Jesse Kass -- How to use matrices to find zeros

A basic math problem is, how to count the real zeros of a polynomial efficiently (no factoring!)? In the 19th century, Jacobi and Borchardt and… proved that the count can be computed using matrix methods. I’ll explain this classical result, show its relation to the topological degree, and then pose the problem of establishing an analogous relation with more complicated degrees in modern algebraic topology.

Patrick McFaddin -- Twisted Linear Algebra

Central simple algebras are a direct generalization of matrix algebras and the study of such objects can be viewed as linear algebra over division rings - fields with not necessarily commutative multiplication. The study of central simple algebras began in the 1930's by Brauer, Noether, Hasse, and Albert, and since then applications have abounded: e.g. class field theory, Galois descent, motivic cohomology. In this talk, I will give a soft introduction to central simple algebras and the Brauer group of a field and discuss some open problems.

Jinjin Liang -- Support Vector Machine Classification

Support Vector Machine (SVM) is an efficient algorithm in data mining, and has been widely used in various areas, such as economics, medical science, social sciences, etc. It separates two kinds of samples by an optimal classification hyperplane, which is determined by the solution to a quadratic program, thus has advantages of few parameters, global optimization, and high precision.

In this talk, I will first present the theory foundation and basic concepts, to make SVM easy to understand. Then I will talk about my research on Support Vector Domain Description as a reformulation of SVM. Also, the possible application area will be discussed. No prior knowledge is assumed.

Jinjin Liang obtained her PhD degree in Applied Mathematics at 2009 from Xidian University in China, then worked at Xi’an Shiyou University, and is now an associate Professor in School of Sciences. She has 12 years researching experiences on optimization theory and applications, her research focuses on Support Vector Machine using pre extraction strategy and optimization method

Matthew Boylan, Doug Meade, and Matt Miller -- Q and A on Department External Review

Upper administration (e.g., Deans, Provost) at institutions of higher education require their academic units to undergo periodic external reviews. To prepare for an external review, academic units prepare a report (self study) summing up the state of the unit. The administration then invites outside experts (external reviewers) to visit the unit. The external reviewers read the self study, talk to faculty and students, and submit a written report to university administrators. Our department will be hosting external reviewers the week of March 19 – 23. In advance of their arrival, we would like to take the opportunity to inform you about the process and to answer any questions that you may have.

Zhiyu Wang -- The method of hypergraph containers and its applications

I will attempt to give a brief introduction to the method of hypergraph containers (developed by Jozsef Balogh, Robert Morris, Wojciech Samotij and independently by Saxton and Thomason) for bounding the number of finite objects with forbidden substructures. It allows one to prove enumerative, structural and extremal results in a wide variety of settings. We will discuss some of its applications in extremal graph theory, additive combinatorics and discrete geometry.

Mohammed Alabbood -- Blowing-up of the projective plane in 6 points over finite and infinite fields

In this seminar, we will discuss the problem of constructing a cubic surface as the blow-up of 6 points in general position in projective plane. We will show that every cubic surface can be obtained in this way by finding generators of the space of plane cubics passing through the six points.

Various -- An Invitation to Upcoming Graduate Courses

Dr. Laszlo Szekely, Dr. Jesse Kass, Dr. Andy Kustin, and Dr. Paula Vasquez will give meet-and-greet talks for what to expect in their courses coming up in the fall. Everyone is welcome.

Harsh Mehta -- Integral Ideas in Analytic Number Theory

We talk about important applications of integration in analytic number theory, essentially, about a theorem of Perron, a theorem of Mellin and the circle method.

Keller Vandebogert -- Random Polygon to Ellipse

Consider any random polygon. We can make a new polygon by connecting the midpoints of the original and rescaling by an appropriate factor. Iterating this process yields a curious result: the sequence of polygons will tend to an ellipse oriented at 45 degrees (check out https://www.jasondavies.com/random-polygon-ellipse/ to see it happen in real time). In this talk I will go over the surprisingly elementary mathematics behind this result, and outline a proof of the more general result; namely, instead of doing the above transformation with midpoints, we can take any point of division of the segments of our polygon and we will still get a polygon oriented at 45 degrees.

Jeremiah Southwick -- An Introduction to Newton Polygons

The Newton Polygon of a polynomial is a geometric object which gives information about how the polynomial factors over the integers. It provides a wealth of irreducibility criteria and hence is a useful topic to begin with studying how polynomials factor. We will define the Newton Polygon and will discuss the pertinent theorem of Dumas. We will also show several criteria for irreducibility, including how the Eisenstein criterion is an elegant consequence of the theory.

Panel -- Everything you always wanted to know about exams (but were afraid to ask)

Graduate students in the 2nd, 3rd and 4th year will talk about their experiences with qualifying and comprehensive exams and answer questions.