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# Binghamton University Math Graduate Student Seminar

## Spring 2024

Usual meetings: Mondays at 4:40pm, WH 309

The Mathematics Graduate Student Seminar seeks to strengthen communication among grad students at Binghamton University, thereby cultivating our community and fostering a friendly environment in which to do mathematics research. We provide a venue for grad student talks on interesting math; we also hope to stimulate discussion around various topics--mathematical and otherwise--relevant to math grad students.

Presentations should be accessible to mathematics grad students, and all are encouraged to contribute a talk!

Email Andrew Velasquez-Berroteran or Tara Koskulitz (math emails velasqua and koskulitz, respectively) to schedule your talk!

For a list of talks from previous semesters, see the link in the top right corner.

### Schedule of Talks

Jan. 24 (4:30PM, Wed.) : Organizational Meeting

Jan. 29 : No Meeting

Feb. 5: No Meeting

Feb. 12 (4:40PM) : Chad Nelson, An Introduction to Index Theory

Abstract: The Gauss-Bonnet Theorem is a fundamental formula in differential geometry. In this talk we will reinterpret this classical

result as an example of an index formula . Such formulas profoundly connect data given by certain differential operators on a manifold (the

analytical index) with the topological and geometric data of the manifold (the topological index). This will lead us into a vast generalization

of the Gauss-Bonnet Theorem, and many other important results, known as the Atiyah-Singer Index Theorem. I will not be assuming any

knowledge of differential geometry.

Feb. 19 (4:40PM): Bruce Phillips, Principal Component Analysis and Distributed estimation of Principal Eigenspaces

Abstract: Principal Component Analysis (PCA) is an important tool for dimension reduction, and data visualization. We will give an

overview of PCA as well as look at a method for estimating principal components when data is large and stored on multiple servers.

Feb. 26 (No Meeting)

Mar. 4: No Meeting (Spring Break)

Mar. 11 (4:40PM): Lucas Williams, Periodic Points and Equivariant Parameterized Cobordism

Abstract: In this talk we investigate invariants that count periodic points of a map, using equivariant parameterized cobordism. Given a self

map $f$ of a compact manifold we could detect $n$-periodic points of $f$ by computing the Reidemeister trace of $f^n$ or by computing

the equivariant Fuller trace. In 2020, Malkiewich and Ponto showed that the collection of Reidemeister traces of $f^k$ for varying $k|n$

and the equivariant Fuller trace are equivalent as periodic point invariants, and they conjecture that for families of endomorphisms the

Fuller trace will be a strictly richer invariant for $n$-periodic points.

In this talk we will explain our new result which confirms Malkiewich and Ponto's conjecture. We do so by proving a new Pontryagin-Thom

isomorphism between equivariant parameterized cobordism and the spectrum of sections of a particular parametrized spectrum.

Mar. 18 (4:40PM): Naftoli Kolodny, Division by Three; or How I Learned to Stop Loving and Start Hating the Axiom of Choice

Abstract: The Axiom of Choice is a very odd and unwieldy axiom, which seeks to generalize perfectly natural intuition with regard to sets of

'small cardinality' to all sets, but whose formulation is, to our understanding, inherently nonconstructive. We use a paper of Conway and

Doyle to demonstrate an application of a powerful result attained without use of AoC, and call for further experimentation in our axiomatic

schemata, both within and without ZF (or, indeed, set theory).

Mar. 25 (4:40PM): Benjamin Warren, Bits and Pieces of Additive Number Theory

Abstract: We will do a general survey of problems in classical additive number theory. We'll briefly cover a range of topics, including the

classical additive bases, the two, three, and four square theorems, Waring's problem, the polygonal number theorem, a little about

quadratic forms, Schnirelmann density, counting problems, the three cube problem, and possibly a little about complete sequences.

Apr. 1: No Meeting (Easter Monday)

Apr. 8 (4:40PM): Yiyi Cao, Beauty in Geometric Analysis: Costa-Hoffman-Meeks surfaces

Abstract: Differential Geometry investigates manifolds using calculus; Geometric Analysis extends this by Partial Differential Equation,

known as “geometric PDE”. The study of Costa-Hoffman-Meeks surfaces as examples of minimal surfaces (defined as surface with zero

mean curvature), shows quite an amazing visualization with the genuine beauty of mathematics, also arising interesting applications in

many science and engineering fields, for an exciting prospect of human future.

The foundation of topological concepts will be explained, hence there is no background needed for audiences. What to expect? The amazing

historical background of minimal surfaces in 200 years, as well as the powerful PDEs for solving the famous Minimal Surface Problem, will

be introduced. Come and pick your favorite minimal surfaces in our math gallery, among helicoid, catenoid, gyroid, Scherk surface, and

more!

Apr. 15 (4:40PM): Meenakshy Jyothis, Crash Course in Geometric Topology of Surfaces

Abstract: In this talk we will go over some ideas and terminology that often show up in geometric topology talks and discussions.

You might hear some of these words at topology conferences, the aim is to make those talks more accessible to graduate students.

In this talk I will define Teichmüller space, moduli space, mapping class groups and curve complexes. I will talk about some influential

and beautiful results from the fields. We will also see some hyperbolic geometry and some examples of combinatorial and algebraic

objects that show up when studying surfaces. If time permits I will talk about my research and some existing open problems.

Apr. 22: No Meeting (Classes Dismiss at 1PM for Passover)

Apr. 29 (4:40PM):

Abstract: