Mathematics & Statistics Department

Student Colloquia

Most Tuesday afternoons during the academic year, the Mathematics students host a math talk. The talks are directed to our mathematics majors but are usually accessible on a variety of levels. Refreshments are served before the talks.

2015-16 Schedule

Fall

4:40 in Physics 123 (unless marked otherwise).

Sept 8Graduate School Q&A
Andrew Bray, Adam Groce, Kyle Ormsby
Time: 4:10 PM
Sept 15Cantor's Theorem, or The Joy of Sets
Riley Thornton '16
Time: 4:10 PM

In this talk, I will introduce some of the basics of set theory: the general notion of a set, the axioms of Zermelo-Fraenkel Set Theory, powersets, products of sets, functions as sets, and the idea of cardinality.

After setting up the basic machinery I will prove Cantor's beautiful theorem on the cardinality of the powerset of a given set, and give a couple applications. At the end of the talk, I'll try to point to some more recent areas of activity in set theory and mathematical logic.

This talk should be accessible to everyone.

Sept 22Verifiably Secure ORAM
Joshua Gancher '16 and Alex Ledger '16
Time: 4:10 PM
Oblivious RAM (ORAM) is a cryptographic technique for reading and writing data on a database without letting the server know the pattern in which you're accessing files. We extended this idea by adding the ability to prove to a third party that the data has not been tampered with. In the first half of the talk, we give a summary of ORAM, and in the second half of the talk, we discuss our extension.
Sept 29Necessary and Sufficient Conditions for the Injectivity of the Dress Map
Ricardo Rojas-Echenique '17
Time: 4:10 PM

The Dress map D_{K/F} : B(G)--->GW(F) maps the Burnside ring of a finite Galois group G = Gal(K/F) to the Grothendieck-Witt Ring of the base field F, taking G/H to tr_{K^H/F} <1>_{K^H}. Using two theorems from Epkenhans, we find that if D_{K/F} is injective then G must be an abelian 2-group of order less than or equal to 4. Some results about the trace forms associated with these groups are then used to show that G\cong \Z/2\Z is a necessary condition for injectivity. Finally, we prove that for a non-trivial Galois extension,

DK/F  is injective  iff K = F(\sqrt{a}})

where a \in F\mbox is not a sum of squares in F.

Oct 6The Homogenous Spectrum of Milnor-Witt K-theory
Riley Thornton '16
Time: 4:10 PM

Milnor-Kitt K-theory is a relatively recently studied ring associated to a field, which relates to the theory of quadratic forms and certain algebraic homotopy theories over the field.

This talk will introduce Milnor-Witt K-theory of a field and explain new work describing how the structure of the space of orderings on the field controls the spectrum of homogenous prime ideals of Milnor-Witt K-theory.

All are encouraged to come, but I will assume some familiarity with abstract algebra.

Oct 13Symbol Length in Milnor and Milnor-Witt K-Theory
Nico Terry '17
Time: 4:10 PM
I will define the symbol length in Milnor and Milnor-Witt K-Theory and present a few results that have already been established on the symbol length of fields modulo a prime p. This talk is half length, so it will assume that you've attended Riley's or Rico's student colloquium talks or the student presentation day in the general colloquium.
Oct 27The Cantor-Schroeder-Bernstein Theorem
Justin Chun '19
Time: 4:10 PM

In this talk I will be giving an overview of the Cantor-Schroeder-Bernstein theorem. After covering some historical background, we shall explore various lines of attack on how to prove CSB, including but not limited to well ordering, the Knaster-Tarski fixed point theorem, and bigraphs. Familiarity with sets and functions on the level of Math 112 is assumed.

Nov 10Persistent Homology and the Gray Scott Model
Joseph Joe '16
Time: 4:10 PM

The Gray Scott model is a simple reaction-di usion system that displays a variety of patterns. In this talk, I will demonstrate how topology can be used to quantitatively distinguish these patterns. Specifically, I will discuss the theory behind homology groups, topological invariants that are easy to compute. Furthermore, I will discuss persistence homology, an extension of homology that is useful in distinguishing "signals" from the "noise" in data sets that approximate a topological space.

Nov 17Descriptive Set Theory
Riley Thornton '16
Time: 4:10 PM

Spring

4:40 in Physics 123 (unless marked otherwise).

Feb 23Borel Complexity
Riley Thornton '16
Time: 4:10 PM
Borel complexity is a continuous analog of the discrete computational complexity theory. In this talk I will motivate the definition of Borel complexity and discuss its relation descriptive set theoretical notion of definability. Time permitting, I will discuss standard benchmarks of the theory and touch on some applications.
Mar 1Fully Homormophic Encryption
Joshua Gancher '16
Time: 4:10 PM

Fully homomorphic encryption (FHE) enables one to carry out computations using ciphertexts, rather than plaintexts. Using this technology, I may offload an arbitrary computation to a server without disclosing the contents of my data. The server would be able to respond to my request with a ciphertext containing the desired result. For example, FHE can be used to conduct encrypted search queries: I encrypt my search query, send the resultant ciphertext to Google, and Google responds with an encryption of my desired search results -- all without Google learning what I had searched.

In this talk, after giving the relevant definitions for public-key encryption and fully homomorphic encryption, I will discuss the construction of a particular FHE scheme introduced by van Dijk et al. in 2010. 

No background is needed for this talk, except for basic familiarity with modular arithmetic.

Apr 5Tree Sampling
Philip Stallworth '16
Time: 4:10 PM

Foresters need to account for the locations of trees within forest plots. They typically perform a census to determine these locations, a process which can be expensive and difficult over very large plots with very small plants. Consequently, many alternative sampling schemes have been proposed to reduce the expensiveness of this procedure. The most simple scheme, k-tree sampling, is cheap, easy to understand, and easy to implement; however, it provides limited information on inter-point interaction making it difficult to perform more valuable inference. In this talk, I explore one method of gaining locational information using k-tree sampling, explore this method's potential faults, and discuss potential expansions. In particular, I will discuss how this method might be used to determine clustering effects that do not depend on underlying environmental heterogeneity. I will focus my attention on results from simulation studies and theory, rather than from applied results derived from forestry. 

The bulk of this talk will be accessible to people at all levels, but some computational and statistical background (equivalent to math 141 and math 121) will be assumed.

Apr 12Improving Secure Computation by Chaining Garbled Circuits
Alex Ledger '16
Time: 4:10 PM

Secure computation is a cryptographic method to securely compute a function between two parties while preserving the privacy of the parties’ inputs. Modern secure computation techniques use an offline/online scheme, where parties pre-process and exchange information in an offline phase prior to computing the function in an online phase. The offline/online setting has many advantages, namely that the online computation is very fast, but the setting is limiting, as the function that is being computed must be determined ahead of time. A new technique, called chaining garbled circuits, provides flexibility to the offline/setting setting and increases the speed of computation of some functions. In this talk, I will introduce secure computation and garbled circuits, explain how to chain garbled circuits and discuss my implementation of chaining. 

Apr 19Partially Ordered Sets and Lower Bounds on Sorting Algorithms
Sarah Brauner '16
Time: 4:10 PM

There is a classical, information-theoretic lower bound on the number of comparisons needed to sort lists. 

However, when it comes to determining the circumstances in which this bound will be tight, we know surprisingly little. A set of size 21, for instance, can always be sorted in exactly the number of comparisons stipulated by the bound, while a set of size 12 cannot. 

It turns out that we get a much better understanding of why this is if we look at sorting algorithms through the lens of partially ordered sets (posets). In this talk, I will present sorting algorithms through this lens, and develop a framework to understand how the information theoretic lower bound interacts with the structure of posets. Time permitting, we will delve into the question of why 12 fails to tightly meet the bound. 

Though this material draws upon some topics covered in abstract algebra, the talk is designed to be accessible to everyone, regardless of mathematical background. 

Apr 26The HHL Algorithm
Alex Salem '16
Time: 4:10 PM

Quantum computing is an exciting field for many reasons, one of which is that quantum computers could theoretically solve some problems exponentially faster than any classical computer. 

In this talk, I will introduce the key tools and concepts of quantum computing. This will include qubits and general quantum states, measurement, the quantum circuit model of computation, the quantum Fourier transform, and the phase estimation subroutine. Time permitting, we will look at Harrow, Hassidim, and Lloyd’s algorithm for solving systems of linear equations, which is one algorithm with exponential speed-up over known classical algorithms.

This talk presumes familiarity with linear algebra and some theory of classical computing, but I will try to make it as accessible as possible.