Mathematics & Computer Science

Seminars and Colloquia

Colloquium

Wednesday, February 06, 2013

04:15 pm - 06:00 pm

Math Colloquium: Clifton Ealy, Western Illinois University "Independence Relations in Non-stable Theories"

Abstract: At the heart of much of the last several decades of model theory is the study of stable structures. One way (of many) in which such structures can be defined is through the existence of an independence relation with certain properties. A prototypical example of a stable structure is an algebraically closed field where the independence relation is given by algebraic independence. Such independence relations exist, albeit in weaker forms, outside stable structures as well. I will provide an introduction to the study of these independence relations, and talk about a few recent applications.

ESC 638

Thursday, November 29, 2012

04:15 pm - 06:00 pm

Math Colloquium, Peter Winkler - Dartmouth College: "Probability and Intuition"

PLEASE NOTE ROOM CHANGE TO EXLEY 121<br/>Abstract: Probability theory was devised in order to understand gambling, but now is the underpinning of statistics, without which we would be clueless in our complex society. Yet probability itself is a mysterious quantity, hard to define, and awkward for our human intuition to cope with. Does it even exist, except in our minds?<br/> Through stories and puzzles, we will attempt to get a slightly better grip on probability and to identify some of the ways in which our intuition tends to lead us astray. <br/>

ESC 638

Thursday, November 15, 2012

04:15 pm - 06:00 pm

Math Colloquium, Keith Conrad - UCONN: "What is the abc conjecture? "

Abstract: The abc conjecture, first posed in the 1980s, is one of the central problems in number theory. Its statement is completely elementary, but it is closely related to numerous conjectures and deep theorems that were already known. In September, Shinichi Mochizuki announced a proof of the abc conjecture, so this seems like a good time to explain to a broader audience than number theorists what the abc conjecture says, why it is plausible, and what consequences it would have if it were proved.<br/> <br/>

ESC 638

Thursday, November 08, 2012

04:15 pm - 06:00 pm

Math Colloquium, Michel Dekking - Delft: "A simple stochastic model for kinetic transport"

Abstract: We introduce a discrete time microscopic single particle model for kinetic transport. The kinetics is modeled by a two-state Markov chain, the transport by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to zero linearly, we obtain a random variable S(t), which is closely connected to a well known (deterministic) PDE reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed part of the solute at time t do exist, and satisfy the partial differential equations.

ESC 638

Thursday, November 01, 2012

04:15 pm - 06:00 pm

Math Colloquium, James Stankewicz - Wesleyan: "Modular Curves and the Birch and Swinnerton-Dyer Conjecture

Abstract: The conjecture of Birch and Swinnerton-Dyer is one of the most important open questions in mathematics today. This talk in three parts will expose the conjecture and some of its trickiest parts from the perspective of modular curves.<br/><br/>1.Tricks and treats in divergent series: In real analysis, we see many series which diverge. There are however a few tricks to finding prospective limits of even divergent series. We explore how to make sense of this and are eventually led to the Hasse Principle.<br/>2.BSD and the Hasse Principle: We introduce elliptic curves and we see how the failure of the Hasse principle stands in the way of the Birch and Swinnerton-Dyer conjecture.<br/>3.Modular Curves: We introduce modular curves and look at their connection to elliptic curves. As a treat, we will finish with some of my own work on modular curves.<br/>

ESC 638

Thursday, September 13, 2012

04:15 pm - 06:00 pm

Math Colloquium, Jayadev Athreya, University of Illinois, Urbana: "Farey Fractions, Gap Distributions, and Homogeneous Dynamics"

Abstract: We'll discuss a way of using tools from hyperbolic geometry and ergodic theory to study problems about gaps between Farey Fractions, and how this can be generalized in many different directions. This talk will be at an elementary level, and the only thing you really need to have is some familiarity with basic linear algebra.

ESC 638

Thursday, April 19, 2012

04:15 pm - 06:00 pm

Mathematics Colloquium, Aaron Levin, Michigan State: Integral points on varieties

Abstract: It is a classical problem, going back to antiquity, to determine the integer solutions to a system of polynomial equations. We will discuss the modern formulation of this problem and give a brief overview of some of the main results and conjectures. I will then concentrate on various recent results and approaches to this problem, focusing on effective aspects and higher-dimensional problems.

ESC 638

Thursday, April 12, 2012

04:15 pm - 06:00 pm

Mathematics Colloquium, Julianna Tymoczko, Smith College: Computing cohomology using combinatorial graphs

A cohomology ring is a ring associated to a topological space that encodes geometric properties of the space---for instance, the dimension, number of puncture holes, number of connected components, and and other invariants of the space. Cohomology is an incredibly useful tool, but also frequently difficult to compute. We will describe a combinatorial method to compute cohomology rings for a family of topological spaces, one that removes many of the computational barriers to hands-on calculations. Surprisingly, we'll do this by computing apparently more complicated rings called equivariant cohomology rings. The construction we describe is often known as GKM theory; the underlying combinatorial objects also generalize splines, piecewise polynomial functions that are important in geometric combinatorics and in applications. Throughout the talk, we'll describe a variety of open questions.

ESC 638

Thursday, April 05, 2012

04:15 pm - 06:00 pm

Math Colloquium, Jennifer Taback, Bowdoin: Groups, Graphs and Trees

Abstract: One goal of geometric group theory is to uncover relationships between the algebraic structure of a group and its "geometry". Given a group G with finite generating set S, one forms a canonical picture of the group called a Cayley graph which can be used to describe the geometry of the group. This process is not difficult, and a natural notion of equivalence exists on these graphs as the generating set varies. I will discuss the reverse implication, whether every "sufficiently nice" graph can be the Cayley graph of a finitely generated group. The answer will lead to an interesting class of groups which provide a geometric generalization to the lamplighter groups $\Z_n \wr \Z$.

ESC 638

Thursday, February 23, 2012

04:15 pm - 06:00 pm

(Mathematics Colloquium) Tom Tucker, University of Rochester: "P-adic parametrization of orbits and the dynamical Mordell-Lang conjecture"

Abstract: Let f : X → X be a morphism of varieties over a field of characteristic 0 and let x be a point on X. In many cases, one can show the orbit of x under f can be "p-adically parametrized"; that is, one can find p-adic analytic map g from a disc in ℂp to X such that g(n)=f n(x) for all n. The existence of such a parametrization allows one to solve the so-called "dynamical Mordell-Lang problem" for f, which states that, given a subvariety W of X, the set of n such that f n(x) is in W forms a finite union of arithmetic sequences. It also allows for the solution of various weak forms of a conjecture of Zhang on the existence of points with Zariski dense orbits.

ESC 638

Thursday, February 09, 2012

04:15 pm - 06:00 pm

Mathematics Colloquium, Ken Ono, Emory University: The legacy of Ramanujan's mock theta functions: Harmonic Maass forms in number theory

Abstract: In his last letter to G. H. Hardy (written from his death bed in 1920), Ramanujan wrote about a new beautiful theory of power series that he refers to as "mock theta functions". This small collection of enigmatic power series puzzled mathematicians for many decades. Then in 2002 Zwegers realized the meaning behind these series; he established that these series are pieces of Maass forms. This understanding has inspired much work. The development of general theorems based on Ramanujan's examples has produced many wonderful theorems on a wide array of subjects such as: L-functions and elliptic curves, Additive number theory (partitions), Donaldson invariants, Representation theory, Explicit class field theory. This talk will be a brief account of this story.

ESC 638

Thursday, December 08, 2011

04:15 pm - 06:00 pm

Raman Parimala, Emory University: Quadratic forms over function fields (Mathematics Colloquium)

Abstract: Abstract: It is a classical theorem of Hasse-Minkowski that every quadratic form in at least five variables over a totally imaginary number field has a nontrivial zero. However it is wide open whether a quadratic form in sufficiently large number of variables over the rational function field in one variable over a totally imaginary number field has a nontrivial zero. We shall review some recent results concerning nontrivial zeros of quadratic forms over function fields.

ESC 638

Thursday, October 27, 2011

04:15 pm - 06:00 pm

Math Colloquium: Overview of the local-global principle in number theory

Speaker: Peter Clark, University of Georgia<br/><br/>Abstract: Here is a truly classic problem in number theory: given a set of polynomial equations with integer coefficients, determine whether or not there is a simultaneous integer solution. This is a difficult problem: in fact, as stated above it is famously algorithmically impossible, although it is known to be possible in many special cases and conjectured to be so in others. In contrast, it is algorithmically possible to determine whether there are solutions over the real numbers and whether there are congruential solutions modulo N for all integers N. The hope that these necessary conditions will be sufficient is called the local-global principle. Although there are some classical successes of the local-global principle -- notably the case of one homogeneous quadratic equation -- most contemporary work has focused on settings in which this principle is not generally valid. In fact the conventional wisdom is that when restricted to, for instance, curves of positive genus, the local-global principle is "most often false". In this talk I will survey attempts to make this precise, discussing theorems, conjectures and wild speculations.

ESC 638

Thursday, October 13, 2011

04:15 pm - 06:00 pm

Mathematics Colloquium: Elliptic curves and Hilbert's Tenth Problem

Speaker: Kirsten Eisentraeger<br/><br/>Abstract: In 1900 Hilbert presented his now famous list of 23 open problems. The tenth problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. Hilbert's Tenth Problem remained open until 1970 when Matijasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by asking the same question for polynomial equations with coefficients and solutions in other commutative rings. In this talk we will discuss how elliptic curves can be used to prove the undecidability of Hilbert's Tenth Problem for various rings and fields.

ESC 638

Thursday, April 21, 2011

04:15 pm - 06:00 pm

Percolation, partitions, and probability

Speaker: Karl Mahlburg<br/><br/>Abstract: I will discuss the surprising connections between finite-size scaling in families of bootstrap percolation models, the combinatorics of integer partitions, and limiting entropy rates for<br/>(Markov-type) stochastic processes. A combinatorial characterization of the percolation processes relates their metastability threshold exponents to the cuspidal asymptotics of generating functions for partitions with restricted sequence conditions. These generating functions are hypergeometric q-series that are of number-theoretic interest, as in many cases they are equal to the product of modular forms and Ramanujan's famous mock theta functions. In other cases, both the percolation processes and partition functions are best understood through entropy rate bounds for probabilistic sequences with gap conditions. These sequences can be understood as stochastic processes with varying transition probabilities, and techniques from the theory of linear operators are used to bound the dominant eigenvalues.<br/>

ESC 638

Thursday, April 14, 2011

04:15 pm - 06:00 pm

Characterizing analyticity by means of Nonlinear Fourier transforms and Applications

Speaker: Shif Barhanu, Temple University<br/><br/>Abstract: In this talk we will discuss a nonlinear Fourier transform which has been used<br/>extensively over the past thirty years to determine the real analyticity of functions<br/>that are solutions of partial differential equations. We will also present some new<br/>classes of nonlinear Fourier transforms obtained in recent joint work with Jorge<br/>Hounie.<br/><br/><br/>

ESC 638

Thursday, April 07, 2011

04:15 pm - 06:00 pm

The Arithmeitc of Quadratic Forms

Speaker: John Hanke, University of Georgia<br/><br/>Abstract: Quadratic forms with integer coefficients are among the oldest objects in mathematics and lie at the crossroads between geometry and arithmetic. They have been studied by mathematicians for thousands of years, but starting with the work of Gauss and Legendre in the 1700s, new perspectives and techniques have been rapidly increasing our understanding of the answers to classical questions like "What numbers can be written as a sum of n squares?". We now know that there are deep connections between these questions and many other interesting objects in mathematics (e.g. modular forms, elliptic curves, abelian varieties, certain zeros of twists of the Riemann zeta function). This talk will describe some of these ideas and techniques, explain what questions they can be used to solve, and list some interesting problems that are still unresolved.

ESC 638

Thursday, March 24, 2011

04:15 pm - 06:00 pm

Liftings of Representations of Finite Groups of Lie Type

Speaker: Joshua Lansky, American University<br/><br/>Abstract: We give an introduction to the theory of reductive groups over finite fields and their representations. The aim of the talk is to present a construction for a lifting of classes of representations in a fairly general setting.<br/><br/>

ESC 638

Thursday, March 03, 2011

04:15 pm - 06:00 pm

Secant Varieties of Smooth Toric Varieties

Speaker: Jessica Sidman, Mount Holyoke College<br/><br/>Abstract: Algebraic geometers have long been interested in secant varieties. The results that I will describe were motivated by work of Sturmfels and others in algebraic statistics. I will discuss joint work with David Cox which gives the dimension and degree of the secant variety of a smooth toric variety in terms of the corresponding polytope under suitably nice conditions.

ESC 638

Thursday, February 10, 2011

04:15 pm - 06:00 pm

Counting Curves Using (Orbifold) Quantum Cohomology

Speaker: Linda Chen, Swarthmore<br/><br/>Abstract: Mathematicians have long been interested in counting the number of geometric objects which satisfy various conditions. A breakthrough occurred in the 1990's, inspired by ideas in physics, leading to surprising and beautiful recursions, for example for the number of degree d rational plane curves passing through 3d-1 general points. I will give a basic introduction to these tools of Gromov-Witten theory and quantum cohomology, discuss developments such as orbifold versions of the theories, and describe further applications to classical problems in enumerative geometry. <br/><br/>The talk will be accessible to graduate students and a general mathematical audience.<br/>

ESC 638

Thursday, December 02, 2010

04:15 pm - 06:00 pm

Cosmetic Surgeries and Heegaard Floer Homology

Speaker: Stanislav Jabuka, University of Nevada, Reno<br/><br/>Abstract: For about 5 decades, it has been known that every 3-dimensional manifold can be obtained by Dehn surgery on a "framed link". Dehn surgery is an operation for building 3-dimensional manifolds that starts with a collection of n disjoint circles (called a "link" if n>1, a "knot" if<br/>n=1) in the 3-sphere, "drills" out tubular neighborhoods of the circles, and re-inserts them in a novel way as specified by the framing. The framing itself is the choice of a rational number, one for each of the components of the link/knot. After explaining these notions in detail, and presenting several examples, we will focus on those 3-dimensional manifolds obtained by surgery on a knot. In this context one can ask both about the uniqueness of the knot as well as the uniqueness of the framing that yields the same 3-dimensional manifold. The "Cosmetic Surgery Conjecture" asserts that no two distinct Dehn surgeries on the same nontrivial knot can ever yield the same 3-dimensional manifold.<br/><br/>While this 30 year old conjecture (whose dynamic name is due to Steve<br/>Bleiler) is still largely open, there has been some recent progress driven by Heegaard Floer theory. The second half of the talk will focus on describing the Heegaard Floer techniques used and detailing the results thus obtained.<br/><br/>Most of the talk will be accessible to a general mathematical audience.<br/><br/>

ESC 638

Thursday, November 18, 2010

04:15 pm - 06:00 pm

TBA

Speaker: Josh Lansky, American University<br/><br/>Abstract: TBA

ESC 638

Thursday, October 28, 2010

04:15 pm - 06:00 pm

A Survey of Modular Curves, and Applications To Torsion Points on Elliptic Curves.

Speaker: Alvaro Lozano-Robledo, UCONN<br/><br/>Abstract: A modular curve X(G) is a Riemann surface, or the corresponding<br/>algebraic curve, constructed as a compactification of the quotient of<br/>the complex upper half-plane by the action of a congruence subgroup G<br/>of the modular group SL(2,Z). This talk will be an introduction to<br/>modular curves, emphasizing the moduli interpretation that relates<br/>points on these curves with elliptic curves that satisfy certain<br/>properties. We will discuss the complete classification of<br/>(non-cuspidal) rational points on the modular curves X_0(N), and apply<br/>this classification to the study of torsion subgroups of elliptic<br/>curves.<br/>

ESC 638

Thursday, October 14, 2010

04:15 pm - 06:00 pm

Annihilators of Elliptical Curves

Speaker: Tom Weston, UMass Amherst<br/><br/>Abstract: TBA

ESC 638

Thursday, October 07, 2010

04:15 pm - 06:00 pm

Kac-Wakimoto Characters and Mock Theta Functions

Speaker: Amanda Folsom, Yale University<br/><br/>Abstract: In this talk I will discuss the role of certain "strange" functions<br/>called "mock theta functions" as a liaison between two different areas of<br/>mathematics: modular forms in number theory, and the representation theory of a<br/>large class of Lie superalgebras. Despite their "strange" appearance, the mock<br/>theta functions in their most classical guises date back to the first part of<br/>the 20th century, however their roles in mathematics were not well understood.<br/>Only within the last 8 years have we finally begun to understand and develop a<br/>greater theory around the mock theta functions in mathematics - relating<br/>modular forms and representation theory is just one of their many interesting<br/>facets. This talk is intended to be an introduction to this theory.<br/>

ESC 638

Thursday, September 09, 2010

04:15 pm - 06:00 pm

The Hopf Algebra of Complex iterated Integrals

Speaker: Sheldon Joyner<br/><br/>Abstract: The space of Chen iterated integrals of holomorphic differentials on a given Riemann surface is known to form a Hopf algebra with the shuffle product. In this talk, I will show that one can interpolate the number of times which any given form is iterated in a Chen iterated integral, to certain complex parameters. The resulting space of homotopy functionals can then be endowed with the structure of Hopf algebra. Some applications, such as a direct (computational) proof of the monodromy of polylogarithms, will be mentioned.

ESC 638

Wednesday, April 07, 2010

04:15 pm - 06:00 pm

Topology Seminar Mark Radosevich, '03

Spin Fillings of Contact 3-Manifolds

ESC 618

Friday, April 02, 2010

04:15 pm - 06:00 pm

Math Colloquium-Prof Todd Quinto, Tufts

"Some thoughts on Limited Data Tomography"

ESC 628