaction of geometry and topology, stress the importance of volume, and describe some manifolds that exhibit these phenomena.
This talk is intended to be accessible to graduate students of all levels; no technical knowledge of hyperbolic geometry is expected (though the definitions of a manifold and its fundamental group may be assumed).
Oct. 15, 2012
Michael Siler :
4:15 p.m. in SEO 636
Abstract
I will introduce carrier graphs, which are a simple tool for using the geometry of a hyperbolic 3-manifold to study generating sets of its fundamental group. I will describe how they have been used to make progress on a long standing 3-manifold problem and present some recent results on their nice geometric properties.
Oct. 22, 2012
Xudong Zheng :
4:15 p.m. in SEO 636
Abstract
Due to results of Mukai, Bondal, Orlov, Bridgeland, and Kawamata, among others, the derived category (of coherent sheaves on a smooth projective variety) turns out to be a very useful invariant of a variety. In this talk, I will briefly introduce the framework of this invariant.
This talk is aimed at graduate students of all levels. I will work through some basics of algebraic geometry, line bundles, sheaves and their cohomology in the first half of the talk. No technical knowledge of algebraic geometry is expected.
Oct. 29, 2012
Luigi Lombardi :
4:15 p.m. in SEO 636
Abstract
Work of Bondal, Kawamata and Orlov has shown that the derived category of coherent sheaves on a variety is a categorical object carrying a plethora of information about the variety itself. I will show which information can be read off from the derived category of a variety, with particular focus on the case of varieties having non-vanishing first Betti number.
Nov. 5, 2012
Yen Duong :
4:15 p.m. in SEO 636
Abstract
I will introduce group cohomology, which concretely connects algebra and algebraic topology. I'll work through examples in lower dimensions and explain how they relate to group properties. I'll also introduce central extensions, which Hao Liang will use in his talk on November 12.
This talk will be accessible to graduate students of all levels; basic concepts from abstract algebra may be assumed, but no technical knowledge of cohomology theory will be expected.
Nov. 12, 2012
Hao Liang :
4:15 p.m. in SEO 636
Abstract
Finding an algorithm to solve equation systems in groups is an
old and hard problem. There is no algorithm which solves arbitrary
equation systems in arbitrary finitely presented groups. However,
algorithms have been shown to exist in many important classes of groups.
I will talk about an algorithm which solves equation systems
in central extensions of hyperbolic groups.
Nov. 26, 2012
Joel Sievert :
4:15 p.m. in SEO 636
Abstract
In this talk we'll be examining the Hodge structure of smooth, compact complex Kahler manifolds (or complex projective algebraic varieties). We'll be concerned primarily with surfaces, but most of the information will be applicable in general dimension. Roughly, Hodge theory is the study of differential forms on a smooth manifold. Our primary goal will be to recall the Hodge Index Theorem and the Lefschetz Theorem on (1,1)-classes.
This talk is intended to be accessible to graduate students of all levels; no technical knowledge of Hodge theory will be expected.
Dec. 6, 2012
Paul Reschke :
3:15 p.m. in SEO 636
Abstract
I will discuss several results regarding invertible dynamical systems on compact Kähler surfaces. The results will highlight the importance of understanding the cohomological actions induced by such systems. I will also present explicit examples of compact Kähler surface automorphisms with infinite-order dynamics.
Jan. 28, 2013
Sam Ziegler :
4:15 p.m. in SEO 636
Abstract
This talk will consist of two parts; first we will discuss some basic descriptive set theory, including Baire Category, the Kuratowski-Ulam theorem, and basic applications. Next we will discuss the basic theory of Polish groups, in particular the locally compact totally disconnected theory, on which Phil will elaborate in the following week.
The talk will assume no background in mathematical logic or descriptive set theory, and should be accessible to all graduate students.
Feb. 4, 2013
Phil Wesolek :
4:15 p.m. in SEO 636
Abstract
In a locally compact, totally disconnected group, $G$, an
element $g\in G$ is \emph{periodic} if $cl(\langle g \rangle)$ is
compact. We denote the collection of all such elements $P_1(G)$.
$P_1(G)$ consists of all the elements which lie in a compact, open
subgroup. Moreover, these elements play an essential role in the
structure of locally compact totally disconnected groups because the
topology has a basis at 1 of compact open subgroups. We study the
structure of $P_1(G)$ and how it interacts with global structure. In
this talk we consider how global homogeneity, algebraic, or finiteness
assumptions interact with the structure of $P_1(G)$ and with the
compact, open subgroups which cover $P_1(G)$.
Feb. 11, 2013
Brad Groff :
4:15 p.m. in SEO 636
Abstract
This talk will have two parts. First, I will discuss some of the basic facts relating to the large-scale geometry of finitely generated groups. Following this, I will discuss how this geometric information impacts many of the algebraic properties of these groups. Particular examples of this phenomenon will come from my work on splittings of groups (in the sense of Bass-Serre theory) that can be seen directly in the group's geometry.
The talk will assume no background in geometric group theory, and should be accessible to all graduate students.
Feb. 18, 2013
Matt Wechter :
4:15 p.m. in SEO 636
Abstract
While classical Galois theory has been established and studied for centuries, Galois correspondences for purely inseparable extensions have only been explored since the 1960s, following work by Jacobson and Gerstenhaber. In this talk we'll give background material on purely inseparable extenions and their modular intermediate subfields. Then we'll see how these subfields can be found by studying the differential operators of the field extension.
Feb. 25, 2013
William Simmons :
4:15 p.m. in SEO 636
Abstract
A differential ring is a ring with an additive homomorphism $\delta$
satisfying the usual product rule. The algebraic and geometric behavior of
differential rings presents interesting variations on standard commutative
algebra and algebraic geometry. The classical fundamental theorem of
elimination theory provides a good case study. It asserts that projective
varieties over an algebraically closed field $K$ are \emph{complete}. That
is, if $V$ is such a projective variety and $W$ is any algebraic variety
defined over $K$, then the projection $V\times W \rightarrow W$ takes
Zariski closed sets to Zariski closed sets. The differential case is more
complicated and has features not present in the original. We discuss how
insights from computational examples and model theory suggest a path
through the complexity.
March 14, 2013
Marc Kjerland :
4 p.m. in SEO 636
Abstract
Most problems in contemporary science are nonlinear and often are sensitive to initial conditions or exhibit structure on multiple scales. As a result, different notions of "solution" must be considered. I will present some notions from nonlinear dynamics, including chaos and perturbation response, with examples from the geosciences. If time permits, we will look at applications of linear response theory to solve multiscale systems.
This talk is intended to be accessible to graduate students of all levels; no background knowledge of nonlinear dynamics is expected.
March 18, 2013
Deniz Bilman :
4:15 p.m. in SEO 636
Abstract
I will briefly introduce the Fermi-Pasta-Ulam (FPU) lattices, and talk on how a humble experiment started a new era in nonlinear evolution equations. Then I will consider the doubly infinite Toda lattice, introduce the notion of complete integrability and soliton solutions along with some other interesting properties of this system. Within this framework, I will briefly cover Lax formalism and the inverse scattering transform which is a method for solving integrable systems. Finally, I will talk on the "soliton resolution conjecture" and discuss the long time asymptotics problem for the lattices obtained by perturbations of the integrable Toda potential. I will mention a few results along with some interesting obstacles towards the solution to this problem.
The talk is intended to be self-contained and accessible to graduate students from different fields. There will be three short movies and two pictures.
April 1, 2013
Ellie Dannenberg :
4:15 p.m. in SEO 636
Abstract
The Teichmüller Space of a surface S is the space of isotopy classes of hyperbolic structures on S. I'll discuss the Teichmüller space and give a dimension count using a pants decomposition and the Fenchel-Nielsen Coordinates. I'll then give a brief introduction to the Curve Complex of a surface.
April 8, 2013
Matthew Durham :
4:15 p.m. in SEO 636
Abstract
We study the action of a finite subgroup of the mapping class group on Teichmüller space and present some results. Following on last week's discussion, we further discuss certain aspects of Teichmüller geometry and some machinery we use in our work.
April 15, 2013
Caroline Terry :
4:15 p.m. in SEO 636
Abstract
We introduce basic notions from first order logic including formulas, theories, and models. We then discuss through examples some model theoretic notions including type spaces and indiscernible sequences.
April 22, 2013
Gabe Conant :
4:15 p.m. in SEO 636
Abstract
After presenting the basic model theoretic notions of type spaces and saturation, we will define forking and dividing. These are geometric/combinatorial measurements of the complexity of a theory, which are designed to detect a notion of independence. We will interpret this independence in several classical examples such as algebraically closed fields and the random graph, as well as some newer examples. Finally, we discuss forking and dividing in the context of dividing lines in the classification of unstable theories.
April 29, 2013
Matthew Bourque :
4:15 p.m. in SEO 636
Abstract
Policy improvement is a well known and popular algorithm for solving Markov decision processes. It has been extended to solve some classes of discounted stochastic games. I will discuss the extension of policy improvement to solve average-payoff stochastic games with additive rewards and additive transitions.
Sept. 9, 2013
Bruno De Mendonca Braga :
5:30 p.m. in SEO 636
Abstract
This talk will lie between the intersection of descriptive set theory and the theory of Banach spaces.
I will introduce some basic definitions from descriptive set theory and the standard coding for the separable Banach
spaces SB. With that in mind, we are able to wonder whether classes of Banach spaces are Borel or not in this special coding.
I will discuss the Borel complexity of some specific classes and talk about how to use its Borel complexity
in order to get some non universality results.
Sept. 16, 2013
Phillip Wesolek :
5:30 p.m. in SEO 636
Abstract
Lie groups over the $p$-adics form an interesting class of non-discrete locally compact totally disconnected groups.
It is well known the local results in connected Lie theory apply equally well in the $p$-adic case. However,
many of the global theorems of connected Lie theory are not known to have $p$-adic counterparts.
We present a global decomposition theorem for $p$-adic Lie groups which is analogous to the familiar solvable by semi-simple
decomposition for connected Lie groups.
Sept. 23, 2013
RJ Stading :
5:30 p.m. in SEO 636
Abstract
Given graphs $G$ and $H$ and a number $m$, let $f(G,H,m)$ be the number of colors
required to edge color $G$ so that every copy of $H$ in $G$ contains at least $m$
distinct colors amongst its edges. Let $Q_n$ denote the the $n$-dimensional
hypercube and $C_k$ denote the cycle of length $k$. For even values of $k$, we
will look at $f(Q_n,C_k,k)$.
Sept. 30, 2013
Cesar Lozano Huerta :
5:30 p.m. in SEO 636
Abstract
In many branches of mathematics there are major guiding problems which are impossible to solve but motivate nonetheless much of the research in the area. In algebraic geometry one such problem is the classification of algebraic varieties.
In this talk I will use tools from two areas devoted to that goal: The Minimal Model Program (MMP) and Theory of Moduli Spaces to understand (classify) polynomials of degree two (and their degeneration.)
Oct. 7, 2013
Lei Song :
5:30 p.m. in SEO 636
Abstract
Given a smooth complex projective variety $X$, the line bundles on X having nontrivial sections form a closed subset $W^0(X)$ of the Picard scheme of X.
I will show that if every divisor $D$ in a fixed linear system $|L|$ is semi-regular, then there exists a Zariski open neighborhood of $L$ on $W^0(X)$ which has at worst rational singularities.
This generalizes Kempf's result for curves.
Oct. 14, 2013
Karen Zaya :
5:30 p.m. in SEO 636
Abstract
We focus on the Euler equations. We will discuss turbulent fluid flows, Kolmogorov's theory or turbulence, Onsager's conjecture, formal conservation of energy, anomalous dissipation of energy, and the inviscid dyadic model.
Oct. 21, 2013
Brian Powers :
5:30 p.m. in SEO 636
Abstract
We will present the background knowledge needed to understand the results in our paper, including anti-coordination games, Nash equilibria and strict Nash equilibria, the price of anarchy, and the theory of NP-completeness.
We present the specific game analyzed in our paper and analyze some of its basic properties.
Oct. 28, 2013
Jeremy Kun :
5:30 p.m. in SEO 636
Abstract
We present a general type of anti-coordination game played on a graph and analyze its stable and strictly stable Nash equilibria. We characterize when such equilibria exist and when their decision problem is NP-hard. Next we consider the directed case, a generalization which captures both coordination and anti-coordination. We prove the decision problem there for non-strict equilibria is also NP-hard.
Nov. 11, 2013
Abel Castillo :
5:30 p.m. in SEO 636
Abstract
Let $f(X, t_1, t_2, \cdots, t_r)$ be a polynomial with integer coefficients that is irreducible over $\mathbb Q$. It can be shown that there exists a specialization $t_i \mapsto a_i$ with $a_i \in \mathbb Z$ such that $f(X, a_1, a_2, \cdots, a_r)$ is also irreducible over $\mathbb Q$. ``Hilbert's Irreducibility Theorem'' refers to the fact that this is actually the case for ``almost all'' integer specializations.
In this talk we will discuss how techniques from probabilistic Galois theory can give rise to so-called ``effective versions'' of Hilbert's Irreducibility Theorem. We will also discuss how Hilbert's Irreducibility Theorem relates to other questions of number-theoretic interest, such as the Inverse Galois problem and conjectures about prime values of polynomials. As time permits, we will also talk about how Hilbert's Irreducibility Theorem can be interpreted in the language of algebraic varieties, using the notion of ``thin'' sets (in the sense of Serre)
Nov. 18, 2013
Caroline Terry :
5:30 p.m. in SEO 636
Abstract
We introduce some questions that drove the development of early model theory, in particular questions of counting the number of models of a theory.
We then discuss the answers to these questions, and some of the techniques developed to do so.
In particular we consider the example of algebraically closed fields of characteristic zero, and the interpretation of some stability theoretic notions there.
Finally we discuss the interpretation of some of these notions in a particular unstable theory, that of the rational Urysohn sphere, i.e. the unique universal
and homogeneous countable rational metric space.
Nov. 25, 2013
Zheng Fang :
4:30 p.m. in SEO 636
Abstract
We introduce some numerical approximation to the solution of ordinary differential equations and partial differential equations. Specifically, the Euler's method and Runge-Kutta method from finite difference, and spectral methods using discrete Fourier basis. Some constraints on applying these methods would also be discussed using linear stiffness equation as an example.
Dec. 2, 2013
Marc Kjerland :
4:30 p.m. in SEO 636
Abstract
Most problems in contemporary science are nonlinear and often display sensitivity to initial conditions or multiscale structure. Due to these difficulties, alternative notions of "solution" must be considered. I will present some concepts from nonlinear dynamics, including chaos and perturbation response, as well as applications to problems in the mathematical geosciences.
Feb. 13, 2014
Alex Austin :
4 p.m. in SEO 636
Abstract
If two metric spaces are bi-Lipschitz equivalent, then they are effectively the same. Using the fundamental question, 'When is a metric space bi-Lipschitz equivalent to the Euclidean space $\mathbb{R}^n$?', as motivation, I will introduce the (still far from solved) quasiconformal Jacobian problem: characterize those weights on $\mathbb{R}^n$ comparable to the Jacobian of a quasiconformal map.
I will discuss a result of Bonk, Heinonen and Saksman which displays a class of weights which are comparable to quasiconformal Jacobians, and a result of Kovalev and Maldonado which gives information on how wide to cast your net if you seek a characterization.
Feb. 20, 2014
Alex Austin :
4 p.m. in SEO 636
Abstract
This is the second of two talks. In the first we saw quasiconformal maps on $\mathbb{R}^n$ generated by vector field flows, and using convex potentials, in order to approach the quasiconformal Jacobian problem. In this talk I will introduce the Heisenberg group, its Lie algebra structure, and explain how the latter gives rise to a horizontal distribution to which we associate a metric. Though this metric renders the space highly non-Euclidean, we can develop a calculus compatible with the horizontal structure and there is a well developed theory of quasiconformal maps. I will explain that the ingredients of the first talk, vector field generators and notions of convexity, are all present, and allow an attempt at the results of the first talk in this new setting.
Feb. 27, 2014
Xudong Zheng :
4 p.m. in SEO 636
Abstract
Given an algebraic variety $X$, the Hilbert scheme of points $Hilb^n(X)$ is the parameter space of all possible collection of $n$ points on $X$. If $X$ is a smooth curve, $Hilb^n(X)$ is isomorphic to the $n$-fold product of $C$
modulo the action of permuting factors by the symmetric group $S_n$. A second extensively studied case is when $X$ is a smooth surface. In particular, M. Haiman utilized geometric properties of the Hilbert scheme of points
to study questions in combinatorics.
Suppose $k[x_1, \dots, x_n]$ is the polynomial ring in $n$ variables. It is a free module of rank $n!$ over the subring $k[x_1, \dots, x_n]^{S_n}$, consisting of permutation invariant polynomials. The most naïve form of the $n!$
conjecture concerns the analogy for the polynomial ring $k[x_1, \dots, x_n; y_1, \dots, y_n]$ with the symmetric group $S_n$ simultaneously permuting the variables $(x_1, \dots, x_n)$ and $(y_1, \dots, y_n)$.
I will explicitly give a system of coordinates on the Hilbert scheme and relate the Hilbert scheme to the $n!$ conjecture. If time permits, I will indicate some generalizations to the case where the underlying variety $X$ has some singularity.
March 13, 2014
Yen Duong :
4 p.m. in SEO 636
Abstract
This is the first of a series of two talks. This will provide the motivation and definitions for the next talk. There'll be algebra and pictures and definitions of the three things in the title. No background is assumed (you should know what a quotient group is, I guess).
March 20, 2014
Matt Durham :
4:15 p.m. in SEO 636
Abstract
I'll introduce some of the Masur-Minsky machinery, which
is at the heart of much of the modern development of our
understanding of the geometry of both Teichmuller spaces and mapping class groups.
I'll try to motivate all this by first discussing a number of big results which use
this machinery (such as the Ending Lamination Theorem),
and after I introduce it, I'll discuss how it is used to prove some of them.
April 3, 2014
Seckin Adali :
4 p.m. in SEO 636
Abstract
Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to a flag satisfying certain tangency conditions with respect to the relevant symmetric bilinear form. In this talk I will introduce restriction varieties without assuming any background on them and go over some examples illustrating a resolution for their singularities.
April 10, 2014
Gabe Conant :
4:15 p.m. in SEO 636
Abstract
We introduce the Fraisse construction, which is a method for building countable homogeneous structures from certain classes of finite structures (called Fraisse classes). The focus of the talk will be on classes of finite metric spaces with distances from a particular finite set, which is fixed for the class. We present a characterization, called the four-values condition, of when theses classes are Fraisse classes, followed by examples and reformulations of this characterization. We then consider the first-order theory of the countable homogeneous Fraisse limit. Of special interest in this context is the strong order property, which is a combinatorial measurement of the complexity of a first order theory.
April 17, 2014
Zheng Fang :
4:15 p.m. in SEO 427
Abstract
The propagation of linear acoustic waves arise in a wide array of applications, for
instance, in mechanical engineering, materials science, and the geosciences. In the frequency domain one must solve a system of coupled elliptic PDEs, however this can by greatly simplified by considering interface unknowns. To realize this one must be able to produce normal stresses at these interfaces and Dirichlet Neumann Operators accomplish this. In this talk, we will discuss a boundary perturbation approach to compute these operators in a rapid, high-order and robust fashion.
May 1, 2014
Edgar A. Bering IV :
4:15 p.m. in SEO 636
Abstract
Recent advances in 3-manifold theory have settled the conjectural picture proposed by Bill Thurston in the 1980's, however these resolutions have a common theme: they tend
not to be effective. For example, Ian Agol's resolution of the Virtual Haken Conjecture is purely existential; every 3-manifold has a finite-index cover that is ``sufficiently large'',
but there are no known bounds on this index in terms of the starting manifold. In this talk we will review some of the history of the result and then discuss recent work in
understanding 3-manifolds and their covers statistically, concluding with some open problems in this direction. The talk will aim to be self contained and broadly accessible.
Aug. 31, 2015
Adam Lelkes :
4 p.m. in SEO 636
Abstract
This talk will be a brief introduction to the beautiful theoretical results in cryptography. Time permitting, we will answer questions such as:
1. Can you have perfect encryption? (Spoiler: yes, but it is impractical.)
2. Is there a notion of security that is practical but still hard enough to break? If so, can we achieve it?
3. Can we algorithmically generate pseudo-random bits? How random is random enough?
4. Is it possible to convince someone you proved the Riemann hypothesis without revealing any information at all about the proof itself?
5. You are talking to your friend on the phone and you want to decide an important question by flipping a coin. How can you make sure that your friend didn't lie about the result of the coin toss?
Sept. 14, 2015
Alex Stathis :
4 p.m. in BH 209
Abstract
A common problem in algebraic geometry is to count the number of objects satisfying given geometric conditions. These could be collections of points, curves, varieties, or almost any conceivable geometric object. Intersection theory is a useful tool in computing these numbers. The easiest and most well understood example is that of the Grassmannian, and more specifically, the first nontrivial Grassmannian of planes in four space. I'll define all the basic objects and we'll compute the intersection product on this Grassmannian. Throughout the talk, I'll attempt to point out how intersection theory is useful in simplifying geometric counting problems.
There will be food.
Sept. 21, 2015
Xudong Zheng :
4 p.m. in BH 209
Abstract
One of the central endeavor in algebraic geometry is to
understand geometric objects in families, hence the study of moduli
spaces. These are parameter spaces where each point corresponds to an
object of some kind. I will focus on one particular such moduli space, the
Hilbert scheme of points. The first part of the talk will concentrate on
its local geometric properties. Later I will put more emphasis on several
of its applications in enumerative geometry, representation theory, and
combinatorics. By the end I will mention some current research on the
Hilbert scheme of points.
There will be food.
Sept. 28, 2015
Jeremy Kun :
4 p.m. in BH 209
Abstract
In this talk I will introduce you to computational learning theory, which is the mathematical foundation for much of applied machine learning. I will discuss two models: the "Probably Approximately Correct" model and the Statistical Query model. I will focus on concrete learning problems and algorithms, and survey some of the deep ideas that have arisen in this field.
Food will be provided.
Oct. 5, 2015
Brian Powers :
4 p.m. in BH 209
Abstract
First proposed in 1966, Final-Offer Arbitration has been adopted by Major League Baseball as well as in the public sector in many states as a means or resolving negotiation impasses. We will discuss the mechanics of this arbitration method and the game theoretic model under which it has been studied. Some of the important results concerning zero-sum games and probability will be addressed. Finally we will look at the problem of extending the basic model to one where multiple issues are in dispute, and some of the surprising results of the extended model.
There will be food.
Oct. 12, 2015
Yajnaseni Dutta :
4 p.m. in BH 209
Abstract
A classical problem in algebraic geometry is to study the space of algebraic subspaces of an algebraic space. A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single algebraic equations of degree 3. One of the oldest and most beautiful statements in algebraic geometry is that on a smooth cubic surface there are exactly 27 lines. We begin with the classical approach, in a terms of cubic forms, and then proceed to the more modern approach, in which the cubic surface is considered as a blow up of the projective planes in 6 points.
Food will be provided.
Oct. 19, 2015
Edgar A. Bering IV :
4 p.m. in BH 209
Abstract
What are the symmetries of a torus? Taking a topological point of view, we look
for self-homeomorphisms that are homotopically non-trivial, that is, not isotopic to the identity. We arrive at the mapping class group: homeomorphisms modulo isotopy. Puncturing the torus and taking a group theoretic point of view, the appropriate symmetries are automorphisms of the fundamental group, which in this case is the free group on two generators. Once again we quotient out by `obvious' automorphisms, in this case the inner automorphism group, and arrive at the outer automorphism group. Returning to the unpunctured torus, we take an algebraic point of view, and consider the torus as the quotient
of $\mathbb{R}^2$ by an integer lattice $\mathbb{Z}^2$. From this point of view the symmetries of the torus are the linear transformations fixing the integer lattice, $SL(2,\mathbb{Z})$.
These three groups are, for the torus, isomorphic. In general this is not the case, instead the three perspectives outlined are the starting point of three areas of study: mapping class groups, outer automorphism groups, and arithmetic groups, respectively. While distinct there is a close analogy between these three families of groups that drives several areas of current research. Focusing on the motivating example of the torus we will sketch this analogy and point to current frontiers.
Nov. 9, 2015
Maxwell Levine :
4 p.m. in BH 209
Abstract
Georg Cantor lost his mind trying to prove his continuum hypothesis: that there is no set with a cardinality strictly between that of the natural numbers and the real numbers. But nearly a century later Paul Cohen demonstrated that the continuum hypothesis could neither be proved nor refuted by the standard Zermelo-Fraenkel axioms of set theory. The technique he invented, called forcing, is a way of constructing new models of set theory by adjoining transcendental elements to the old models. In this talk I will outline the general methodology of forcing and give a sense of the technical obstacles it must overcome.
Nov. 16, 2015
Karen Zaya :
4 p.m. in BH 209
Abstract
The Navier-Stokes equations, known since the 19th century, describe the motion of a viscous fluid. They are used for many applications in science and engineering, such as modeling weather and designing aircrafts. However, fundamental mathematical questions about solutions to the equations remain unanswered, such as the question of the regularity of solutions, which is one of the Clay Mathematics Institute Millennium Prize problems. We will discuss the context, meaning, and challenges of the problem, as well as key concepts and results that have advanced progress toward answers.
Nov. 30, 2015
Cara Mullen :
4 p.m. in BH 209
Abstract
The goal of this talk is to introduce the subject of p-adic dynamics: the study of dynamics over a field that is complete with respect to a nonarchimedean absolute value. We will discuss some of the major results, with a little motivation from classical complex dynamics. No previous knowledge of dynamical systems or number theory will be assumed.
Sept. 12, 2016
Maxwell Levine :
1 p.m. in SEO 636
Abstract
Gödel's Incompleteness Theorems tell us that there are true statements in mathematics that the Zermelo-Fraenkel axioms of set theory cannot prove, and some of these questions require the use of large cardinal axioms to resolve them. I will motivate the classical development of large cardinals with some naturally-occurring questions. Some light proofs will be presented to give a sense of combinatorial flavor. The audience does not need any detailed knowledge of set theory, but should be familiar with Cantor's diagonal argument.
Sept. 19, 2016
Janis Lazovskis :
1 p.m. in SEO 636
Abstract
This will be an entry-level talk at analyzing abstract data by applying topological tools to it. I will introduce persistent homology and describe how that answers some (but not all) problems in this area. Different settings and assumptions will be presented, as the ideas of topological data analysis are very general. We'll end with a discussion of the current directions and techniques of the field.
Sept. 26, 2016
Cloie McClellan :
1 p.m. in SEO 636
Abstract
In this talk, I will introduce the field of geometric group theory and discuss some classic problems and their solutions.
Oct. 3, 2016
Edgar A. Bering IV :
1 p.m. in SEO 636
Abstract
Continuing the theme of the first talk, I will introduce the classical Banach Tarski paradox, its algebraic origins, and the geometry of non-paradoxical groups.
This talk will conclude with a map of the universe of finitely generated groups, as sketched in these two talks. The map to-date contains much uncharted territory.
Oct. 10, 2016
Alexander Stathis :
1 p.m. in SEO 636
Abstract
Enumerative geometry is the theory of counting geometric objects. For simple questions, one can find so-called "classical solutions" by manipulating the geometry in such a way so that the count is easy to obtain. For more complicated questions, one must use more advanced tools such as moduli spaces and intersection theory. I will give a brief history and introduction to the subject and discuss some of the key questions and tools in its development.
Oct. 17, 2016
Nathan Lopez :
1 p.m. in SEO 636
Abstract
Consider a surface S of genus g>1 with n\geq 0 punctures. A crucial tool in the study of automorphisms of S is its Teichmüller Space \mathcal{T}(S), i.e., the space of isotopy classes of hyperbolic structures on S. My first goal in this talk is to explain what \mathcal{T}(S) is, examine its topology, and compute a few examples. Secondly, I'll provide some context and motivation for Teichmüller theory by giving a brief survey of the Dehn-Thurston classification and of Thurston's asymmetric metric on \mathcal{T}(S). To cover so much in just an hour, I'll focus less on rigor and more on intuition.
Oct. 24, 2016
Ellie Dannenberg :
1 p.m. in SEO 636
Abstract
The Koebe-Andreev-Thurston Circle Packing Theorem says that given a triangulation $\tau$ of a surface $S$, there is a unique pair $(g,P)$, where $g$ is a constant curvature Riemannian Metric on $S$. $P$ is a circle packing of circles with respect to $g$ and combinatorics given by $\tau$. If, instead of constant curvature Riemannian metrics on $S$, we consider complex projective structures on $S$, there is a deformation space of complex projective circle packings with fixed combinatorics.
In this talk, I'll discuss circle packings and complex projective structures on surfaces. I'll then discuss the deformation space of complex projective circle packings with combinatorics given by $\tau$. Much is still unknown about this space.
Oct. 31, 2016
Ben Fish :
1 p.m. in SEO 636
Abstract
In this talk, I will introduce learning theory, which is the theory of learning in an algorithmic setting. We will discuss what it means to learn in this context, and discuss what can and can't be learned. Recall Occam's razor is the philosophical principle that says that all else being equal, the simpler hypothesis that describes a given phenomenon should be preferred over a more complicated one. However, 'simpleness' assumes that there is some notion of complexity of a hypothesis. In the process of this talk, I will introduce concrete theorems that formalize Occam's razor and provide concrete notions of complexity.
Nov. 14, 2016
Trevor Leslie :
1 p.m. in SEO 636
Abstract
Let $u$ be a solution to the classical 3D Navier-Stokes equations. If $u$ is smooth, then we know that $u$ satisfies a certain energy balance relation. But what if $u$ is not smooth? It has been known since the 1960s (by a result of Lions) that u satisfies the energy balance relation if $u$ is $L^4$ integrable in both time and space. We give new regularity conditions on $u$ that guarantee energy balance; our conditions depend on both the integrability of $u$ and the Hausdorff dimension $d$ of the singularity set. We recover Lions’ result when $d = 1$ and improve upon it when $d < 1$ or when the singularity set is confined to a single time-slice. This is joint work with Roman Shvydkoy.
Nov. 21, 2016
Janet Page :
1 p.m. in SEO 636
Abstract
One of the difficulties that arises in algebraic geometry is the existence of singularities (or "non-manifold" points). I will introduce singularities and some of the methods we use to study them. In particular, I will compare methods in characteristic 0 and characteristic p, and discuss when they match up.
Jan. 9, 2017
Tori Noquez :
1 p.m. in SEO 636
Abstract
In this talk, we will introduce the basics of model theory and use examples from algebraic geometry to demonstrate notions of definability and theories.
With this in mind, we will introduce the notion of categoricity and its role in the describing the well-behavedness of various classes of mathematical objects. Time permitting, we will discuss some geometric approaches to determining if a theory is categorical.
Jan. 25, 2018
Nathan Lopez :
5 p.m. in SEO 636
Feb. 1, 2018
Cloie McClellan :
5 p.m. in SEO 636
Feb. 8, 2018
Jānis Lazovskis :
5 p.m. in SEO 636
Abstract
I'll give an introduction into my area of research, topological data analysis at the intersection of algebraic topology and algebraic geometry. Visual aids will be provided so that we don't get bogged down with "technicalities" and "definitions" and "proofs."
Feb. 15, 2018
Hunter Chase :
5 p.m. in SEO 636
Feb. 22, 2018
Janet Page :
5 p.m. in SEO 636
Abstract
Combinatorial objects often serve as a natural testing ground to answer questions in algebraic geometry and commutative algebra. In this talk, I'll introduce a few different such objects and explain why they are useful to work with. No background will be required, although a basic knowledge of undergraduate abstract algebra will be extremely helpful.
March 8, 2018
Keaton Quinn :
5 p.m. in SEO 636
March 15, 2018
Wes Townsend :
5 p.m. in SEO 636
Abstract
This talk will introduce the audience to the completion functor on a local ring $(R,\mathfrak{m})$ and the Cohen Structure Theorem.
March 22, 2018
Jay Kopper :
5 p.m. in SEO 636
Abstract
Given two integers, it is a classical problem to find their sum. The modern perspective is to note that the operation of addition endows the integers with the structure of a group. However, the classical algorithm relies on a complicated procedure which obscures the underlying group structure. People typically treat an integer as an element of $\mathbb{Z}/10\mathbb{Z}^{\oplus n}$, and add elements of this group via an arcane process called "carrying." In this talk I will attempt to clarify this confusing apparatus using the group cohomology of certain cyclic groups. To conclude the talk, I will give an explicit example of a sum and discuss applications of addition.
April 12, 2018
Noah Schoem :
5 p.m. in SEO 636
Abstract
In 1878, Georg Cantor proved that there are more real numbers than natural numbers,
and asked whether there is a size strictly in between.
Cantor conjectured that there isn't, and called this conjecture the Continuum Hypothesis.
Nearly a century later, the combined works of Gödel and Cohen
proved this question mathematically impossible to resolve;
the methods they used shaped the foundations of modern set theory into what it is today.
After a brief background on what we mean by mathematically impossible to resolve,
we will explore Gödel's contribution, The Constructible Universe (and the foundations of Inner Model Theory),
in which the Continuum Hypothesis is true.
April 19, 2018
Daniel Ingebretson :
5 p.m. in SEO 636
Abstract
This will be an elementary and accessible introduction to the dimension theory of a simple class of fractals, the attractors of iterated function systems.
Feb. 6, 2019
Joel Stapleton :
5 p.m. in 636 SEO
Abstract
In the 50's, Grothendieck defined $K_0$ of a scheme in order to build a generalization of Riemann-Roch. We can examine the quality of information $K_0$ holds by using its universal property: all additive functions on your exact category (with codomain an abelian group) factor through it. We will describe higher K-theories (probably spending some time on $K_1$) and end with some modern perspectives.
Feb. 13, 2019
Sam Dodds :
5 p.m. in 636 SEO
Abstract
In this talk we will discuss some basic notions in the fields of ergodic theory and geometric group theory. By way of providing some organizing priniciple, we will introduce these notions as they appear in a very rough outline of Mostow's rigidity theorem
Feb. 20, 2019
Christopher Perez :
5 p.m. in 636 SEO
Abstract
Infinite groups can be studied by looking at how they split, i.e. how they can be expressed as amalgamated free products and HNN extensions, and more generally as graphs of groups. Splittings arise in a very geometric way when considering fundamental groups of spaces, and they can be found by studying how groups act on trees. In this talk we will give an overview of Bass-Serre theory and briefly discuss some applications in group theory, geometry and topology, and logic.
Feb. 27, 2019
Keaton Quinn :
5 p.m. in 636 SEO
Abstract
We introduce some basic objects that let us discuss the geometry of smooth manifolds. We show ways the geometry and topology of the manifolds are tied together and investigate some related inverse problems that lead to the field of geometric analysis.
March 6, 2019
Sayan Mukherjee :
5 p.m. in 636 SEO
Abstract
Combinatorics is a diverse field that takes inputs from several other areas of math. In this talk, I plan on presenting three different techniques that are borrowed from other areas and that have been used to produce surprising advances in combinatorics. The tentative topics that I wish to cover are:
1) Polynomials on the finite field Kakeya needle,
2) Probability on diagonal Ramsey numbers,
3) Semidefinite programming on Erdos-Ko-Rado theorem.
If time doesn't permit, we'll skip one or more of the topics.
March 13, 2019
Tom Dean :
5 p.m. in 636 SEO
Abstract
After proving that the real numbers are uncountable, one is naturally led to the Continuum Hypothesis (denoted CH), proposed originally by Georg Cantor. CH asserts that every infinite subset of the reals is either countable or has the same cardinality as the reals. Cantor spent much of his life trying to prove CH true, but to no avail. In fact, resolving CH became the first of Hilbert’s 23 problems, proposed in 1900.
Around 40 years later, progress was made towards a solution when Gödel showed that the CH could not be proven false. However, in 1963, Paul Cohen shocked the world and proved that CH could not proven true either. In doing so, Cohen developed an incredibly powerful mathematical technique called forcing, still widely used today in modern set theoretic research.
In this talk, we discuss the independence phenomena, discuss the background of forcing, and sketch how Cohen’s forcing was used to prove the independence of CH.
March 20, 2019
Fumiaki Suzuki :
5 p.m. in 636 SEO
Abstract
Poncelet’s theorem is a beautiful result of the classical projective geometry: for conics C and D, if there is an n-gon inscribed in C and circumscribed around D, then there is such an n-gon having a given point of C as a vertex. An elliptic curve, on the other hand, is a one-dimensional Abelian variety, which is basic but important in algebraic geometry. The aim of this talk is to see how Poncelet’s theorem follows from the group law of an elliptic curve.
April 3, 2019
Aida Alibek :
5 p.m. in 636 SEO
Abstract
This will be a very gentle introduction into what Mathematics Education Research is, how one conducts such research, why mathematicians should care about it and how they could benefit from it. We will look into an example of a very big study on College Calculus to explore this topic more in-depth.
April 10, 2019
Dani Tucker :
5 p.m. in 636 SEO
Abstract
Every statistician has heard of R.A. Fisher. Modern experimental design was born with his 1935 publication, The Design of Experiments. The goal of this talk will be to first introduce design fundamentals, as developed by Fisher, as well as a general model framework, the Gauss-Markov Theorem for finding the Best Linear Unbiased Estimator (BLUE), and properties of this estimator. Then we will move toward discussing what much of current experimental design research involves, that is how to choose an “optimal” design. Finally, we will explore several commonly used criteria which “minimize” the variance-covariance matrix of the BLUE.
April 17, 2019
Nick Reynolds :
5 p.m. in 636 SEO
Abstract
We will discuss various models for describing the phenomena known as collective behavior. We start with a brief history of the area and how certain models were created. One can separate the models into two different camps. Those that emphasize long-range interactions, and those that emphasize short-range interactions. We will discuss the benefits and drawbacks of both of these types of models. With the second half of the talk we derive a Topological Model for describing collective motion, and discuss results associated with this model.
April 24, 2019
Eun Hye Lee :
5 p.m. in 636 SEO
Abstract
In this talk, I will give a brief introduction on analytic number theory starting from the prime number theorem and how it developed till now for the first half, and I will talk a little bit about my thesis research for the second half.
May 1, 2019
Matthew DeVilbiss :
5 p.m. in 636 SEO
Abstract
TBA
Feb. 5, 2020
Jacob Mayle :
5 p.m. in 636 SEO
Feb. 12, 2020
Yanlong Hao :
5 p.m. in 636 SEO
Abstract
In this talk, we will give some basic invariants of group actions on the unit circle, and some obstruction theories related to this.
Feb. 19, 2020
Noah Schoem :
5 p.m. in 636 SEO
Abstract
In 1963, Paul Cohen introduced the method of forcing, definitively proving that the longstanding open question about the size of the real numbers is undecidable.
Although first formulated by Cantor and later popularized by Hilbert, neither would have anticipated the result, or the method used to obtain it.
We will exposit a variation on Cohen's method to usher new real numbers into existence and change the number of real numbers. No set theory background will be assumed.
Feb. 26, 2020
Nick Reynolds :
5 p.m. in 636 SEO
March 4, 2020
Sam Dodds :
5 p.m. in 636 SEO
March 11, 2020
Greg Taylor :
5 p.m. in 636 SEO
March 18, 2020
Nathan Lopez :
5 p.m. in 636 SEO
Jan. 10, 2022
Ben Tighe :
5 p.m. in Zoom
Abstract
"All...regular points are locally the same; every singular point is singular in its own way." -Mark Spivakovsky. We will attempt to motivate this philosophy.
Jan. 24, 2022
Jacob Mayle :
5 p.m. in Zoom
Abstract
Famous for their role in the proof of Fermat’s last theorem, Galois representations are a powerful tool for studying elliptic curves. In this talk, we give an introduction to the topic. Further, we'll discuss recent research on computational aspects and an application to local-global phenomena for elliptic curves.
Jan. 31, 2022
Ben Gould :
5 p.m. in 636 SEO
Abstract
Last GSC I gave a talk about triangles. This time, circles. What could go wrong? PS: There might be algebraic geometry.
Feb. 7, 2022
Dani Tucker :
5 p.m. in 636 SEO
Abstract
Global Fr\'echet regression is an extension of linear regression to cover more general types of responses, such as distributions, networks and manifolds, which are becoming more prevalent. In such models, predictors are Euclidean while responses are metric space valued. Predictor selection is of major relevance for regression modeling in the presence of multiple predictors but has not yet been addressed for Fr\'echet regression. Due to the metric space valued nature of the responses, Fr\'echet regression models do not feature model parameters, and this lack of parameters makes it a major challenge to extend existing variable selection methods for linear regression to global Fr\'echet regression. In this talk, I will share my work, in collaboration with Yichao Wu (UIC) and Hans-Georg Mueller (UC Davis), which addresses this challenge and proposes a novel variable selection method with good practical performance. We provide theoretical support and demonstrate that the proposed variable selection method achieves selection consistency. We also explore the finite sample performance of the proposed method with numerical examples and real data illustrations. If time permits, I will also briefly share my more recent research that developed as a result of this paper, research which seeks to further generalize global Fr\'echet regression and allow for not only the responses to come from a general metric space, but also the predictors.
Feb. 14, 2022
Matthew Kehoe :
5 p.m. in 636 SEO
Abstract
The scattering of electromagnetic radiation by a layered periodic diffraction grating is an important model in engineering and the sciences. The numerical simulation of this experiment has been widely explored in the literature and we advocate for a novel interfacial method which is perturbative in nature. More specifically, we extend a recently developed High--Order Perturbation of Surfaces/Asymptotic Waveform Evaluation (HOPS/AWE) algorithm to utilize a stabilized numerical scheme which also suggests a rigorous convergence result. An implementation of this algorithm is described, validated, and utilized in a sequence of challenging and physically relevant numerical experiments. This is joint work with David Nicholls.
Feb. 21, 2022
Xizhi Liu :
5 p.m. in 636 SEO
Abstract
This talk is a gentle introduction to extremal problems in Hypergraph theory including the Hypergraph Turan problem and its difference from the Graph Turan problem, Extremal set theory, and Extremal graph theory (graphs are special hypergraphs!). Everything will be finite in this talk!
Feb. 28, 2022
Shravan Patankar :
5 p.m. in 636 SEO
Abstract
I will talk about my recent work while touching upon several important techniques and objects of study in commutative algebra and singularities in algebraic geometry (not to be confused with geometric algebra).
March 7, 2022
Kejia Zhu :
5 p.m. in 636 SEO
Abstract
When the river of the morphisms between complex smooth varieties flows in you, you will see Kodaira and Atiyah's smile (Kodaira fibration). The white-haired shadow of the mapping class group is roaming like a dream, which brings you the aroma of the monodromy representation. "The woods decay, the woods decay, and fall", the curvature withers slowly in thine arms, Alexandrov and Gromov weep their burthen to the ground. "And after many a summer dies the swan." Non-positively curved space is only cruel in immortality. In the end, even you are still too mean to show love and sympathy to the hyperbolic groups, you will still hear "Va, je ne te hais point~" from the acylindrically hyperbolic groups.
March 14, 2022
Sam Dodds :
5 p.m. in 636 SEO
Abstract
Given an orientable surface S with genus at least 2, it is always possible to fix a metric on S which is locally isometric to the hyperbolic plane. Such a metric space is called a hyperbolic surface. Take a geodesic circle which separates S into two pieces, and spin one of the pieces. This results in a continuous family of hyperbolic surfaces that are homeomorphic but not isometric. One of the landmark geometric theorems of the 20th century was Mostow’s Rigidity Theorem: that for higher dimensions this is impossible; if two compact hyperbolic manifolds of dimension greater than 2 are homeomorphic, then they are isometric. It turns out that the flexibility and rigidity of these spaces really originates with their fundamental groups. The fundamental groups of surfaces are in some sense flexible groups, but in higher dimensions fundamental groups of hyperbolic manifolds are rigid. More precisely, they have flexible/rigid group actions. We will survey various rigidity theorems for some groups, outline some proofs (including a proof of Mostow’s theorem), and talk about flexibility results for other groups, (including some constructions).
March 28, 2022
Matthew DeVilbiss :
5 p.m. in 636 SEO
Abstract
I will be giving a proof that 2 is my favorite number using facts and logic.
And Vaught's Theorem.
April 4, 2022
Adam Pratt :
5 p.m. in 636 SEO
Abstract
I will discuss $G$-CW complexes and their $RO$-graded, Mackey functor-valued cohomology. I'll cover some examples of constructions and computations along with an overview of some of the major results in equivariant homotopy theory.
April 11, 2022
Jake Maranzatto :
5 p.m. in 636 SEO
Abstract
In this talk I'll discuss the problem of learning evolving concepts over a combinatorial structure. First I give a framework that captures previous models considered by Emamjomeh-Zadeh et al. (2020). I'll briefly discuss how to use this to close upper and lower query complexity bounds, and finally discuss a simple efficient algorithm for learning concepts on low-diameter graphs and analyze it using markov chains.
April 18, 2022
Evelyn Richman :
5 p.m. in 636 SEO
Abstract
Classically, charged quantum particles in so-called axisymmetric magnetic fields display interesting, helical trajectories. Within these trajectories are drifting circular orbits whose radii shrink as the magnetic field strength increases. However, since quantum particles are small (very very small), their dynamics are governed by partial differential equations called Schrodinger equations. Thus, the behavior of such particles can be very difficult to predict, and it is often impossible to find exact solutions to these systems. In this talk, I will show that in the strong magnetic field limit, we can in fact derive an effective solution to the Schrodinger equation with axisymmetric fields. Furthermore, the effective solution will display the same behavior that we expect from classical dynamics, thus demonstrating that, sometimes, quantum physics isn't all that weird.
Feb. 22, 2023
Anish Chedalavada :
4 p.m. in 612 SEO
Abstract
This talk will assume some background in algebra; background in algebraic geometry is a plus, but not strictly necessary. We'll go through a quick tour of Morita theory and Tannakian Formalism, and discuss how to reconstruct geometric objects from their categories of sheaves. If time permits, I'll discuss tensor-triangular geometry, which is a way to apply these techniques to derived settings; notably modular representation theory and stable homotopy theory.
Oct. 3, 2025
MSCS Grad Intern Alums :
3 p.m. in 636 SEO
Abstract
Five Internship Alums will discuss their internship experiences with other MSCS Grads. Please bring your questions!