This is the website for the Math Graduate Student Association and the Math Grad Student Seminar.
Math Department TShirts
The math department now has tshirts! The design can be viewed here. In addition to the grey shirt, there is a red shirt with white lettering. Shirts are $5 and can be purchased in MW (math tower) 216. Cash only, please, and exact change is appreciated but not usually necessary. Sizes range from small to XXL. The shirts are a 5050 cottonpolyester blend. The design on the back is the Zassenhaus butterfly.
Mathematical Research Lectures presented by MGSA (Grad Seminar)
All talks will be given in CH 240 at 5:15 pm unless otherwise noted. The talks will last between 40 minutes and an hour. All Ohio State undergraduates and graduate students are welcome to attend!
Fall 2017  Spring 2017  Fall 2016  Past Talks 

August 29: Evan Nash
The Convex Hull of Two Circles in $\mathbb{R}^3$ (click for abstract)
This is an algebraic geometry talk, but it is a somewhat unorthodox algebraic geometry talk in that not only do you recognize all the words in the title, but they mean exactly what you think they do. We're really going to be looking at two circles in 3space and trying to describe the properties of their convex hull. On the way, we'll get a glimpse at some of the algebraic geometry machinery we use to get a perspective on this question. There will be many pretty pictures.

September 26: Yongxiao Lin
Research area: Analytic Theory of $L$Functions (click for abstract)
Abstract forthcoming.

October 24: Marissa Renardy
Research area: Computational Biology (click for abstract)
Abstract forthcoming.

November 7: Alex Beckwith
Research area: Analytic Number Theory (click for abstract)
Abstract forthcoming.

November 28: Reeve Garrett
Research area: Commutative Algebra and Ideal Theory (click for abstract)
Abstract forthcoming.

January 17: Linh Huynh  Scott Lab E 105
Clustering Sleep Wake Transitions in Electromyography Data (click for abstract)
Contrary to the common perception that sleep is continuous, sleep is actually fragmented by brief awakenings throughout the night  even in healthy people. Experiments have shown that sleep interval lengths are exponentially distributed in both infants and adults, while the distribution of wake interval lengths changes from exponential in infants to power law in adults. To understand this phenomenon of wake intervals and fundamental mechanisms in sleepwake cycle dynamics, I analyze the transitions between sleep and brief awakening states. I apply machine learning methods on electromyography data to identify clusters of sleepwake transitions and compare results with activity of stochastic models. In this talk, I will discuss the background of my research as well as the methods that I use.

January 31: Mike Belfanti  Scott Lab E 105
Some Examples of Langlands' "principle of functoriality" coming from Galois Theory (click for abstract)
One aspect of classical Fourier analysis is the study of the space of squareintegrable functions on the compact quotient Z\R. Two of the main theorems are the decomposition of this Hilbert space into a sum of one dimensional subspaces and the Poisson summation formula, which relates the values of a function f on the lattice Z with the values of the Fourier transform $\hat{f}$ on Z. An important feature of this situation is that Z sits inside of R as a discrete subgroup. Let F be a finite extension of Q and let G=GL(n). There is a topological ring A_F, the adele ring of F, which contains F as a discrete subring. Then G(F) is a discrete subgroup of G(A_F) and we can consider the space of squareintegrable functions on the quotient G(F)\G(A_F). This function space carries an action of G(A_F) and one can ask how it decomposes into irreducible representations. The setup is similar to classical Fourier analysis, with F playing the role of Z and A_F playing the role of R. It turns out that if G is a classical group (SO(n), Sp(n), U(n)...), or more generally a reductive linear algebraic group, then we can ask the same question about the space of squareintegrable functions on G(F)\G(A_F). Namely, how do the squareintegrable functions decompose into irreducible representations of G(A_F)? The irreducible constituents of this representation are known as "automorphic representations" of G. Langlands' "principle of functoriality" is a broad collection of conjectures which predicts how automorphic representations for different groups are related. The goal of this talk is to explain some special cases of these conjectures which are motivated by Galois theory. Along the way, we will discuss some aspects of representation theory for GL(n), Galois representations and the local Langlands correspondence. With the remaining time, we will try to show how Arthur's trace formula, a generalization of the Poisson summation formula, can be used to prove some of Langlands' conjectures.

February 14: Jacob Miller  Scott Lab E 105
Introduction to Disentanglement Puzzles (click for abstract)
We will explore the notion of a topological puzzle and some necessary conditions for its disentanglement. In addition, we will explore two techniques used to encode the complexity of solving these puzzles.

February 28: Irfan Glogic  Scott Lab E 105
Singularity Formation in Nonlinear Partial Differential Equations (click for abstract)
Can water in motion spontaneously explode? How is a black hole formed? These physical questions are mathematically interpreted in terms of the singularity formation (also called ``blowup'') of the solution to the underlying evolution equation. The first part of the talk will serve as a beginners guide to the world of nonlinear partial differential equations and singularity formation in them. Later, I will demonstrate how the concept of blowup plays a central role in some of the most popular problems in modern mathematics (e.g. NavierStokes equations, Poincaré Conjecture, Einstein's equations). Finally, I will mention my own work on the topic in regards to the wave maps equation, a toy model for Einsteins equations of general relativity.

March 21: Willa Del Negro Skeehan  Scott Lab N 054
Staged Progression and Retrogression Model of Influenza (click for abstract)
Diseases and mathematics: two of people's least favorite things. In this talk, I will give an overview of the history of modeling infectious diseases, how to create a mathematical model for a disease, and discuss the typical mathematical analysis that can be performed. I will also discuss a new model that I have developed in conjunction with my advisor and our collaborators. This work includes creating the model, standard analysis, and numerical fitting to data provided by the HIVE study at the University of Michigan.

April 4: Sohail Farhangi  Scott Lab N 054
On Refinements of Van der Waerden's Theorem on Arithmetic Progressions (click for abstract)
In my talk I will discuss one possible refinement of Van der Waerden's Theorem which leads to what is known as the 2large is large conjecture in Ramsey theory. We will be examining the statements of this conjecture and its variants in higher dimensions, as well as discussing some of what is known and unknown. Lastly, we will close with some dynamical reformulations of these conjectures.

April 18: Michael Horst  Scott Lab N 054
The Best Category: An introduction to higher dimensional algebra (click for abstract)
We analyze the prototypical example motivating the study of higher categories and generalize to the definition of bicategories. To highlight the difference between these new objects and oldschool categories, we show how even basic notions cease to have a single, obvious definition. Examples will be given, fun will be had, perhaps the word "tricategory" will be whispered.

August 30: Florian Richter
On the structure and randomness of multiplicative functions (click for abstract)
We will talk about old and new results in the intersection of Multiplicative Number Theory and Additive Combinatorics. We will start with discussing different notions of structure and pseudorandomness for arithmetic functions. We then explore the beginnings of the so called "pretentious approach" to Number Theory and learn how to catalogue bounded multiplicative functions using classical theorems of Halasz, Daboussi and Delange. A main topic will also be the dichotomy between the "additive structure" and the "additive pseudorandomness" of multiplicative functions. This relates to Sarnak's Mobius randomness conjecture.

September 13: Keshav Aggarwal
Subconvexity bound problems (click for abstract)
Modern number theory centers around the study of $L$functions and their various arithmetic and analytic properties. One such analytic property is the rate of growth of an $L$function $L(f,s)$, described in terms of a power of the "conductor" $Q$. Fairly basic complex analysis shows $L(f,s)\ll Q^{1/4}$ (under correct normalization). This is called the convexity bound. The subconvexity bound problem aims to beat the bound to a smaller exponent. In this presentation, we talk about some methods of beating the bound for low degree $L$functions associated with holomorphic and Maass cusp forms. Towards the end, we'll try to describe some applications, like equidistribution results and Quantum chaos.

September 27: Neil DeBoer
Mathematics Without Excluded Middle (click for abstract)
The Law of Excluded Middle (LEM) is the most controversial rule in mathematics. I will explain the reasons for rejecting it. But what happens to mathematics if we disallow LEM? To understand this we will first need a new interpretation of logic, and this will lead to several distinct ways to rebuild mathematics without using LEM. As it turns out, a given theorem can be true if you are using one kind of mathematics, false if you are using another, or even neither. Even basic concepts like the ordering of the real numbers aren't as straightforward as you'd expect.

October 11: Dan McGregor
Kronecker Function Rings (click for abstract)
Factorization and ideals are some of the main objects of study in commutative ring theory. Ideals were first defined by Dedekind in the context of rings of integers and Fermat's last theorem. Kronecker developed an early theory of divisors as an alternative to ideals, which was later used by Krull to construct what are now called Kronecker function rings. In this talk we will explore some of the history and intuition behind these objects, as well as examine some of their modern applications. This includes connections to valuation rings, spectral spaces, and complete integral closure. We will also look at some recent generalizations.

October 25: Hanbaek Lyu
Discrete excitable media on graphs (click for abstract)
An excitable medium is a nonlinear dynamical system over a network which has the capacity to propagate a wave of fluctuations, examples of which in nature include neural networks, chemical reactions, and coupled oscillators. Spontaneous generation of such waves often leads to surprising selforganizations in the system. We study three discrete models of excitable media in probabilistic aspects, emphasizing their common nature as well as interesting discrepancies. GreenbergHastings Model (GHM) and Cyclic cellular automaton (CCA) are two particular discrete excitable media which have been studied extensively from 90s. Recently, we proposed a discrete model for coupled oscillators which we call the firefly cellular automaton (FCA), and studied its limiting behavior on finite trees.
A classic technique for studying GHM and CCA on the one dimensional lattice is to relate the limiting behavior with random walks with i.i.d. increments by considering their embedded particle system structure. An extra complication arises for FCA, since the associated random walk has longrange correlation. We handle this complication through a combination of Markov chain and generating function methods and show that all sites gets fluctuated less and less frequently, as known for GHM and CCA. However, this particle system method does not easily carry over to general network topology. To overcome this limitation, we develop a new technique to construct a monotone comparison process in the universal covering space of the underlying graph. Roughly speaking, it is to compare original model with an easier metamodel but constructed on larger graph. This enables us to characterize limiting behavior of 3color GHM and CCA on arbitrary graphs, including the ErdosRenyi random graphs. On infinite rooted trees, our method in particular relates the average rate of fluctuation of the root to a certain notion of speed of a treeindexed random walk. Due to a largedeviations effect, all sites get fluctuated at a linear average rate.
In two or more dimensions, both CCA and GHM show spontaneous emergence of spiral waves by which all sites fluctuates periodically, for any available number of colors for each site. This behavior is shared in FCA for all but one case: the 4color FCA shows mysterious clustering behavior in all higher dimensions. Time permitted, we remark on an application of a continuous generalization of the 4color FCA to distributed clock synchronization algorithms. 
November 7: Evan Nash  Monday
Hodge Theory in Combinatorics (click for abstract)
In 1968, Read conjectured that the coefficients of the chromatic polynomial of a (finite) graph form a log concave sequence. Over 40 years later Adiprasito, Huh, and Katz provided a proof of this conjecture using purely combinatorial methods. What makes this story interesting is that the ideas they used were inspired by constructions from algebraic geometry. Their technique is to develop a Hodge theory for matroids, which are generalizations of graphs. Proving analogs of theorems from algebraic geometry, they are able to deduce Read's conjecture. We will give a few snippets of this conversation between the two fields.

November 28: Linh Huynh  Monday  MOVED TO NEXT SEMESTER
Classifying SleepWake Transitions in Rat Electromyography Data (click for abstract)
Contrary to the common perception that sleep is continuous, sleep is actually fragmented by brief awakenings throughout the night even in healthy people. Experiments have shown that sleep bouts duration follows an exponential distribution in both infants and adults, while wake bouts duration distribution changes from exponential in infants to power law in adults. To understand this phenomenon of wake bouts and fundamental mechanisms in sleep cycle dynamics, I analyze the transitions between sleep and wake states throughout a night. I apply machine learning methods on rat electromyography data to identify clusters of sleepwake transitions and compare results with activity of stochastic mathematical models. In this talk, I am going to discuss the background of my research as well as the machine learning methods that I use.
Happy Hour
Happy hour schedule forthcoming!
Everyone is welcome to come, drinkers and nondrinkers alike!