*info*Introduction

Hello, my name is Marc Schilder and I'm a mathematics graduate student at UB! This is my website, which includes resources, such as notes and images/animations, for my students and anyone interested in studying mathematics.

*help*Questions?

I'm usually available for mathematics help -- if you have any questions or want a concept reexplained, feel free to email me or stop by during my office hours.

**Office Hours:** Monday 10:00a -- 11:00a, Wednesday 09:00a -- 10:00a (or by appointment)

**Office Room:** Mathematics Building Rm 139 (or Rm 240 "The Lounge")

*equalizer*My Interests

My mathematical interests include:

#### Knot Theory

Knot Theory broadly studies mathematical knots, embeddings of spheres of codimension 2. We typically think of knots as embeddings of closed circles into 3-dimensional spheres \(S^{1}\hookrightarrow S^{3}\), and we look for ways to understand whether or not two knots are equivalent.

#### Geometry

Geometry studies many problems including curvature, shape, and surfaces. An interesting idea of geometry is that planar triangles' angles sum to 180º, but triangles on surfaces with positive curvature will sum to a value greater than 180º and triangles on surfaces with negative curvature will have angles that sum to less than 180º.

#### Topology

Topology extends the ideas of "closeness" by defining open sets on spaces. Properties such as non-orientability (like the Möbius strip) allow us to understand how some spaces are different from others. Other common tools for distinguishing spaces are algebraic invariants, such as homotopy and homology groups.

#### Analysis

Analysis broadly seeks to extend the ideas of calculus to more abstract spaces, which can be understood usually in terms of multivariable calculus. Other interesting topics of study are convergence and measure theory within the realm of analysis.

#### Logic

Logic is the study of abstract ("quantitative") truth, and mathematical logic axiomatically approaches what is and is not deducible from these axioms.

Outside of mathematics, my interests include music, art, writing, and learning new things!

*star*Something Cool

Ordinarily, my home page is pretty empty, so here's a brief discussion of one of my undergraduate topics of interest: Fractional Calculus and the **Fractional Derivative**.

This animation shows the Fractional Derivative of the simple identity function \[f(x)=x\] varying ''smoothly'' between the derivative and the antiderivative -- here, the derivative and the antiderivative correspond to orders \(\alpha=1\) and \(\alpha=-1\), and what this animation shows is the what the intermediate orders of differentiation look like, coming from the Cauchy Formula for general iterated integration:\[\mathrm{J}^{n}f(x)=\frac{1}{\Gamma(n)}\int_{0}^{x}(x-t)^{n-1}f(t)\hspace{3pt}\mathrm{d}t.\]

However, this is just the first quadrant -- what happens when we include negative inputs? We get **complex-valued outputs**! So using the formula for the general \(\alpha\)th-order derivative of \(f(x)=x\),\[\frac{\mathrm{d}^{\alpha}}{\mathrm{d}x^{\alpha}}(x)=\frac{1}{\Gamma(2-\alpha)}\hspace{2pt}x^{1-\alpha},\] we can understand the imaginary component of the output as pushing off of the plane in the first animation and giving us a more 3-dimensional complex function of the real numbers. It is important to understand Euler's Gamma Function \(\Gamma(z)\) as a complex generalization of the factorial function \(n!=\Gamma(n+1)\). This formula still works to compute the \(1\)- and \(-1\)-order derivatives as known from Calculus:
\[\frac{\mathrm{d}}{\mathrm{d}x}(x)=\frac{1}{\Gamma(1)}\hspace{2pt}x^{0}=1\hspace{15pt}\frac{\mathrm{d}^{-1}}{\mathrm{d}x^{-1}}(x)=\frac{1}{\Gamma(3)}\hspace{2pt}x^{2}=\tfrac{1}{2}\hspace{2pt}x^{2}\]
However, we can use any complex order derivative (the orders of our derivatives don't have to be **real**), so the following two animations show this more complete picture. First by varying \(\alpha\) between \(1\) and \(-1\) along the real-axis and second by orbiting \(\alpha\) around the unit circle \(e^{i\theta}=\cos\theta+i\sin\theta\) in the complex plane.